Mathematics is the door and key to the sciences.

— Roger Bacon

It is time to return to great papers of the past. The past has clear lessons about how progress can be achieved. Here, I will discuss a trio of papers that came at a critical juncture in the history of numerically solving hyperbolic conservation laws. In a sense, these papers were nothing new, but provided a systematic explanation and skillful articulation of the progress at that time. In a deep sense these papers represent applied math at its zenith, providing a structural explanation along with proof to accompany progress made by others. These papers helped mark the transition of modern methods from heuristic ideas to broad adoption and common use. Interestingly, the depth of applied mathematics ended up paving the way for broader adoption in the engineering world. This episode also provides a cautionary lesson about what holds higher order methods back from broader acceptance, and the relatively limited progress since.

The three papers I will focus on are:

Harten, Ami. “High resolution schemes for hyperbolic conservation laws.” Journal of computational physics 49, no. 3 (1983): 357-393.

Harten, Ami. “On a class of high resolution total-variation-stable finite-difference schemes.” SIAM Journal on Numerical Analysis 21, no. 1 (1984): 1-23.

Sweby, Peter K. “High resolution schemes using flux limiters for hyperbolic conservation laws.” SIAM journal on numerical analysis 21, no. 5 (1984): 995-1011.

The first two are by the late Ami Harten providing a proof of the monotone behavior seen with the heuristic methods existing at that time. The proofs provided some confidence to many that had been lacking from the truly innovative, but largely heuristic invention of the methods. The third paper by Peter Sweby provided a clear narrative and an important graphical tool for understanding these methods and displaying limiters, the nonlinear mechanism that produced the great results. The “Sweby diagram” was the reduction of these complex nonlinear methods to a nonlinear function. The limiter was then a switch between two commonly used classical methods. The diagram produced a simple way of seeing whether any given limiter was going to give second-order non-oscillatory results. Together these three papers paved the way for common adoption of these methods.

Mathematics is the art of giving the same name to different things.

– Henri Poincaré

In the 1970’s three researchers principally invented these nonlinear methods, Jay Boris, Bram Van Leer, and Vladimir Kolgan.  Of these three Boris and Van Leer achieved fame and great professional success. The methods were developed heuristically and worked very well. Each of these methods explicitly worked to overcome Godunov’s barrier theorem that says a second-order linear method cannot be monotone. Both made the methods nonlinear through adapting the approximation based on the local structure of the solution. Interestingly Boris and Van Leer were physicists, Kolgan was an engineer (Van Leer went on to work extensively in engineering). Kolgan was a Russian in the Soviet Union and died before his discovery could take its rightful place next to Boris and Van Leer (Van Leer has gone to great effort to correct the official record).

[Mathematics] is security. Certainty. Truth. Beauty. Insight. Structure. Architecture. I see mathematics, the part of human knowledge that I call mathematics, as one thing—one great, glorious thing. Whether it is differential topology, or functional analysis, or homological algebra, it is all one thing. … They are intimately interconnected, they are all facets of the same thing. That interconnection, that architecture, is secure truth and is beauty. That’s what mathematics is to me.

― Paul R. Halmos

The problem with all these methods was a lack of mathematical certainty on the quality of results along with proofs and structured explanations of their success. This made the broader community a bit suspicious of the results. In a flux corrected transport (FCT, Boris’ invention) commemorative volume this suspicion is noted. At conferences, there were questions raised about the results that implied that the solutions were faked. The breakthrough with these new methods was that good, too good to be true. Then the explanations came and made a strong connection to theory. The behavior seen in the results had a strong justification in mathematics, and the trust in the methodology grew. Acceptance came on the heals of this trust and widespread adoption.

Harten and others continued to search for even better methods after introducing TVD schemes. The broad category of essentially non-oscillatory (ENO) methods was invented. It has been a broad research success, but never experienced the wide spread adoption that these other methods enjoyed. Broadly speaking, the TVD methods are used in virtually every production code for solving hyperbolic conservation laws. In the physics world, many use Van Leer’s approach and engineering uses Harten-Sweby’s formalism broadly. FCT is used somewhat in the physics world, but its adoption is far less common. Part of the reason for this disparity comes down to the power of mathematical proof and the faith it gives. The lack of success of follow-on methods to get adoption and have success comes from the lack of strong theory with its requisite confidence. Faith, confidence and systematic explanation are all provided by well executed applied mathematics.

What is TVD the theory and how does it work?

(Note: WordPress’ Latex capability continues to frustrate, I cannot get them to typeset so if you can read TeX the equations will make sense)

In a nutshell, TVD is a way of extending the behavior of monotone methods (upwind for the purposes of this discussion) to high-order nonlinear methods. Upwind methods have the benefit of positive coefficients in their stencil. If we write this down for a scalar advection equation, , we get the following form, $u_j^{n+1} = u_j^n – C_{j-1/2} \left( u_j^n – u_{j-1}^n \right) + D_{j+1/2} \left(u_{j+1}^n – u_j^n \right) $. The key for the methods is the positivity of the functions and . For example, an upwind method will give constants for these functions, $latex  C_{j-1/2}  = a \Delta t/\Delta x = \nu $ and for . The coefficient is the famous CFL (Courant-Friedrichs-Lewy) number. For the TVD methods, these functions become nonlinear functions of the solution itself, but satisfy the inequalities. Harten had done other work that connected monotone methods to entropy satisfying (i.e., physically relevant solutions), which then implies that TVD methods would be a route to similar results (this would seem to be true, but definitive proofs are lacking). Still the connections are all there and close enough to provide faith in the methodology. This is where Sweby’s work comes in and provides a crucial tool for broad acceptance of this methodology.

What Sweby did was provide a wonderful narrative description of TVD methods, and a graphical manner to depict them. In the form that Sweby described, TVD methods were a nonlinear combination of classical methods: upwind, Lax-Wendroff and Beam Warming. The limiter was drawn out of the formulation and parameterized by the ratio of local finite differences. The limiter is a way to take an upwind method and modify it with some part of the selection of second-order methods and satisfy the inequalities needed to be TVD. This technical specification took the following form, $ C_{j-1/2}  = \nu \left( 1 + 1/2\nu(1-\nu) \phi\ledt(r_{j-1/2}\right) \right) $ and for and $r_{j-1/2} = \frac{ u_{j}^{n} – u_{j-1}^{n} }{ u_{j-1}^{n}  – u_{j-2}^{n}} $. This produced a beautiful and simple diagram that usefully displayed how any given method compared to others. This graphical means was probably the essential step for broad acceptance (my opinion, but for visual people it was essential and a lot of technical folks are visual).

Beyond the power of applied mathematics, other aspects of the technical problem have contributed to the subsequent lack of progress. The biggest issue is the quantum leap in performance from first- to second-order accuracy. The second order methods produce results that seem turbulent because first-order methods produce a truncation error that laminarizes flows. The second-order method produces results for complex problems that have the look and feel of real flows (this may also be quantitatively true, but the jury is out). Important flows are turbulent, high energy with very large Reynolds numbers. First-order schemes cannot produce these realistically at all. Second-order methods can, and for this reason the new schemes unleashed utility upon the World. With these methods, the solutions took on the look, feel and nature of reality. For this reason, these schemes became essential for codes.

The second reason is the robustness of these methods. First-order monotone methods like upwind are terribly robust. These methods produce physically admissible solutions and do not fail often. Codes run problems to completion. The reason is their extremely dissipative nature. This makes them very attractive for difficult problems and almost guarantees a solution for the calculation. The same dissipation also destroys almost every structure in the solution and smears out all the details that matter. You get answer, but an answer that is fuzzy and inaccurate. These first order methods end up being as extremely expensive when accuracy is desired. Harten’s TVD methods provided a systematic connection of the new second-order methods to the old reliable first-order methods. The new methods were almost as reliable as the first-order methods, but got rid of much of the smearing dissipation that plagued them. Having a structured and expertly produced explanation for the behavior of these methods with clear connections to things people already knew produced rapid adoption by practitioners.

Mathematics is the cheapest science. Unlike physics or chemistry, it does not require any expensive equipment. All one needs for mathematics is a pencil and paper.

― George Pólya

The follow-on efforts with higher than second-order methods have lacked these clear wins. It is clear that going past second-order does not provide the same sort of quantum leap in results. The clear connection and expectations of robustness is also lacking. The problems do not stop there. The essentially non-oscillatory methods select the least oscillatory local approximation, which also happens to be quite dissipative by its very nature. Quite often the high-order method is actually not threatening oscillations at all yet a less accurate approximation is chosen needlessly reducing accuracy. Furthermore, the adaptive approximation selection can preferentially choose unstable approximation in an evolutionary sense, which can result in catastrophe. The tendency to produce the worst of both Worlds has doomed their success and broad adoption. Who wants dissipative and fragile? No one! No production code would make these choices, ever!

Recent efforts have sought to rectify this shortcoming. Weighted ENO methods (WENO) have provided far less intrinsically dissipative methods that also enhance the accuracy. These methods are still relatively dissipative compared to the best TVD methods and invoke their expensive approximations needlessly in regions of the solution where the nonlinear mechanisms are unnecessary. Efforts have produced positivity preserving methods that avoid the production of inherently unphysical results with high-order methods. These developments are certainly a step in the right direction. The current environment of producing new legacy codes is killing any other the energy to stewart these methods into broad adoption. The expense, overly dissipative nature and relatively small payoff all stand in the way.

What might help in making progress past second-order methods?

The first thing to note is that TVD methods are mixed in their order of accuracy. They are second-order in a very loose sense and only when one takes the most liberal norm for computations (L1 for you nerds out there). For the worst-case error, TVD methods are still first-order (L-infinity, and multiple dimensions). This is a pretty grim picture until one also realizes that for nonlinear PDEs with general solutions, first-order accuracy is all you get anyway unless you are willing to track all discontinuities. These same conditions hold for high-order methods we might like to adopt. The accuracy from the new methods is always quite limited and puts a severe constraint on the efficiency of the methods, and a challenge to development and progress. The effort that it takes to get full accuracy for nonlinear problems is quite large, and if this accuracy is not realized, the effort is not worth it. We do know that some basic elements of high-order methods yield substantial benefits, but these benefits are limited (an example are high-order edge values used in the piecewise parabolic method – PPM).

I asked myself, what worked so well for TVD? To me there is a clear and unambiguous connection to what worked in the past. The past was defined by the combination of upwind, Lax-Wendroff, and Beam-Warming methods. These methods along with largely ad hoc stabilization mechanisms provided the backbone of production codes preceding the introduction of these methods. Now TVD schemes form the backbone of production codes. It would seem that new higher order methods should preserve this sort of connection. ENO and WENO methods did not do this, which partially explains their lack of adoption. My suggestion would be a design of methods where one uses a high-order method that can be shown to be TVD, or the high-order method closest to a chosen TVD scheme. This selection would be high-order accurate by construction, but would also produce oscillations at third-order. This is not the design principle that ENO methods use where the unproven assertion is oscillations at the order of approximation. The tradeoff between these two principles is larger potential oscillations with less dissipation and a more unambiguous connection to the backbone TVD methods.

1. Everyone is entitled to their opinion about the things they read (or watch, or listen to, or taste, or whatever). They’re also entitled to express them online.

2. Sometimes those opinions will be ones you don’t like.

3. Sometimes those opinions won’t be very nice.

4. The people expressing those may be (but are not always) assholes.

5. However, if your solution to this “problem” is to vex, annoy, threaten or harrass them, you are almost certainlya bigger asshole.

6. You may also be twelve.

7. You are not responsible for anyone else’s actions or karma, but you are responsible for your own.

8. So leave them alone and go about your own life.

[Bad Reviews: I Can Handle Them, and So Should You(Blog post, July 17, 2012)]

― John Scalzi

My own connection to this work is a nice way of rounding out this discussion. When I started looking at modern numerical methods, I started to look at the selection of approaches. FCT was the first thing I hit upon and tried. Compared to the classical methods I was using, it was clearly better, but its lack of theory was deeply unsatisfying. FCT would occasionally do weird things. TVD methods had the theory and this made is far more appealing to my technically immature mind. After the fact, I tried to project FCT methods onto the TVD theory. I wrote a paper documenting this effort. It was my first paper in the field. Unknowingly, I walked into a veritable mine field and complete shit show. All three of my reviewers were very well-known contributors to the field (I know it is supposed to be anonymous, and the shit show that unveiled itself, unveiled the reviewers too).

The end result was that the paper was never published. This decision occurred five years after it was submitted, and I had simply moved on. My first review was from Ami Harten who basically said this paper is awesome and publish it. He signed the review and sent me some lecture notes on the same topic. I was over the moon, and did call Ami and talk briefly. Six months later my second review came in. It was as different as possible from Ami’s. It didn’t say this exactly, but in a nutshell, it said the paper was a piece of shit. It still remains the nastiest and most visceral review I’ve ever gotten. It was technically flawless on one hand and thoroughly unprofessional in tone on the other. My third review came a year later and was largely editorial in nature. I revised the paper and resubmitted. While all this unfolded Ami died, and the journal it was submitted to descended into chaos partially due to the end of the cold war and its research largess. When it emerged from chaos, I decided to publish the work was largely pointless and not worth the effort.

Some commentary about why this shit show happened is worth explaining. It is all related to the holy war between two armed camps that arose via the invention of these methods and who gets the credit. The paper was attempting to bridge the FCT and TVD worlds, and stepped into the bitter fighting around previous publications. In retrospect, it is pretty clear that FCT was first, and others like Kolgan and Van Leer came after. Their methodologies and approaches were also fully independent, and the full similarity was not clear at the time. While the fullness of time sees these approaches are utterly complementary, at the time of development it was seen as a competition. It was definitely not a collaborative endeavor, and the professional disagreements were bitter. They poisoned the field and people took sides viewing the other side with vitriolic fury. A friend and associate editor of the Journal of Computational Physics quipped that this was one of the nastiest sub-communities in the Journal, and why did I insist on working in this area. It is also one of the most important areas in computational physics working on a very difficult problem. The whole field also hinges upon expert judgement and resists a firm quantitative standard of acceptance.

What an introduction to the field and its genuinely amazing that I continue to work in it at all. If I didn’t enjoy the technical content so much, and not appreciated the importance of the field, I would have run. Perhaps greater success professionally would have followed such a departure. In the long run this resistance and the rule of experts works to halt progress.

If you can’t solve a problem, then there is an easier problem you can solve: find it.

― George Pólya

Kolgan, V. P. “Application of the principle of minimum values of the derivative to the construction of finite-difference schemes for calculating discontinuous gasdynamics solutions.” TsAGI, Uchenye Zapiski 3, no. 6 (1972): 68-77.

Boris, Jay P., and David L. Book. “Flux-corrected transport. I. SHASTA, a fluid transport algorithm that works.” Journal of computational physics 11, no. 1 (1973): 38-69.

Van Leer, Bram. “Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second-order scheme.” Journal of computational physics 14, no. 4 (1974): 361-370.\

Van Leer, Bram. “Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method.” Journal of computational Physics 32, no. 1 (1979): 101-136.

Harten, Ami, Bjorn Engquist, Stanley Osher, and Sukumar R. Chakravarthy. “Uniformly high order accurate essentially non-oscillatory schemes, III.” Journal of computational physics 71, no. 2 (1987): 231-303.

Harten, Ami, and Stanley Osher. “Uniformly high-order accurate nonoscillatory schemes. I.” SIAM Journal on Numerical Analysis 24, no. 2 (1987): 279-309.

Harten, Amiram, James M. Hyman, Peter D. Lax, and Barbara Keyfitz. “On finite‐difference approximations and entropy conditions for shocks.” Communications on pure and applied mathematics 29, no. 3 (1976): 297-322.

 

What I cannot create, I do not understand.

– Richard Feynman

We are in deep danger of relying upon science and associated software we do not understand because we have stopped the active creation of knowledge so broadly. I open with one of my favorite quotes by the great physicist Richard Feynman, who also wrote about Cargo Cult Science (https://en.wikipedia.org/wiki/Cargo_cult_science). It is a bold, but warranted assertion to note that much of our science work today is taking on the character of Cargo Cult Science. We are not all the way there, but we have moved a long way toward taking on all of the characteristics of this pathology. In this assertion money is the “cargo” that pseudo-scientific processes are chasing. It is noexaggeration to say that getting funding for science has replaced the conduct and value of that science today. This is broadly true, and particularly true in scientific computing where getting something funded has replaced funding what is needed or wise. The truth of the benefit of pursuing computer power above all else is decided upon a priori. The belief was that this sort of program could “make it rain” and produce funding because this sort of marketing had in the past. All results in theprogram must bow to this maxim, and support its premise. All evidence to the contrary is rejected because it is politically incorrect and threatens the attainment of the cargo, the funding, the money. A large part of this utterly rotten core of modern science is the ascendency of the science manager as the apex of the enterprise. The accomplished scientist and expert is merely now a useful and necessary detail, the manager reigns as the peak of achievement.

The first principle is that you must not fool yourself — and you are the easiest person to fool.

We’ve learned from experience that the truth will come out. Other experimenters will repeat your experiment and find out whether you were wrong or right. Nature’s phenomena will agree or they’ll disagree with your theory. And, although you may gain some temporary fame and excitement, you will not gain a good reputation as a scientist if you haven’t tried to be very careful in this kind of work. And it’s this type of integrity, this kind of care not to fool yourself, that is missing to a large extent in much of the research in cargo cult science.

– Richard Feynman

If one looks at the scientific computing landscape today, one sees a single force for progress: the creation of a new more powerful supercomputer that is much faster than anything we have today. The United States, Europe and China are all pursuing this path for advancing scientific computing. It is a continuation of a path we have pursued for the last 25 years, but our future is not remotely like the last 25 years. This approach to progress can be explained simply and marketed to the naïve and untechnical. This works because our National leadership is increasingly naïve, witless and obsessively anti-intellectual lacking any technical sophistication. We are in the midst of a tide of low information leadership who are swayed by sweet sounding bullshit far more easily than hard-nosed facts.

The farther backward you can look, the farther forward you are likely to see.

― Winston S. Churchill

In this putrid environment, faster computers seem an obvious benefit to science. They are a benefit and pathway to progress, this is utterly undeniable. Unfortunately, it is an expensive and inefficient path to progress, and an incredibly bad investment in comparison to alternative. The numerous problems with the exascale program are subtle, nuanced, highly technical and pathological. As I’ve pointed out before the modern age is no place for subtlety or nuance, we live it an age of brutish simplicity where bullshit reigns and facts are optional. In such an age, exascale is an exemplar, it is a brutally simple approach tailor made for the ignorant and witless. If one is willing to cast away the cloak of ignorance and embrace subtlety and nuance, a host of investments can be described that would benefit scientific computing vastly more than the current program. If we followed a better balance of research, computing to contribute to science far more greatly and scale far greater heights than the current path provides.

Applications that matter to something big would create a great deal of this focus naturally. The demands of doing something real and consequential would breed a necessity to focus progress in an organic way. Last week I opined that such big things are simply not present today in science or society’s broader narrative. Society is doing nothing big or aspirational or challenging to drive progress forward with genuine purpose. To be more pointed, the push for exascale is not big at all, it is rather an exemplar of the lack of vision and consequence. There is a bit of chicken and egg argument to all this. The bottom line is a general lack of underlying and defining purpose to our efforts in computing. Exascale is what we do when we want to market something as “feeling” big, when it is actually doing something small and inconsequential.

Those who do not move, do not notice their chains.

― Rosa Luxemburg

How can I say such a thing?

In a nutshell computing speed is one of the least efficient and least effective ways to improve computational science. It has only been an enabler because computing speed came for free with Moore’s law for most of the last half century. That free lunch is over and past, yet we mindfully ignore this reality (http://herbsutter.com/welcome-to-the-jungle/ ). Even with Moore’s law fully in effect, it was never the leading contributor for progress, progress was paced by numerical methods and algorithmic scaling. Moreover, computing speed cannot fix modeling that is wrong (methods and algorithms don’t fix this either). If a model is wrong, the wrong answer is simply computed much faster. Of course, we know that every model is wrong and the utility of any model is determined via V&V. Issues associated with the use of computing, naïve code users, the loss of expertise, and understanding are simply overlooked, or worse yet made more intractable due to inattention.

Each of these advances has been mentioned before in the guise of a full blog post, but it is useful to put things together to see the wealth of unused opportunity.

80% of results come from 20% of effort/time

― Vilfredo Pareto

1. Modernizing modeling ought to be a constant and consistent emphasis in science. Computational science is no different. For some reason, the modeling advances have simply stopped. Our basic models of reality are increasingly fixed and immutable, and ever less fit for future purpose. The models of reality have become embedded in computer codes, and ultimately central to the codes structure in numerous respects. As such we start to embed a framework for modeling whose foundation becomes invariant. We can’t change the model without developing an entirely different code. We reduce our modeling to submodels and closure of existing models while the staying within a fundamental modeling framework. This is another area where progress is phenomenally risky to approach and substantially prone to failures and misguided efforts. Without the failure, the ability to learn and produce new and improved model is virtually impossible. https://williamjrider.wordpress.com/2015/02/02/why-havent-models-of-reality-changed-more/, https://williamjrider.wordpress.com/2015/07/03/modeling-issues-for-exascale-computation/ , https://williamjrider.wordpress.com/2017/07/07/good-validation-practices-are-our-greatest-opportunity-to-advance-modeling-and-simulation/
2. Modernizing methods is not happening. Since methods are one of the best ways to improve the efficiency and effective solution of models, progress is harmed in a manner that cannot be easily recovered by other means. Usually when a model is decided upon, a method is used to solve the model numerically. The numerical method is only slightly less code specific and invariant than the model itself. By virtue of this character, the basic numerical method for a model becomes indistinguishable from the code. If we preserve the code base, we preserve old methods, which means no progress. We are stuck using relatively low-order methods with crude stability mechanisms. The ability to use high-order methods with enhanced accuracy and efficiency is not advancing. The research in numerical methods and the practical application of numerical methods is becoming increasingly divorced from one another. The gap has grown into a chasm, and numerical methods research is losing relevance. Part of the problem is related to the standards of success where methods research allows success to be found on easier problems rather than keeping the problem difficulty fixed. This is yet another place where the inability to accept failure as a necessary element (or even fuel) for success is fatal. https://williamjrider.wordpress.com/2016/06/14/an-essential-foundation-for-progress/, https://williamjrider.wordpress.com/2016/07/25/a-more-robust-less-fragile-stability-for-numerical-methods/,
3. Algorithmic scaling is the most incredible thing we could achieve in terms of computational performance. The ability to change the scaling exponent on how much work it takes to solve a problem can have a magical impact. Linear algebra is the posterchild for this effect. A breakthrough in scaling can make the impossible problem, possible and even routine to solve. The classical naïve scaling for matrix inversion has the work scaling with the cube of the problem size. Even small problems quickly become utterly intractable and almost no amount of computer power can fix this. Change the scaling to quadratic and new problems suddenly become routine, change the scaling to linear and the problems that can be tackled routinely were unimaginable before. We are stuck at linear, although some fields are starting to see sublinear algorithms. Could these breakthroughs be more common and useful? If they could the impact on computational science would overwhelm the capacity of exascale easily. Today we aren’t even trying to make these advances. In my view, such work is generically risky and prone to failure, can failure is something that has become intolerable, thus success if sacrificed. https://williamjrider.wordpress.com/2015/05/29/focusing-on-the-right-scaling-is-essential/
4. Today supercomputing is completely at odds with the commercial industry. After decades of first pacing advances in computing hardware, then riding along with increases in computing power, supercomputing has become separate. The separation occurred when Moore’s law died at the chip level (in about 2007). The supercomputing world has become increasingly disparate to continue the free lunch, and tied to an outdated model for delivering results. Basically, supercomputing is still tied to the mainframe model of computing that died in the business World long ago. Supercomputing has failed to embrace modern computing with its pervasive and multiscale nature moving all the way from mobile to cloud. https://williamjrider.wordpress.com/2017/12/15/scientific-computings-future-is-mobile-adaptive-flexible-and-small/
5. Verification & validation – If the scientific computing efforts are to be real scientific endeavors, V&V is essential. Computational modeling is still modeling and comparison with experiment is the gold standard for modeling, but with computational work the comparison has numerous technical details needing serious attention.  In a very complete way V&V is the scientific method in action within the context of modeling and simulation. This energizes a top to bottom integration of scientific activities and essential feedback up and down this chain. The process produces actionable evidence of how progress is being made and where the bottlenecks to progress exist. The entirety of the V&V work provides a deep technical discourse on the breadth of computational science. The whole of computational science can be improved by its proper application. By weakly supporting V&V, current efforts are cutting themselves off from the integration of the full scientific enterprise and impact into the use of computation scientifically. https://williamjrider.wordpress.com/2016/12/22/verification-and-validation-with-uncertainty-quantification-is-the-scientific-method/
6. Expansive uncertainty quantification – too many uncertainties are ignored rather than considered and addressed. Uncertainty is a big part V&V, a genuinely hot topic in computational circles, and practiced quite incompletely. Many view uncertainty quantification as only being a small set of activities that only address a small piece of the uncertainty question. Too much benefit is achieved by simply ignoring a real uncertainty because the value of zero that is implicitly assumed is not challenged. This is exacerbated significantly by a half funded and deemphasized V&V effort in scientific computing. Significant progress was made several decades ago, but the signs now point to regression. The result of this often willful ignorance is a lessening of impact of computing and limiting the true benefits. https://williamjrider.wordpress.com/2016/04/22/the-default-uncertainty-is-always-zero/
7. Data integration and analysis – one of the latest hot topics is big data and data analysis. The internet and sensors are creating massive amounts of data, and its use is a huge technical problem. The big data issue is looking for significant and actionable understanding from the oceans of data. A related and perhaps more difficult problem is small data where there isn’t enough data, or the enough of the data you want. Lots of science and engineering is data limited to a degree that scientific understanding is limited. Modeling and simulation offers a vehicle to augment this data and fill in the gaps. Doing this in a manner that is credible will be huge challenge. The ways forward with credibility use V&V and intensive uncertainty quantification. The proper use of codes and the role of calibration also becomes critical to success.  https://williamjrider.wordpress.com/2016/07/10/10-big-things-for-the-future-of-computational-science/
8. Multidisciplinary, multiscale science – one of the hot topics a quarter century ago was better Multiphysics methods to replace the pervasive use of operator splitting for complex codes. This effort has utterly failed. We have made very little progress forward. Part of the issue is the inability to produce computational algorithms that are efficient enough to compete. A fully coupled method ends up being so expensive that any accuracy increases from the improved coupling are rendered ineffective. A second and perhaps more powerful reason for lack of progress are the computer codes. Old computer codes are still being used, and most of them use operator splitting. Back in the 1990’s a big deal was made regarding replacing legacy codes with new codes. The codes developed then are still in use, and no one is replacing them. The methods in these old codes are still being used and now we are told that the codes need to be preserved. The codes, the models, the methods and the algorithms all come along for the ride. We end up having no practical route to advancing the methods. https://williamjrider.wordpress.com/2016/09/16/is-coupled-or-unsplit-always-better-than-operator-split/
9. Complete code refresh – we have produced and now we are maintaining a new generation of legacy codes. A code is a storage for vast stores of knowledge in modeling, numerical methods, algorithms, computer science and problem solving. When we fail to replace codes, we fail to replace knowledge. The knowledge comes directly from those who write the code and create the ability to solve useful problems with that code. Much of the methodology for problem solving is complex and problem specific. Ultimately a useful code becomes something that many people are deeply invested in. In addition, the people who originally write the code move on taking their expertise, history and knowledge with them. The code becomes an artifact for this knowledge, but it is also a deeply imperfect reflection of the knowledge. The code usually contains some techniques that are magical, and unexplained. These magic bits of code are often essential for success. If they get changed the code ceases to be useful. The result of this process is a deep loss of expertise and knowledge that arises from the process of creating a code that can solve real problems. If a legacy code continues to be used it also acts to block progress of all the things it contains starting with the model and its fundamental assumption. As a result, progress stops because even when there is research advances, it has no practical outlet. This is where we are today. https://williamjrider.wordpress.com/2015/10/30/preserve-the-code-base-is-an-awful-reason-for-anything/ https://williamjrider.wordpress.com/2016/01/01/are-we-really-modernizing-our-codes/ https://williamjrider.wordpress.com/2016/01/14/a-response-to-criticism-are-we-modernizing-our-codes/ https://williamjrider.wordpress.com/2014/03/20/legacy-code-is-terrible-in-more-ways-than-advertised/
10. Democratization of expertise – the manner in which codes are applied has a very large impact on solutions. The overall process is often called a workflow, encapsulating activities starting with problem conception, meshing, modeling choices, code input, code execution, data analysis, visualization. One of the problems that has arisen is the use of codes by non-experts. Increasingly code users are simply not sophisticated and treat codes like black boxes. Many refer to this as the democratization of the simulation capability, which is generally beneficial. On the other hand, we increasingly see calculations conducted by novices who are generally ignorant of vast swaths of the underlying science. This characteristic is keenly related to a lack of V&V focus and loose standards of acceptance for calculations. Calibration is becoming more prevalent again, and distinctions between calibration and validation are vanishing anew. The creation of broadly available simulation tools must be coupled to first rate practices and appropriate professional education. In both of these veins the current trends are completely in the wrong direction. V&V practices are in decline and recession. Professional education is systematically getting worse as the educational mission of universities is attacked, and diminished along with the role of elites in society. https://williamjrider.wordpress.com/2016/12/02/we-are-ignoring-the-greatest-needs-opportunities-for-improving-computational-science/

One of the key aspects of this discussion is recognizing that these activities are all present to some small degree in exascale, but all of them are subcritical. The program basically starves all of these valuable activities and only supports them in fashion that creates a “zombie-like” existence. As a result, the program is turning its back on a host of valuable avenues for progress that could actually make an exascale computer actually far more useful. Our present path has genuine utility, but represents an immense opportunity cost if you factor in what could have been accomplished instead with better leadership, vision and technical sophistication. The way we approach science more broadly is permeated with these inefficiencies meaning our increasingly paltry investments in science are further undermined by our pathetic execution. At the deepest level our broader societal problems revolving around trust, expertise, scandal and taste for failure may doom any project unless they are addressed. For example, the issues related to the preservation of code bases (i.e., creating new legacy codes) are creating deep problems with advancing on the essential fronts of modeling, methods and algorithms. Everything is woven together into a tapestry whose couplings cannot be ignored. This is exactly the sort of subtlety and nuance our current time finds utterly incomprehensible.

Postscript:

It is sometimes an appropriate response to reality to go insane.

― Philip K. Dick

Healey’s First Law Of Holes: When in one, stop digging.

― Denis Healey

Last week I tried to envision a better path forward for scientific computing. Unfortunately, a true better path flows invariably through a better path for science itself and the Nation as a whole. Ultimately scientific computing, and science more broadly is dependent on the health of society in the broadest sense. It also depends on leadership and courage, two other attributes we are lacking in almost every respect. Our society is not well, the problems we are confronting are deep and perhaps the most serious crisis since the Civil War. I believe that historians will look back to 2016-2018 and perhaps longer as the darkest period in American history since the Civil War. We can’t build anything great when the Nation is tearing itself apart. I hope and pray that it will be resolved before we plunge deeper into the abyss we find ourselves. We see the forces opposed to knowledge, progress and reason emboldened and running amok. The Nation is presently moving backward and embracing a deeply disturbing and abhorrent philosophy. In such an environment science cannot flourish, it can only survive. We all hope the darkness will lift and we can again move forward toward a better future; one with purpose and meaning where science can be a force for the betterment of society as a whole.

Everything passes, but nothing entirely goes away.

― Jenny Diski

The purpose of life is not to be happy. It is to be useful, to be honorable, to be compassionate, to have it make some difference that you have lived and lived well.

― Ralph Waldo Emerson

It would really be great to be starting 2018 feeling good about the work I do. Useful work that impacts important things would go a long way toward achieving this. I’ve put some thought into considering what might constitute work having these properties. This has two parts, what work would be useful and impactful in general, and what would be important to contribute to. As a necessary subtext to this conversation is a conclusion that most of the work we are doing in scientific computing today is neither useful, nor impactful and nothing important is at stake. This alone is a rather bold assertion. Simply put, as a Nation and society we are not doing anything aspirational, nothing big. This shows up in the lack of substance in the work we are paid to pursue. More deeply, I believe that if we did something big and aspirational, the utility and impact of our work would simply sort itself out as part of a natural order.

The march of science is the 20^th Century was deeply impacted by international events, several World Wars and a Cold (non) War that spurred National interests in supporting science and technology. The twin projects of the atom bomb and the nuclear arms race along with space exploration drove the creation of much of the science and technology today. These conflicts steeled resolve, purpose and granted resources needed for success. They were important enough that efforts were earnest. Risks were taken because risk is necessary for achievement. Today we don’t take risks because nothing important is a stake. We can basically fake results and market progress where little or none exists. Since nothing is really that essential bullshit reigns supreme.

There is only one thing that makes a dream impossible to achieve: the fear of failure.

― Paulo Coelho

One of the keys to these conflicts was the presence of a worthy adversary to steel ourselves for the push forward. Both Nazi Germany and Soviet Russia were worthy enemies whose competence meant putting our best foot forward. In reality and rhetorically we lack such an adversary today to push us. We needed to fully commit and faithfully execute our endeavors to achieve victory against these enemies. These opponents had the clear capacity to destroy the United States and the West if the resistance was not real. Ironically the Soviets were ultimately defeated by bullshit. The Strategic Defense Initiative, or Star Wars bankrupted the Soviets. It was complete bullshit and never had a chance to succeed. This was a brutal harbinger of today’s World where reality is optional, and marketing is the coin of the realm. Today American power seems unassailable. This is partially true and partially over-confidence. We are not on our game at all, and far to much of our power is based on bullshit. As a result, we can basically just pretend to try, and actually not execute anything with substance and competence. This is where we are today; we are doing nothing important, and wasting lots of time and money in the process.

How do you defeat terrorism? Don’t be terrorized.

― Salman Rushdie

Again, I freely admit that this is a bold assertion. In scientific computing, we have a National exascale program that underpins National security and economic interests.  It contributes to all of these things in massive ways, at least rhetorically. This support for these National goals is pure marketing, or less generously absolute bullshit. This is simply trotting out a bunch of tired sales pitches for scientific computing that lack any soul and increasingly lack substance. The Nation has no large objectives to support, the entire system is drifting along on auto-pilot. It is brimming with over-confidence and a feeling of superiority that only needs a worthy opponent to expose our largess. We have no enemies that are remotely worthy. We have created some chicken-shit paper tigers like Iran, North Korea and the amorphous and largely toothless Islamic fundamentalism. None of these enemies is even the remotest threat to the United States, or the West in general. If they were a worthy threat then we are in awful shape and far worse than we actually are. Terrorism is only as much of a threat as we make it. We have stoked fear and let ourselves we terrorized because it is useful for the defense-intelligence Industrial complex. It has put trillions of dollars into their coffers, and done little or nothing to build a future. We could simply defeat these enemies by refusing to be terrorized. Some courage and resilience as a Nation would be sufficient to render these pathetic enemies utterly impotent. The greatest damage and threat from these enemies is our response to it, not the actual carnage. Our “leaders” are using them to spread fear among the populace to further their own agendas.

The result of the current model is a research establishment that only goes through the motions and does little or nothing. We make lots of noise and produce little substance. Our nation deeply needs a purpose that is greater. There are plenty of worthier National goals. If war-making is needed, Russia and China are still worthy adversaries. For some reason, we have chosen to capitulate to Putin’s Russia simply because they are an ally against the non-viable threat of Islamic fundamentalism. This is a completely insane choice that is only rhetorically useful. If we want peaceful goals, there are challenges aplenty. Climate change and weather are worthy problems to tackle requiring both scientific understanding and societal transformation to conquer. Creating clean and renewable energy that does not create horrible environmental side-effects remains unsolved. Solving the international needs for food and prosperity for mankind is always there. Scientific exploration and particularly space remain unconquered frontiers. Medicine and genetics offer new vistas for scientific exploration. All of these areas could transform the Nation in broad ways socially and economically. All of these could meet broad societal needs. More to the point of my post, all need scientific computing in one form or another to fully succeed. Computing always works best as a useful tool employed to help achieve objectives in the real World. The real-World problems provide constraints and objectives that spur innovation and keep the enterprise honest.

Reality is that which, when you stop believing in it, doesn’t go away.

― Philip K. Dick

Instead our scientific computing is being applied as a shallow marketing ploy to shore up a vacuous program. Nothing really important or impactful is at stake. The applications for computing are mostly make believe and amount to nothing of significance. The marketing will tell you otherwise, but the lack of gravity for the work is clear and poisons the work. The result of this lack of gravity are phony goals and objectives that have the look and feel of impact, but contribute nothing toward an objective reality. This lack of contribution comes from the deeper malaise of purpose as a Nation, and science’s role as an engine of progress. With little or nothing at stake the tools used for success suffer, scientific computing is no different. The standards of success simply are not real, and lack teeth. Even stockpile stewardship is drifting into the realm of bullshit. It started as a worthy program, but over time it has been allowed to lose its substance. Political and financial goals have replaced science and fact, the goals of the program losing connection to objective reality.

Scientific computing came to maturity being an important supporting player for large enterprises. Originally born in the Cold War as a key tool for science and engineering supporting defense science. Scientific computing spread from this base toward more general science, and more recently into broad use by business and the society as a whole. The kernel from which computing sprang was an interwoven set of large National objectives providing the technical foundation that powers our economy today. Computing was a key contributing player in these endeavors. These endeavors also supported a broad phalanx of other technologies and scientific explorations that formed the broad basis of modernizing the world. Such over-arching goals are breathtakingly missing today. We are lacking a World with any vision of a better future and limitless progress.

If we could marshal our efforts into some worthy efforts, what would we work on?

We would still be chasing faster computers, but the faster computers would not be the primary focus. We would focus on using computing to solve problems that were important. We would focus on making computers that were useful first and foremost. We would want computers that were faster as long as they enabled progress on problem solving. As a result, efforts would be streamlined toward utility. We would not throw vast amounts of effort into making computers faster, just to make them faster (this is what is happening today there is no rhyme or reason to exascale other than, faster is like better, Duh!). Utility means that we would honestly look at what is limiting problem solving and putting our efforts into removing those limits. The effects of this dose of reality on our current efforts would be stunning; we would see a wholesale change in our emphasis and focus away from hardware. Computing hardware would take its proper role as an important tool for scientific computing and no longer be the driving force. The fact that hardware is a driving force for scientific computing is one of clearest indicators of how unhealthy the field is today.

Thinking something does not make it true. Wanting something does not make it real.

― Michelle Hodkin

If scientific computing was taking its role in a healthy National enterprise, the focus would be entirely different. Invariably we would see a very strong emphasis on modeling. In almost every serious endeavor using computing to get real design and analysis results, the physical modeling is the greatest limiting factor. A faster computer is always welcome, but a faster computer never fixes a faulty model. This maxim seems to be utterly and completely ignored in the current scientific computing narrative. The most effective way to improve modeling is also different than current emphasis. Better numerical methods and algorithms provide faster and more accurate solutions to models than computing hardware. This is another area where progress is completely stalled.

Current computing focus is only porting old codes to new computers, a process that keeps old models, methods and algorithms in place. This is one of the most corrosive elements in the current mix. The porting of old codes is the utter abdication of intellectual ownership. These old codes are scientific dinosaurs and act to freeze antiquated models, methods and algorithms in place while acting to squash progress. Worse yet, the skillsets necessary for improving the most valuable and important parts of modeling and simulation are allowed to languish. This is worse than simply choosing a less efficient road, this is going backwards. When we need to turn our attention to serious real work, our scientists will not be ready. These choices are dooming an entire generation that could have been making breakthroughs to simply become caretakers. To be proper stewards of our science we need to write new codes containing new models using new methods and algorithms. Porting codes turns our scientists into mindless monks simply transcribing sacred texts without any depth of understanding. It is a recipe for transforming our science into magic. It is the recipe for defeat and the passage away from the greatness we once had.

Without Your Opponent, You are no Victor.

― Anajo Black

 

 

 

 

Things which matter most must never be at the mercy of things which matter least.

― Johann Wolfgang von Goethe

My work day is full of useless bullshit. There is so much bullshit that it has choked out the room for inspiration and value. We are not so much managed as controlled.  This control comes from a fundamental distrust of each other to a degree that any independent ideas are viewed as dangerous. This realization has come upon me in the past few years. It has also occurred to me that this could simply be a mid-life crisis manifesting itself, but the evidence might seem to indicate that it is something more significant (look at the bigger picture of the constant crisis my Nation is in). My mid-life attitudes are simply much less tolerant of time-wasting activities with little or no redeeming value. You realize that your time and energy is limited, why waste it on useless things.

You and everyone you know are going to be dead soon. And in the short amount of time between here and there, you have a limited amount of fucks to give. Very few, in fact. And if you go around giving a fuck about everything and everyone without conscious thought or choice—well, then you’re going to get fucked.

― Mark Manson

I read a book that had a big impact on my thinking, “The Subtle Art of Not Giving a Fuck” by Mark Manson . In a nutshell, the book says that you have a finite number of fucks to give in life and you should optimize your life by mindfully not giving a fuck about unimportant things. This gives you the time and energy to actually give a fuck about things that actually matter. The book isn’t about not caring, it is about caring about the right things and dismissing the wrong things. What I realized is that increasingly my work isn’t competing for my fucks, they just assume that I will spend my limited fucks on complete bullshit out of duty. It is actually extremely disrespectful of me and my limited time and effort. One conclusion is that the “bosses” (the Lab, the Department of Energy) not give enough of a fuck about me to treat my limited time and energy with respect and make sure my fucks actually matter.

Maturity is what happens when one learns to only give a fuck about what’s truly fuckworthy.

― Mark Manson

I’ve realized recently that a sense of being inspired has departed from work. I’ve felt this building for years with the feeling that my work is useful and important ebbing away. I’ve been blessed for much of my career with work that felt important and useful where an important component of the product was my own added creativity. The work included a distinct element of my own talents and ideas in whatever was produced.

Superficially speaking, the element of inspiration seems to be present, work with meaning and importance with a sense of substantial freedom. As I implied, these elements are superficial, the reality is that each of these pieces has eroded away, and it is useful to explore how this has happened. The job I have would be a dream to most people, but conditions are degrading. It isn’t just my job, but most Americans are experiencing worsening conditions. The exception is the top of the management class, the executives. This is a mirror to broader societal inequalities logically expressed in the working environment. The key is recognizing that my job used to be much better, and that is something worth exploring in some depth.

At one level, I should be in the midst of a glorious time to be working in computational science and high-performance computing. We have a massive National program focused on achieving “exascale” or at the very least a great advance in computing power. Looking more closely, we can see deep problems that produce an inspiration gap. On the one hand, we have the technical objectives for the program being obsessively hardware focused for progress. We have been on this hardware path for 25 years producing progress, but no transformation in science has actually occurred (the powers that be will say it has, but the truth is that is hasn’t). Our computations are still not predictive, and the hardware is not the limiting aspect of computational science. Worse yet, the opportunities for massive hardware advances has passed and advancing now is fraught with difficulties, roadblocks and will be immensely costly. Aside from hardware, the program is largely focused on low-level software focused while porting old codes, methods and models (note: the things being ported and not invested in are the actual science!). It is not focused on the more limiting aspects of predictive modeling because they are subtle and risky to work on. They cannot be managed like a construction project using off the shelf management practices better suited for low wage workers, and unsuitable for scientists. The hardware path is superficial, easy to explain to the novice and managed as a project similarly to building a bridge or road.

This gets to the second problem with the current programs, how they are managed. Science cannot be managed like a big construction project, at least not successfully. The result of this management model is a stifling level of micromanagement. Our management model is defined by overwhelming suspicion and lack of trust resulting in massive inefficiency. The reporting requirements for this mode of management are massive and without value except to bean-counters. At the same time, there is no appetite for risk, and no capacity to tolerate failure. As a result, the entire program loses an ability to inspire, or reach for greatness.

If the Apollo Program had been managed in this fashion, we would have never made it to the Moon while spending vastly greater sums of money. If we had managed the Manhattan project in this way, we would have failed to create the atomic bomb. Without risk, there is no reward. There is a huge amount of resource and effort wasted. We do not lack money as much as we lack vision, inspiration and competent management. This is not to say that the United States does not have an issue investing in science and technology, we do. The current level of commitment to science and technology will assure that some other nation becomes the global leader in science and technology. A compounding issue to the lack of investment is how appallingly inefficient our investment is because of how science is managed today. A complimentary compounding element is the lack of trust in the scientists and engineers. Without trust, no one will take any risk and without taking risks nothing great will ever be achieved. If we don’t solve these problems, we will not produce greatness, plain and simple; we will create decline and decay into mediocrity.

But until a person can say deeply and honestly, “I am what I am today because of the choices I made yesterday,” that person cannot say, “I choose otherwise.

― Stephen R. Covey

None of these problems suddenly appeared. They are the consequence of decades of evolution toward the current completely dysfunctional management approach. Once great Laboratories have been brought to heel with a combination of constraints, regulations and money. There is more than enough money and people to accomplish massive things. The problem is that the constraints and regulatory environment have destroyed any chance for achievement. With each passing year our scientific programs sound more expansive, but less capable of achieving anything of substance. Our management approach is undermining achievement at every turn. The focus of the management is not producing results, but producing the appearance of success without regard for reality. The workforce must be complaint, and never make any mistakes. The best way to avoid mistakes is low-balling results. You always aim low to avoid the possibility of failing. Each year we aim a little lower, and achieve a little less. This has produced a steady erosion of capability much like an interest-bearing account, but in reverse.

If we look at work, it might seem that an inspired workforce would be a benefit worth creating. People would work hard and create wonderful things because of the depth of their commitment to a deeper purpose. An employer would benefit mightily from such an environment, and the employees could flourish brimming with satisfaction and growth. With all these benefits, we should expect the workplace to naturally create the conditions for inspiration. Yet this is not happening; the conditions are the complete opposite. The reason is that inspired employees are not entirely controlled. Creative people do things that are unexpected and unplanned. The job of managing a work place like this is much harder. In addition, mistakes and bad things happen too. Failure and mistakes are an inevitable consequence of hard working inspired people. This is the thing that our work places cannot tolerate. The lack of control and unintended consequences are unacceptable. Fundamentally this stems from a complete lack of trust. Our employers do not trust their employees at all. In turn, the employees do not trust the workplace. It is vicious cycles that drags inspiration under and smothers it. The entire environment is overflowing with micromanagement, control suspicion and doubt.

In the end that was the choice you made, and it doesn’t matter how hard it was to make it. It matters that you did.”

― Cassandra Clare

How do we change it?

One clear way of changing this is giving the employees more control over their work. It has become very clear to me that we have little or no power to make choices at work. One of the clearest ways of making a choice is being given the option to say “NO”. Many articles are written about the power of saying NO to things because it makes your “YES” more powerful. The problem is that we can’t say NO to so many things. I can’t begin to elaborate on all the functionally useless things that don’t have to option of skipping. I spend a great deal of effort on mandatory meetings, training, and reporting that has no value whatsoever. None of it is optional, and most of it is completely useless. Each of these useless activities drains away energy from something useful. All of the useless things I do are related to a deep lack of trust in me and my fellow scientists.

Let’s take the endless reporting and tracking of work as a key example. There is nothing wrong with planning a project and getting updates on progress. This is not what is happening today. We are seeing a system that does not trust its employees and needs to continually look over their sholders. A big part of the problem is that the employees are completely uninspired because the programs they work on are terrible. The people see very little of themselves in the work, or much purpose and meaning in the work. Rather than make the work something deeper and more collaborative, the employers increase the micromanagement and control. A big part of the lack of trust is the reporting. Somehow the whole concept of quarterly progress used for business has become part of science creating immense damage. Lately quarterly progress isn’t enough, and we’ve moved to monthly reporting. All of this says, “we don’t trust you,” “we need to watch you closely” and “don’t fuck up”.

The smart way to keep people passive and obedient is to strictly limit the spectrum of acceptable opinion, but allow very lively debate within that spectrum….

― Noam Chomsky

If we can’t say NO to all this useless stuff, we can’t say YES to things either. My work and time budget is completely stocked up with non-optional things that I should say NO to. They are largely useless and produce no value. Because I can’t say NO, I can’t say YES to something better. My employer is sending a message to me with very clear emphasis, we don’t trust you to make decisions. Your ideas are not worth working on. You are expected to implement other people’s ideas no matter how bad they are. You have no ability to steer the ideas to be better. Your expertise has absolutely no value. A huge part of this problem is the ascendency of the management class as the core of organizational value. We are living in the era of the manager; the employee is a cog and not valued. Organizations voice platitudes toward the employees, but they are hollow. The actions of the organization spell out their true intent. Employees are not to be trusted, they are to be controlled and they need to do what they are told to do. Inspired employees would do things that are not intended, and take organizations in new directions, focused on new things. This would mean losing control and changing plans. More importantly, the value of the organization would move away from the managers and move to the employees. Managers are much happier with employees that are “seen and not heard”.

If something is not a “hell, YEAH!”, then it’s a “no!

― James Altucher

What should I be saying YES to?

If I could say YES then I might be able to put my focus into useful, inspired and risky endeavors. I could produce work that might go in directions that I can’t anticipate or predict. These risky ideas might be complete failures. Being a failure I could learn invaluable lessons, and grow my knowledge and expertise. Being risky these ideas might produce something amazing and create something of real value. None of these outcomes are a sure thing. All of these characteristics are unthinkable today. Our managers want a sure thing and cannot deal with unpredictable outcomes. The biggest thing our managers cannot tolerate is failure. Failure is impossible to take and leads to career limiting consequences. For this reason, inspired risks are impossible to support. As a result, I can’t say NO to anything, no matter how stupid and useless it is. In the process, I see work as an increasingly frustrating waste of my time.

Action expresses priorities.

― Mahatma Gandhi

We all have limits defined our personal time and effort. Naturally we have 24 hours a day, 7 days a week and 365 days a year, along with our own personal energy budget. If we are managed well, we can expand our abilities and create more. We can be more efficient and work more effectively. If one looks honestly at how we are managed expanding our abilities and personal growth has almost no priority. Creating an inspiring and exciting place to work is equally low on the list. Given the pathetic level of support for creation and inspiration attention naturally turns elsewhere. Everyone needs a level of balance in their lives and we obviously gravitate toward places where a difference can be made.

As Mark Manson writes we only have so many fucks to give and my work is doing precious little to give them there. I have always focused on personal growth and increasingly personal growth is resisted by work instead of resonated with. It has become quite obvious that being the best “me” is not remotely a priority. The priority at work is to be compliant, take no risks, fail at nothing and help produce marketing material for success and achievement. We aren’t doing great work anymore, but pretend we are. My work could simply be awesome, but that would require giving me the freedom to set priorities, take risks, fail often, learn continually and actually produce wonderful things. If this happened the results would speak for themselves and the marketing would take care of itself. When the Labs I’ve worked at were actually great this is how it actually happened.  The Labs were great because they achieved great things. The labs said NO to a lot of things, so they could say YES to the right things. Today, we simply don’t have this freedom.

We are our choices.

― Jean-Paul Sartre

If we could say NO to the bullshit, and give our limited fucks a powerful YES, we might be able to achieve great things. Our Labs could stop trying to convince everyone that they were doing great things and actually do great things. The missing element at work today is trust. If the trust was there we could produce inspiring work that would generate genuine pride and accomplishment. Computing is a wonderful example of these principles in action. Scientific computing became a force in science and engineering contributing to genuine endeavors for massive societal goals. Computing helped win the Cold War and put a man on the moon. Weather and climate has been modeled successfully. More broadly, computers have reshaped business and now societally massively. All of these endeavors had computing contributing to solutions. Computing focused on computers was not the endeavor itself like it is today. The modern computing emphasis was originally part of a bigger program of using science to support the nuclear stockpile without testing. It was part of a focused scientific enterprise and objective. Today it is a goal unto itself, and not moored to anything larger. If we want to progress and advance science, we should focus on great things for society, not superficially put our effort into mere tools.

Most of us spend too much time on what is urgent and not enough time on what is important.

― Stephen R. Covey

Say no to everything, so you can say yes to the one thing.

― Richie Norton

Better to get hurt by the truth than comforted with a lie.

― Khaled Hosseini

Being honest about one’s shortcomings is incredibly difficult. This is true whether one is looking at their self, or looking at a computer model. It’s even harder to let someone else be honest with you. This difficulty is the core of many problems with verification and validation (V&V). If done correctly, V&V is a form of radical honesty that many simply cannot tolerate. The reasons are easy to see if our reward systems are considered. Computer modeling desires to get great results on the problems they want to solve. Computer modelers are rated on their ability to get seemingly high-quality answers (https://williamjrider.wordpress.com/2016/12/22/verification-and-validation-with-uncertainty-quantification-is-the-scientific-method/ ). As a result, there is significant friction with honest V&V assessments, which provide uncertainty and doubt on the quality of results. The tension between good results and honesty will always favor the results. Thus V&V is done poorly to conserve the ability of modelers to believe their results are better than they really are. If we want V&V to be done well an additional level of emphasis needs to be placed on honesty.

If you do not tell the truth about yourself you cannot tell it about other people.

― Virginia Woolf

V&V is about assessing capability. It is not about getting great answers. This distinction is essential to recognize. V&V is about collecting highly credible evidence about the nature of modeling capability. By its very nature, the credibility of the evidence means that the results are whatever the results happen to be. If the results are good the evidence will show this persuasively. If the results are poor, the evidence will indicate the quality (https://williamjrider.wordpress.com/2017/09/22/testing-the-limits-of-our-knowledge/ ). The utility of V&V is providing a path to improvement along with evidence to support this path. As such, V&V provides a path and evidence for getting to improved results. This improved result would then be supported by V&V assessments. This entire process is predicated on the honesty of those conducting the work, but the management of these efforts is a problem. Management is continually trying to promote the great results outcomes for modeling. Unless the results are actually great, this promotion provides direction for lower quality V&V. In the process, honesty and evidence are typically sacrificed.

If we want to do V&V properly, something in this value system needs to change. Fundamentally, honesty and a true understanding of the basis of computational modeling must surpass the desire to show great capability. The trends in management of science are firmly arrayed against honestly assessing capability. With the prevalence of management by press release, and a marketing based sales pitch for science money both act to promote a basic lack of honesty and undermine disclosure of problems. V&V provides firm evidence of what we know, and what we don’t know. The quantitative and qualitative aspects of V&V can produce exceptionally useful evidence of where modeling needs to improve. These characteristics conflict directly with the narrative that modeling has already brought reality to heel. Program after program is sold on the basis that modeling can produce predictions of what will be seen in reality. Computational modeling is seen as an alternative to expensive and dangerous experiments and testing. It can provide reduced costs and cycle times for engineering. All of this can be a real benefit, but the degree of current mastery is seriously oversold.

Doing V&V properly can unmask this deception (I do mean deception even if the deceivers are largely innocent of outright graft). The deception is more the product of massive amounts of wishful thinking, and harmful group think focused on showing good results rather than honest results. Sometimes this means willfully ignoring evidence that does not support the mastery. In other cases, the results are based on heavy-handed calibrations, and the modeling is far from predictive. In the naïve view, the non-predictive modeling will be presented as predictions and hailed as great achievements. Those who manage modeling are largely responsible for this state of affairs. They reward the results that show how good the models are and punish honest assessment. Since V&V is the vehicle for honest assessment, it suffers. Modelers will either avoid V&V entirely, or thwart any effort to apply it properly. Usually the results are given without any firm breakdown of uncertainties, and simply assert that the “agreement is good” or the “agreement is excellent” without any evidentiary basis save plots that display data points and simulation values being “close”.

If you truly have faith in your convictions, then your convictions should be able to stand criticism and testing.

― DaShanne Stokes

This situation can be made better by changing the narrative about what constitutes good results. If we value knowledge and evidence of mastery as objectives instead of predictive power, we tilt the scales toward honesty. One of the clearest invitations to hedge toward dishonesty is the demand of “predictive modeling”. Predictive modeling has become a mantra and sales pitch instead of an objective. Vast sums of money are allotted to purchase computers, and place modeling software on these computers with the promise of prediction. We are told that we can predict how our nuclear weapons work so that we don’t have to test them. The new computer that is a little bit faster is the key to doing this (they always help, but are never the lynchpin). We can predict the effects of human activity on climate to be proactive about stemming its effects. We can predict weather and hurricanes with increasing precision. We can predict all sorts of consequences and effect better designs of our products. All of these predictive capabilities are real, and all have been massively oversold. We have lost our ability to look at challenges as good things and muster the will to overcome them. We need to tilt ourselves to be honest about how predictive we are, and understand where our efforts can make modeling better. Just as important we need to unveil the real limits on our ability to predict.

A large part of the conduct of V&V is unmasking the detailed nature of uncertainty. Some of this uncertainty comes from our lack of knowledge of nature, or flaws in our fundamental models. Other uncertainty is simply intrinsic to our reality. This is phenomena that is variable even with seemingly identical starting points. Separating these types of uncertainty, and defining their magnitude should be greatly in the service of science. For the uncertainties that we can reduce through greater knowledge, we can array efforts to affect this reduction. This must be coupled to the opportunity for experiment and theory to improve matters. On the other hand, if uncertainty is irreducible, it is important to factor it into decisions and accommodate its presence. By ignoring uncertainty with the practice of default of ZERO uncertainty (https://williamjrider.wordpress.com/2016/04/22/the-default-uncertainty-is-always-zero/ ), we become powerless to assert our authority, or practically react to it.

In the conduct of predictive science, we should look to uncertainty as one of our primary outcomes. When V&V is conducted with high professional standards, uncertainty is unveiled and estimated in magnitude. With our highly over-promised mantra of predictive modeling enabled by high performance computing, uncertainty is almost always viewed negatively. This creates an environment where willful or casual ignorance of uncertainty is tolerated and even encouraged. Incomplete and haphazard V&V practice becomes accepted because it serves the narrative of predictive science. The truth and actual uncertainty is treated as bad news, and greeted with scorn instead of praise. It is simply so much easier to accept the comfort that the modeling has achieved a level of mastery. This comfort is usually offered without evidence.

The trouble with most of us is that we’d rather be ruined by praise than saved by criticism.

― Norman Vincent Peale

Somehow a different narrative and value system needs to be promoted for science to flourish. A starting point would be a recognition of the value of highly professional V&V work and the desire for completeness and disclosure. A second element of the value system would be valuing progress in science. In keeping with the value on progress would be a recognition that detailed knowledge of uncertainty provides direct and useful evidence to steer science productively. We can also use uncertainty to act proactively in making decisions based on actual predictive power. Furthermore, we may choose not to use modeling to decide if the uncertainties are too large and informing decisions. The general support for the march forward of scientific knowledge and capability is greatly aided by V&V. If we have a firm accounting of our current state of knowledge and capability, we can mindfully choose where to put emphasis on progress.\

This last point gets at the problems with implementing a more professional V&V practice. If V&V finds that uncertainties are too large, the rational choice may be to not use modeling at all. This runs the risk of being politically incorrect. Programs are sold on predictive modeling, and the money might look like a waste! We might find that the uncertainties from numerical error are much smaller than other uncertainties, and the new super expensive, super-fast computer will not help make things any better. In other cases, we might find out that the model is not converging toward a (correct) solution. Again, the computer is not going to help. Actual V&V is likely to produce results that require changing programs and investments in reaction. Current management often looks to this as a negative and worries that the feedback will reflect poorly on previous investments. There is a deep-seated lack of trust between the source of the money and the work. The lack of trust is driving a lack of honesty in science. Any money spent on fruitless endeavors is viewed as a potential scandal. The money will simply be withdrawn instead of redirected more productively. No one trusts the scientific process to work effectively.  The result is an unwillingness to engage in a frank and accurate dialog about how predictive we actually are.

It’s discouraging to think how many people are shocked by honesty and how few by deceit.

― Noël Coward

It wouldn’t be too much of a stretch to say that technical matters are a minor aspect of improving V&V. This does not make light of, nor minimize the immense technical challenges in conducting V&V. The problem is that the current culture of science is utterly toxic for progress technically. We need a couple of elements to change in the culture of science to make progress. The first one is trust. The lack of trust is pervasive and utterly incapacitating (https://williamjrider.wordpress.com/2013/11/27/trust/, https://williamjrider.wordpress.com/2016/04/01/our-collective-lack-of-trust-and-its-massive-costs/, https://williamjrider.wordpress.com/2014/12/11/trust-and-truth-in-management/  ). Because of the underlying lack of trust, scientists and engineers cannot provide honest results or honest feedback on results. They do not feel safe and secure to do either. This is a core element surrounding the issues with peer review (https://williamjrider.wordpress.com/2016/07/16/the-death-of-peer-review/ ). In an environment where there is compromised trust, peer review cannot flourish because honesty is fatal.

Nothing in this world is harder than speaking the truth, nothing easier than flattery.

― Fyodor Dostoyevsky

The second is a value on honesty. Today’s World is full of examples where honesty is punished rather than rewarded. Speaking truth to power is a great way to get fired. Those of us who want to be honest are left in a precarious position. Choose safety and security while compromising our core principles, or stay true to our principles and risk everything. Over time, the forces of compromised integrity, marketing and bullshit over substance wear us down. Today the liars and charlatans are winning. Being someone of integrity is painful and overwhelming difficult. The system seems to be stacked against honest discourse and disclosure. Of course, honesty and trust are completely coupled. Both need to be supported and rewarded. V&V is simply one area where these trends play out and distort work.

It is both jarring and hopeful that the elements holding science back are evident in the wider world. The new and current political discourse is full of issues that are tied to trust and honesty. The degree to which we lack trust and honesty in the public sphere is completely disheartening. The entire system seems to be spiraling out of control. It does not seem that the system can continue on this path much longer (https://williamjrider.wordpress.com/2017/10/20/our-silence-is-their-real-power/ ). Perhaps we have hit bottom and things will get better. How much worse can things get? The time for things to start getting better has already passed. This is true in the broader public World as well as science. In both cases trust for each other, and a spirit of honesty would go a long way to providing a foundation for progress. The forces of stagnation and opposition to progress have won too much ground.

Integrity is telling myself the truth. And honesty is telling the truth to other people.

― Spencer Johnson

 

Nothing is so difficult as not deceiving oneself.

― Ludwig Wittgenstein

 

Without deviation from the norm, progress is not possible.

― Frank Zappa

There is something seriously off about working on scientific computing today. Once upon a time it felt like working in the future where the technology and the work was amazingly advanced and forward-looking. Over the past decade this feeling has changed dramatically. Working in scientific computing is starting to feel worn-out, old and backwards. It has lost a lot of its sheen and it’s no longer sexy and fresh. If I look back 10 years everything we then had was top of the line and right at the “bleeding” edge. Now we seem to be living in the past, the current advances driving computing are absent from our work lives. We are slaving away in a totally reactive mode. Scientific computing is staid, immobile and static, where modern computing is dynamic, mobile and adaptive. If I want to step into the modern world, now I have to leave work. Work is a glimpse into the past instead of a window to the future. It is not simply the technology, but the management systems that come along with our approach. We are being left behind, and our leadership seems oblivious to the problem.

For most of the history of computing in the 20^th and into the 21^st Century, scientific computing was at the forefront of technology. That is starting to change. Even today scientific computing remains exotic in terms of hardware and some aspects of software, but it also feels antiquated and antique. We get to use cutting edge computer chips and networking hardware that demand we live on the ragged edge technologically. This is only half the story. We also remain firmly entrenched in the “mainframe” era with corporate computing divisions that seem more “Mad Men” and less “Star Trek” than ever. The distance between the computers we use to execute our leading edge scientific investigations and our offices or our personal lives are diverging at warp speed. It has become hopelessly ironic in many ways. Worse than ironic, the current state of things is unhealthy and lessens the impact of scientific computing on today’s World.

Even worse than the irony is the price this approach is exacting on scientific computing. For example, the computing industry used to beat a path to scientific computing’s door, and now we have to basically bribe the industry to pay attention to us. A fair accounting of the role of government in computing is some combination of being a purely niche market, and partially pork barrel spending. Scientific computing used to be a driving force in the industry, and now lies as a cul-de-sac, or even pocket universe, divorced from the day-to-day reality of computing. Scientific computing is now a tiny and unimportant market to an industry that dominates the modern World. In the process, scientific computing has allowed itself to become disconnected from modernity, and hopelessly imbalanced. Rather than leverage the modern World and its technological wonders many of which are grounded in information science, it resists and fails to make best use of the opportunity. It robs scientific computing of impact in the broader World, and diminishes the draw of new talent to the field.

It would be great to elaborate on the nature of the opportunities, and the cost of the present imbalances. If one looks at the modern computing industry and its ascension to the top of the economic food chain, two things come to mind: mobile computing – cell phones – and the Internet. Mobile computing made connectivity and access ubiquitous with massive penetration into our lives. Networks and apps began to create new social connections in the real world and lubricated communications between people in a myriad of ways. The Internet became both a huge information repository, and commerce. but also an engine of social connection. In short order, the adoption and use of the internet and computing in the broader human World overtook and surpassed the use by scientists and business. Where once scientists used and knew computers better than anyone, now the World is full of people for whom computing is far more important than for science. Science once were in the lead, and now they are behind. Worse yet, science is not adapting to this new reality.

Those who do not move, do not notice their chains.

― Rosa Luxemburg

The core of the problem with scientific computing is its failure to adapt and take advantage of the opportunity defined by this ascendency of computing. A core of science’s issue with computing is the lost sense that computers are merely a tool. Computers are a tool that may be used to do science. Instead of following this maxim, we simply focus on the older antiquated model of scientific computing firmly grounded in the mainframe era. Our mindset has not evolved with the rest of the World. One of the clear consequences of the mindset is a creeping degree of gluttony and intellectual laziness with high performance computing. All problems reduce to simply creating faster computers and making problems submit to the raw power of virtually limitless computations. We have lost sight of the lack of efficiency of this approach. A renewed focus on issues of modeling, methods and algorithms could be deeply enlivened by the constraints imposed by limited computing resources. Moreover, the benefits of solving problems more efficiently with smaller computing resources would yield innumerable benefits in the setting of big iron. This could be achieved without the very real limitations of having big iron be the sole focus of our efforts.

Scientific computing could be arranged to leverage the technology that is advancing the World today. We could look at a mobile, adaptive platform for modeling, simulation and data analysis that harnessed the best of technology. We could move through the cloud using technology in an adaptive, multiscale manner. One of the biggest challenges is letting go of the power dynamic that drives thinking today. Scientific computing has been addicted to Moore’s law for too long. The current exascale push is symptomatic of this addiction. Like any addiction it is unhealthy and causes the subject to avoid real cures for their problem. We see progress as equivalent to raw power with a single computer. The huge stunt calculation as a vehicle for science is a manifestation of this addiction. Science is done with many calculations along with an adaptive examination of problems or mindful interrogation of results. Power can also be achieved through mobility, ubiquity and flexibility. The big iron we pursue has become tantamount to progress because it’s the only route we can envision. The problem is that technology, and the arc of progress is working against us instead of with us. It is past time to change our vision of what the future can be. The future needs to be different by embracing a different technological path. On one hand, we won’t be swimming against the current of computing technology, but on the other hand we will need to invest in different solutions to make it work.

Flexibility is an art of creating way outs within the cul-de-sacs!
― Mehmet Murat ildan

Mobility is power, and it has made computing ubiquitous. When the broader computing industry embraced the death of Moore’s law, it switched its attention to cell phones. Instead of simply being phones, they became mobile computers and mobile extensions of the Internet. In doing so we unleashed a torrent of creativity and connection. All of a sudden, we saw computers enable the level of social connection that the Internet always had promised, but never delivered. The mobile computing revolution has reshaped the World in a decade. In the process, the mobile market overwhelmed the entire computing industry and created economic dominance on an unparalleled scale. The killer piece of technology was the iPhone. It combined a focus on user interface along with software that enabled everything. We also need to recognize that each phone is more powerful than the fastest computer in the World 25 years ago. We have tremendous power at our fingertips.

One of the really clear messages of the recent era in computing is a change in the nature of value and power. For a long time, power was measured by hardware gains in speed, memory and capability, but now application innovation and flexibility rule. Hardware is largely a fixed and slowly changing commodity and represents a level playing field. The software in the applications and the user interface are far more important. Algorithms that direct information and attention are dominating the success in computing. Providing the basis of connection and adaption to the needs of the users has become the medium for creating new markets. At the same time these algorithms have come under fire for how they manipulate people and data. These mobile computers have become a massive issue for society as a whole. We are creating brand new social problems and side-effects we need to effectively solve. The impact of this revolution in computing on society as a whole has been incredible.

A whole cadre of experts is fading from the field of play in computing. In taking the tact of focusing on mainframe computing, scientific computing is sidelining itself. Instead of this enormously talented group of people playing in the area that means the most to society, they are focused on a cul-de-sac grounded in old and outdated models of success. Our society would benefit by engaging these experts in making mobile computing more effective in delivering value in new innovative ways. We could be contributing to solving some of the greatest problems facing us rather than seeing our computing as a special niche serving a relatively small segment of society’s needs. In the past, scientific computing has provided innovative and dynamic solutions that ultimately made their way into the general computing. A perfect example is Google. The problem that Google solved is firmly grounded in scientific computing and applied mathematics. It is easy to see how massive the impact of this solution is. Today we in scientific computing are getting further and further from relevance to society. This niche does scientific computing little good because it is swimming against a tide that is more like a tsunami. The result is a horribly expensive and marginally effective effort that will fail needlessly where it has the potential to provide phenomenal value.

You never change things by fighting the existing reality.

To change something, build a new model that makes the existing model obsolete.

― R. Buckminster Fuller

We are long passed the time to make a change in scientific computing’s direction and strategy. Almost everywhere else the mainframe era died decades ago. Why is scientific computing tied to this model? Why are scientists resisting the conclusions so nakedly obvious? In today’s risk, adverse environment making a change to the underlying model of this branch of science is virtually impossible. Even when the change is dramatically needed and overdue by years the resistance is strong. The status quo is safe and firmly entrenched. In a time when success can be simply asserted and largely manufactured, this unacceptable state of affairs will persist far longer than it should. Sooner or later someone will take the plunge, and success will follow them. They will have the winds of progress at their backs solving most of the problems easily that we throw billions of dollars at with meager success.

The measure of intelligence is the ability to change.

― Albert Einstein

 

If I had an hour to solve a problem I’d spend 55 minutes thinking about the problem and 5 minutes thinking about solutions.

― Albert Einstein

A few years ago, I was challenged to examine the behavior of void in continuum hydrocodes. A senior colleague suggested looking at problems that might allow us to understand how the absence of material would be treated in a code. The simplest version of this problem would solve the expansion of a real gas into a void. With an ideal gas this problem has an exact solution that can be found with a Riemann solution. In the process, we have discovered that these problems are not solved well by existing methods. We approximate the void with a very low density and pressure material, and we have found as the material approaches an actual void, the solutions seem to become non-convergent, and prone to other significant numerical difficulties. Even when using extremely refined meshes with many 1000’s of cells in one dimension, convergence is not observed for a broad class of methods. These methods have solved many difficult problems and we believe them to be robust and reliable. These problems persist for all methods tested including our fail-safe methods (e.g., first order Godunov).

What is going on?

I’ll just say in passing that this post is a bit of a work in progress conversation with myself (or myself to you). My hope is that it will shake lose my thinking. It is patterned on the observation that sometimes you can solve a problem by carefully explaining it to someone else.

I suppose it is tempting, if the only tool you have is a hammer, to treat everything as if it were a nail.

― Abraham H. Maslow

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One of the difficulties of this problem is the seemingly bad behavior coming from our most reliable and robust methods. When we want a guaranteed a good solution to a problem, we unleash a first-order Godunov method on it, and if use an exact Riemann solver we can expect the solution to be convergent. The results we see with void seemingly violate this principle. We are getting terrible solutions in a seemingly systematic manner. To make matters worse, the first-order Godunov method is the basis, and fallback position for the more important second- or third-order methods we practically want to use. We can conclude that this problem is exposing some rather serious problems with our workhorse methods and the potential for wholesale weakness in our capability.

There are no facts, only interpretations.

― Friedrich Nietzsche

Let’s look at what happens for the approximate-void problem. We approximate the void with a gas that has a density and pressure of twelve orders of magnitude smaller than the “real” material. This problem has a solution that almost gives the expansion into vacuum solution to the Euler equations (where the head of the rarefaction and the contact discontinuity collapse into a single structure that separates material from nothing). The problem is dominated by an enormous rarefaction that takes the density down by many orders of magnitude. What we see is a solution that appears to get worse and worse under mesh refinement. In other words, it diverges under mesh refinement. Actually, the behavior we see is a bit more complex than this. At very low resolutions, the solution is behind the exact solution, and as we refine the mesh, the solution catches up to and, then passes the exact solution. Then as we add more and more mesh, the solution just gets worse and worse. This is not supposed to happen. This is a very bad thing that needs focused attention.

So maybe backing away from the extreme problem is worth doing. I ran a sequence of shock tube problems varying the jump in pressure and density starting at 10:1 and slowly going up to the extreme jump that approximates an expansion into void. The shock tube is a self-similar problem, meaning that we can swap time and space through a similarity transformation. Thus, the very early time evolution on a very fine grid is essentially the same as a late time solution on a very coarse grid. What I noticed is the same pattern over and over. More importantly, the problem gets worse and worse as the jumps get larger and larger. By examining the trend as the jumps become very large, we start to see the nature of our problem. As the jump becomes larger and larger, the solution converges more and more slowly. We can start to estimate the mesh resolution needed for a good result and we can see that the situation becomes almost hopeless in the limit. I believe the solution will eventually converge given enough mesh, but the size of the mesh needed to get a convergent solution becomes completely absurd.

In summary, the problem with a factor of a million jump converges with modestly unreasonable mesh. As the jump grows in size, the convergence requires a mesh that is prohibitive for any practical work. If we are going to accurately solve this class of problems some other approach is needed. To make things worse the when the problem converges, the rate of convergence under refinement of the mesh is painfully slow, and incredibly expensive as a result.

Everywhere is walking distance if you have the time.

― Steven Wright

The second issue we see is a persistent velocity glitch at the head of the rarefaction. It is fair to say that the glitch has heretofore been viewed as a cosmetic problem. This velocity peak looks like a meaningfully wrong solution to the equations locally. It produces a shock like solution in the sense that produces a violation of Lax’s entropy condition, where the characteristics locally converge in a shock-like manner in a rarefaction where the characteristics should diverge locally. We might expect that this problem would hurt the physically meaningful solution. Not all-together surprisingly the solution can also violate the second law of thermodynamics when using higher than first-order methods. Moreover, this character simply gets worse and worse as the problem gets closer to a void. A reasonable supposition is that this feature in the numerical solution is a symptom of difficulties in rarefactions. Usually this feature can be categorized as a nuisance and relatively small contributor to error, but may be a sign of something deeper. Perhaps this nuisance becomes a significant issue as the rarefaction becomes stronger, and ultimately dominates the numerical character of the solution. We might be well-served by removing it from the solution. One notion we might add to the treatment of the glitch is its diminishing size as the mesh is refined. Having this anomalous shock-like character allows dissipation to damp the spike and improve the solution. The counter-point to this solution is not creating the glitch in the first place.

At this point it’s useful to back away from the immediate problem to a broader philosophical point. The shock capturing methods are naturally focused on computing shocks. Shock waves were a big challenge for numerical methods. They remain a large challenge, and failure to treat them effectively can be fatal for a calculation. If the shock wave was not treated with care, the numerical can fail catastrophically, or significantly damaged. Even when the results are not catastrophic, poor treatment of a shock can result in significant corruption of the solution that often spreads from the shock to other areas in the solution. For this reason, the shock wave and its numerical treatment has been an enduring focus of numerical methods for compressible flows. Conversely rarefactions have largely been an afterthought. Rarefactions are benign smooth structures that do not directly threaten a calculation. A few bad things can happen in rarefactions, but they are rarely fatal to the calculation. A few have been so cosmetically problematic that major effort has ensued (the rarefaction shock). Problems in rarefactions are generally just a nuisance, and only become a focal point when the details of the solution are examined. One aspect of the details is the convergence character of the solution. Shock tube problems are rarely subjected to a full convergence analysis. The problem we focus on here is dominated by a rarefaction thus magnifying any problems immensely. What we can conclude is that strong rarefactions are not computed with high fidelity.

The trick to forgetting the big picture is to look at everything close up.

― Chuck Palahniuk

One of the key ways of dealing with shock waves are upwind methods. A clear manner of treating these waves and getting an upwind solution is the use of a discontinuous basis to define the spatial discretization. This discontinuous basis is also used with high-order methods, and the first order solution becomes the fallback position for the methods. This approach is very well suited to computing shocks; a discontinuous approximation for a discontinuous phenomenon. By the same token, a discontinuous basis is not well suited for a continuous phenomenon like a rarefaction. One hypothesis to explore is different types of approximations to the problem where the rarefaction dominates the solution. We may find that we can solve this class of problem far more efficiently with a continuous basis getting asymptotically convergent solutions far sooner. What we observe is an ever slower approach to a convergent behavior in the code. For this class of problems we see a consistent pattern, the solution starts out being under-resolved and the velocity rises, it then overshoots the correct analytical result, then slowly decays toward the correct solution. As the rarefaction becomes stronger and stronger, we see that the mesh resolution needed to capture the full rise, its achievement of the peak overshoot value take place at a finer and finer mesh.  Ultimately, the mesh required to get a solution that converges becomes absurdly refined.

If this proposition is indeed correct, it implies that we need to define a hybrid approach where the basis is adaptively chosen. At discontinuous structures, we want to choose discontinuous approaches, and at continuous structures we want continuous structures. This is almost obvious, but carrying this out in practice is difficult. Clearly the current adaptive approaches are not working well enough as evidenced by the painful and absurd degree of mesh needed to get a reasonable solution. It would seem that the answer to this problem lies in developing a new method capable of solving extreme rarefactions on reasonable meshes.  We need to have methods that can solve strong, but continuous waves with higher fidelity. In all reality, these methods might need to effectively compute shocks albeit less effectively than methods using a discontinuous basis. The bottom line from attacking a challenging problem like this is the demonstration that our methods today are not sufficient to all our challenges.

Creativity consists of coming up with many ideas, not just that one great idea.

― Charles Thompson

Yes.

Of course, it’s not really that simple, but yes, the code is part of the model. If it isn’t, one has to provide a substantial burden of proof.

We have no idea about the ‘real’ nature of things … The function of modeling is to arrive at descriptions which are useful.

– Richard Bandler and John Grinder

Ideally, it should not be, but proving that ideal is a very high bar that is almost never met. A great deal of compelling evidence is needed to support an assertion that the code is not part of the model. The real difficulty is that the more complex the modeling problem is, the more the code is definitely and irreducibly part of the model. These complex models are the most important uses of modeling and simulation. The complex models of engineered things, or important physical systems have many submodels each essential to successful modeling. The code is often designed quite specifically to model a class of problems. The code then becomes are clear part of the definition of the problem. Even in the simplest cases, the code includes the recipe for the numerical solution of a model. This numerical solution leaves its fingerprints all over the solution of the model. The numerical solution is imperfect and contains errors that influence the solution. For a code, there is the mesh and geometric description plus boundary conditions, not to mention the various modeling options employed. Removing the specific details of the implementation of the model in the code from consideration as part of the model becomes increasingly intractable.

The word model is used as a noun, adjective, and verb, and in each instance it has a slightly different connotation. As a noun “model” is a representation in the sense in which an architect constructs a small-scale model of a building or a physicist a large-scale model of an atom. As an adjective “model” implies a degree of perfection or idealization, as in reference to a model home, a model student, or a model husband. As an adjective “model” implies a degree or perfection or idealization, as in reference to a model home, a model student, or a model husband. As a verb “to model” means to demonstrate, to reveal, to show what a thing is like.

– Russell L. Ackoff

The word model itself is deeply problematic. Model is one of those words that can mean many different things whether its used a noun or verb (I’ll note in passing much like the curse word, “fuck” is so flexible as to be wonderful and confusing all at once). Its application in a scientific and engineering context is common and pervasive. As such, we need to inject some precision into how it is being used. For this reason, some discourage the use of “model” in discussion. On the other hand, models and modeling is so central to the conduct of science and engineering that it should be dealt with head on. It isn’t going away. We model our reality when we want to make sure we understand it. We engage in modeling when we have something in the Real World, we want to demonstrate an understand of. Sometimes this is for the purpose of understanding, but ultimately this gives way to manipulation, the essence of engineering. The Real World is complex and effective models are usually immune to analytical solution.

Essentially, all models are wrong, but some are useful.

– George E. P. Box, Norman R. Draper

You view the world from within a model.

― Nassim Nicholas Taleb

Computational science comes to the rescue, and opens the doors to solving these complex models via numerical approximations. It is a marvelous advance, but brings new challenges because the solutions are imperfect. This adds a new layer of imperfection to modeling. We already should recognize that models are generically approximate versions of reality (i.e., wrong), and necessarily imperfect mathematical representations of the Real World. Solving this imperfect model, imperfectly via an approximate method makes the modeling issue even more fraught. Invariably for any model with complexity, the numerical solution of the model, and its detailed description are implemented in computer code, or “a computer code”. The details and correctness of the implementation become inseparable from the model itself. It becomes quite difficult to extract the model as any sort of pure mathematical construct; the code is part of it intimately.

Evidence of the model’s nature and correctness is produced in the basic conduct of verification and validation with uncertainty quantification. Doing a full accounting of the credibility of modeling, including pedigree of the model will not help to exclude the code from the model, simply define the extent of this connection. Properly speaking, the code is always part of the model, but the extent or magnitude of its impact can be small, or even considered minor or negligible. This evidence is contained within the full assessment of the predictive quality of the simulation including a quantitative assessment. Among these activities verification is the most important for the question at hand. Do we have evidence that the mathematical model desired is correctly solved? Do we have evidence that the numerical errors in the solution are small? Can all the aspects of the model be well described by clearly articulated mathematics?

Any physical theory is always provisional, in the sense that it is only a hypothesis: you can never prove it. No matter how many times the results of experiments agree with some theory, you can never be sure that the next time the result will not contradict the theory.

― Stephen Hawking

A model is not the operating system for the universe. Reality is not determined by these mathematical abstractions; the mathematics is designed to describe what we observe. As such, the models are always flawed and imperfect representations to some level. Determining the flaws and the quantitative level of imperfection is difficult work requiring detailed verification and validation. It is an abstraction and representation of the processes we believe produce observable physical effects. We theorize that the model explains how these effects are produced. Some models are not remotely this high minded; they are nothing, but crude empirical engines for reproducing what we observe. Unfortunately, as phenomena become more complex, these crude models become increasingly essential to modeling. They may not play a central role in the modeling, but still provide necessary physical effects for utility. These submodels necessary to produce realistic simulations become ever more prone to include these crude empirical engines as problems enter the engineering realm. As the reality of interest becomes more complicated, the modeling becomes elaborate and complex being a deep chain of efforts to grapple with these details.

It is a capital mistake to theorize before one has data. Insensibly one begins to twist facts to suit theories, instead of theories to suit facts.

― Arthur Conan Doyle

Validation of a model occurs when we take the results of solving the model and compare them directly with observations from the Real World. A key aspect of the validation exercise is characterizing the uncertainty in both the observations and the model. When all this assessment is in hand, we can render a judgment of whether the model represents the observed reality well enough for the purposes we intend. This use is defined by a question we want to answer with the modeling. The answer needs to have a certain fidelity, and certainty that provides the notion of precision to the exercise. The certainty of the observations defines a degree of agreement that can be demanded. The model’s uncertainties define the model’s precision, but includes the impact of numerical approximation. The numerical uncertainty needs to be accounted for to isolate the model. This uncertainty defines the level of approximation in the solution to the model, and a deviation from the mathematical idealization the model represents. In actual validation work, we see a stunning lack of this essential step from validation work presented. Another big part of the validation is recognizing the subtle differences between calibrated results and predictive simulation. Again, calibration is rarely elaborated in validation to the degree that it should.

We should always expect the model to deviate from observations to some degree. If we are capable of producing more accurate observations of reality, we can more accurately determine how wrong the model is. In a sense, we can view this as a competitive race. If our model is quite precise, we are challenged in being able to observe nature well enough to expose its innate flaws. Conversely, if we can observe nature with extreme precision, we can define the model’s imperfections clearly. Progress can be made by using this tension to push one or the other. The modeling uncertainty is compounded by approximate numerical solution implemented in a computer code (including the correctness of the code). Verification and validation activities are a systematic manner to collect evidence so that the comparison can be made in a complete and compelling manner.

Computer codes serve two very important roles in modeling: the model is contained in the code including geometry, boundary condition, and a host of ancillary models for complex situations, and solving the model numerically. Both of these characteristics are essential in the conduct of modeling, but numerical solutions are far more subtle and complex. Many people using codes for modeling do not have a background sufficient to understand the subtleties of numerical methods and their impact on solutions. Moreover, the fiction that numerical methods and codes are so reliable that detailed understanding is not essential, persists and grows. Our high performance computing programs work to fuel this fiction. The most obvious aspect of the numerical solution is the meshing and the time integration with the error’s proportionality to this detail. Producing evidence of the correctness and error characteristics is produced through verification. In addition, most advanced codes solve linear and nonlinear equations in an iterative manner. Iterative solutions have a finite tolerance in their solution, which can impact solutions. This is particularly true for nonlinear equation solvers where the error tolerance that can be achieved by some popular solvers is extremely loose. This looseness can produce significant physical effects in solutions. Most verification work does not examine these aspects closely although they should. Again, the code and its capabilities and methods are extremely important, if not essential, to the model produced. In many cases fantastic modeling work is polluted by naïve numerical methods, thus a wonderful model can be wiped out by a terrible code.

You’ve baked a really lovely cake, but then you’ve used dog shit for frosting.

― Steve Jobs

So, when can we exclude the code? The big thing to focus on in this question is verification evidence. Code verification is necessary to be confident that the mathematical model intended is provably present in the code. It asks whether the mathematical abstraction that the model is based on is correctly solved by the code. Code verification can be completely satisfactory and successful, and the code can still be important. Code verification does not say that the numerical error is small, it says that numerical error is ordered and the model equations desired to be solved are indeed solved. The second half of verification is solution (calculation) verification determines the errors in solving the model. The question is how large (or small) the numerical errors in the solution of the model are? Ultimately, these errors are a strong function of the discretization and solver used in the code. The question of whether the code matters comes down to asking if another code used skillfully would produce a significantly different result. This is rarely, if ever, the case. To make matters worse, verification evidence tends to be flimsy and half-assed. Even if we could make this call and ignore the code, we rarely have evidence that this is a valid and defensible decision.

Truth can only be found in one place: the code.

― Robert C. Martin

In closing, the code IS part of the model unless evidence can be found otherwise. This can happen more easily where the model is simple. In general, the exclusion of the code is an ideal that cannot be reached. As models become complex detaching the model from the code becomes nearly intractable, and indefensible. Evidence will almost invariably point to the code being an important contributor to the model’s picture of reality.

For the scientist a model is also a way in which the human though processes can be amplified. This method often takes the form of models that can be programmed into computers. At no point, however, the scientist intend to loose control of the situation because off the computer does some of his thinking for him. The scientist controls the basic assumptions and the computer only derives some of the more complicated implications.

– C. West Churchman

 

 

From the bad things and bad people, you learn the right way and right direction towards the successful life.

― Ehsan Sehgal

Computational science is an extremely powerful set of disciplines for conducting scientific investigations. The end result of computational science is usually grounded in the physical sciences, and engineering, but depends on a chain of expertise spanning much of modern science. Doing computational science well completely depends on all of these disparate disciplines working in concert. A big area of focus these days are the supercomputers being used. The predicate for acquiring a these immensely expensive machines is the improvement in scientific and engineering product arising from their use. While this should be true, getting across this finish line requires a huge chain of activities to be done correctly.

Let’s take a look at all the things we need to do right. Computer engineering and computer science are closest to the machines needed for computational science. These disciplines make these exotic computers accessible and useful for domain science and engineering. A big piece of this work is computer programming and software engineering. The computer program is a way of expressing mathematics in a way for the computer to operate on. Efficient and correct computer programs are a difficult endeavor all by themselves. Mathematics is the language of physics and engineering and essential for the conduct of computing. Mathematics is a middle layer of work between the computer and their practical utility. It is a deeply troubling and ironic trend that applied mathematics is disappearing from computational science. As the bridge between the computer and its practical use, it forms the basis for conducting and believing the computed results. Instead of being an area of increased focus, the applied math is disappearing into either the maw of computer programming or domain science/engineering. It is being lost as a separate contributor. Finally, we have the end result in science and engineering. Quite often we lose sight of computers and computing as a mere tool that must follow its specific rules for quality, reliable results. Too often the computer is treated like it is a magic wand.

Another common thread to horribleness is the increasing tendency for science and engineering to be marketed. The press release has given way to the tweet, but the sentiment is the same. Science is marketed for the masses who have no taste for the details necessary for high quality work. A deep problem is that this lack of focus and detail is creeping back into science itself. Aspects of scientific and engineering work that used to be utterly essential are becoming increasingly optional. Much of this essential intellectual labor is associated with the hidden aspects of the investigation. Things related to mathematics, checking for correctness, assessment of error, preceding work, various doubts about results and alternative means of investigation. This sort of deep work has been crowded out by flashy graphics, movies and undisciplined demonstrations of vast computing power.

Some of the terrible things we discuss here are simply bad science and engineering. These terrible things would be awful with or without a computer being involved. Other things come from a lack of understanding of how to add computing to an investigation in a quality focused manner. The failure to recognize the multidisciplinary nature of computational science is often at the root of many of the awful things I will now describe.

Fake is the new real, You gotta keep a lot a shit to yourself.

― Genereux Philip

Without further ado, here are some terrible things to look out for. Every single item on the list will be accompanied by a link to a full blog post expanding on the topic.

1. If one follows high performance computing online (institutional sites, Facebook, Twitter) you might believe that the biggest calculations on the fastest computers are the very best science. You are sold that these massive calculations have the greatest impact on the bottom line. This is absolutely not the case. These calculations are usually one-off demonstrations with little or no technical value. Almost everything of enduring value happens on the computers being used by the rank and file to do the daily work of science and engineering. These press release calculations are simply marketing. They almost never have the pedigree or hard-nosed quality work necessary for good science and engineering. – https://williamjrider.wordpress.com/2016/11/17/a-single-massive-calculation-isnt-science-its-a-tech-demo/, https://williamjrider.wordpress.com/2017/02/10/it-is-high-time-to-envision-a-better-hpc-future/
2. The second thing you come across is the notion that a calculation with larger-finer mesh is better than one with a coarser mesh. In the naïve pedestrian analysis, this would seem to be utterly axiomatic. The truth is that computational modeling is an assembly of many things all working in concert. This is another example of proof by brute force. In the best circumstances this would hold, but most modeling is hardly taking places under the best conditions. The proposition is that the fine mesh allows one to include all sorts of geometric details, so the computational world looks more like reality. This is a priori What isn’t usually discussed is where the challenge is in modeling. Is geometric detail driving uncertainty? What is biggest challenge, and is the modeling focused there? – https://williamjrider.wordpress.com/2017/07/21/the-foundations-of-verification-solution-verification/, https://williamjrider.wordpress.com/2017/03/03/you-want-quality-you-cant-handle-the-quality/, https://williamjrider.wordpress.com/2014/04/04/unrecognized-bias-can-govern-modeling-simulation-quality/
3. In concert with these two horrible trends, you often see results presented as the result of single massive calculation that magically unveils the mysteries of the universe. This is computing as a magic wand, and has very little to do with science or engineering. This simply does not happen. Real science and engineering takes 100’s or 1000’s of calculations to happen. There is an immense amount of background work needed to create high quality results. A great deal of modeling is associated with bounding uncertainty or bounding the knowledge we possess. A single calculation is incapable of this sort of rigor and focus. If you see a single massive calculation as the sole evidence of work, you should smell and call “bullshit”. – https://williamjrider.wordpress.com/2016/11/17/a-single-massive-calculation-isnt-science-its-a-tech-demo/
4. One of the key elements in modern computing is the complete avoidance of discussing how the equations in the code are being solved. The notion is that this detail has no importance. On the one hand, this is evidence of progress, our methods for solving equations are pretty damn good. The methods and the code itself is still an immensely important detail, and constitute part of the effective model. There seems to be a mentality that the methods and codes are so good that this sort of thing can be ignored. All one needs are a sufficiently fine mesh, and the results are pristine. This is almost always false. What this almost willful ignorance shows are lack of sophistication. The methods are immensely important to the results, and we are a very long way from being able to apply the sort of ignorance of this detail that is rampant. The powers that be want you to believe that the method disappears from importance because the computers are so fast. Don’t fall for it. – https://williamjrider.wordpress.com/2017/05/19/we-need-better-theory-and-understanding-of-numerical-errors/, https://williamjrider.wordpress.com/2017/05/12/numerical-approximation-is-subtle-and-we-dont-do-subtle/
5. The George Box maxim about models being wrong, but useful is essential to keep in mind. This maxim is almost uniformly ignored in the high-performance computing bullshit machine. The politically correct view is that the super-fast computers will solve the models so accurately that we can stop doing experiments. The truth is that eventually, if we are doing everything correct, the models will be solved with great accuracy and their incorrectness will be made evident. I strongly expect that we are already there in many cases; the models are being solved too accurately and the real answer to our challenges is building new models. Model building as an enterprise is being systematically disregarded in favor of chasing faster computers. We need far greater balance and focus on building better models worthy of the computers they are being solved on. We need to build the models that are needed for better science and engineering befitting the work we need to do. –https://williamjrider.wordpress.com/2017/09/01/if-you-dont-know-uncertainty-bounding-is-the-first-step-to-estimating-it/
6. Calculational error bars are an endangered species. We never see them in practice even though we know how to compute them. They should simply be a routine element of modern computing. They are almost never demanded by anyone, and their lack never precludes publication. It certainly never precludes a calculation being promoted as marketing for computing. If I was cynically minded, I might even day that error bars when used are opposed to marketing the calculation. The implicit message in the computing marketing is that the calculations are so accurate that they are basically exact, no error at all. If you don’t see error bars or some explicit discussion of uncertainty you should see the calculation as flawed, and potentially simply bullshit. – https://williamjrider.wordpress.com/2017/07/07/good-validation-practices-are-our-greatest-opportunity-to-advance-modeling-and-simulation/, https://williamjrider.wordpress.com/2017/09/22/testing-the-limits-of-our-knowledge/, https://williamjrider.wordpress.com/2017/04/06/validation-is-much-more-than-uncertainty-quantification/
7. One way for a calculation to seem really super valuable is to declare that it is direct numerical simulation (DNS). Sometimes this is an utterly valid designator. The other term that follows DNS is “first principles”. Each of these terms seeks to endow the calculation with legitimacy that it may, or may not deserve. One of the biggest problems with DNS is the general lack of evidence for quality and legitimacy. There is a broad spectrum of the technical World that seems to be OK with treating DNS as equivalent (or even better) with experiments. This is tremendously dangerous to the scientific process. DNS and first principles is still based on solving a model, and models are always wrong. This doesn’t say that DNS isn’t useful, but this utility needs to be proven and bounded by uncertainty. – https://williamjrider.wordpress.com/2017/11/02/how-to-properly-use-direct-numerical-simulations-dns/
8. Most press releases are rather naked in the implicit assertion that the bigger computer gives a better answer. This is treated as being completely axiomatic. As such there is no evidence provided to underpin this assertion. Usually some colorful graphics, or color movies beautifully rendered accompany the calculation. Their coolness is all the proof we need. This is not science or engineering even though this mode of delivery dominates the narrative today. –https://williamjrider.wordpress.com/2017/01/20/breaking-bad-priorities-intentions-and-responsibility-in-high-performance-computing/, https://williamjrider.wordpress.com/2014/09/19/what-would-we-actually-do-with-an-exascale-computer/, https://williamjrider.wordpress.com/2014/10/03/colorful-fluid-dynamics/
9. Modeling is the use of mathematics to connect reality to theory and understanding. Mathematics is translated into methods and algorithms implemented in computer code. It is ironic that the mathematics that forms the bridge between physical world and the computer is increasingly ignored by science. Applied mathematics has been a tremendous partner for physics, engineering and computing throughout the history of computational science. This partnership has waned in priority over the last thirty years. Less and less applied math is called upon and happens being replaced by computer programming or domain science and engineering. Our programs seem to think that the applied math part of the problem is basically done. Nothing could be further from the truth. – https://williamjrider.wordpress.com/2014/10/16/what-is-the-point-of-applied-math/, https://williamjrider.wordpress.com/2016/09/27/the-success-of-computing-depends-on-more-than-computers/
10. A frequent way of describing a computation is to describe the mesh as defining the solution. Little else is given about the calculation such as the equations being solved or how the equations are being approximated. Frequently, the fact that the solutions are approximated is left out. This fact is damaging to the accuracy narrative of massive computing. The designed message is that the massive computer is so powerful that the solution to the equations is effectively exact. The equations themselves basically describe reality without error. All of this is in service of saying computing can replace experiments, or real-world observations. The entire narrative is anathema to science and engineering doing each great disservice. – https://williamjrider.wordpress.com/2015/07/03/modeling-issues-for-exascale-computation/
11. Computational science is often described in terms that are not consistent with the rest of science. We act like it is somehow different in a fundamental way. Computers are just tools for doing science, and allowing us to solve models of reality far more generally than analytical methods. With all of this power comes a lot of tedious detail needed to do things with quality. This quality comes from the skillful execution of this entire chain of activities described at the beginning of this Post. These details all need to be done right to get good results. One of the biggest problems in the current computing narrative is ignorance to the huge set of activities bridging a model of reality and the computer itself. The narrative wants to ignore all of this because it diminishes the sense that these computers are magical in their ability. The power isn’t magic, it is hard work, success is not a forgone conclusion, and everyone should ask for evidence, not take their word for it. – https://williamjrider.wordpress.com/2016/12/22/verification-and-validation-with-uncertainty-quantification-is-the-scientific-method/

Taking the word of the marketing narrative is injurious to high quality science and engineering. The narrative seeks to defend the idea is that buying these super expensive computers is worthwhile, and magically produces great science and engineering. The path to advancing the impact of computational science dominantly flows through computing hardware. This is simply a deeply flawed and utterly naïve perspective. Great science and engineering is hard work and never a foregone conclusion. Getting high quality results depends on spanning the full range of disciplines associated with computational science adaptively as evidence and results demand. We should always ask hard questions of scientific work, and demand hard evidence of claims. Press releases and tweets are renowned for simply being cynical advertisements and lacking all rigor and substance.

One reason for elaborating upon things that are superficially great, but really terrible is cautionary. The current approach allows shitty work to be viewed as successful by receiving lots of attention. The bad habit of selling horrible low-quality work as success destroys progress and undermines accomplishing truly high-quality work. We all need to be able to recognize these horrors and strenuously reject them. If we start to effectively police ourselves perhaps this plague can be driven back, and progress can flourish.

The thing about chameleoning your way through life is that it gets to where nothing is real.

― John Green

 

 

A method which can solve this problem well should be able to handle just about anything which can arise in one-dimensional pure hydrodynamic flow. PPM is such a scheme.

– P.R. Woodward

Colella, Phillip, and Paul R. Woodward. “The piecewise parabolic method (PPM) for gas-dynamical simulations.” Journal of computational physics 54, no. 1 (1984): 174-201.

This is one of the most important methods in the early history of the revolutionary developments for solving hyperbolic PDEs in the 1980’s. For a long time this was one of the best methods available to solve the Euler equations. It still outperforms most of the methods in common use today. For astrophysics, it is the method of choice, and also made major inroads to the weather and climate modeling communities. In spite of having over 4000 citations, I can’t help but think that this paper wasn’t as influential as it could have been. This is saying a lot, but I think this is completely true. This partly due to its style, and relative difficulty as a read. In other words, the paper is not as pedagogically effective as it could have been. The most complex and difficult to understand version of the method is presented in the paper. The paper could have used a different approach to great effect by perhaps providing a simplified version to introduce the reader and deliver the more complex approach as a specific instance. Nonetheless, the paper was a massive milestone in the field.

It was certainly clear that high-order schemes were not necessarily bringing greater accuracy so physics would have to step in to shore up the failing numerics.

– Jay Boris

Part of the problem with the paper is the concise and compact introduction to the two methods used in the accompanying review article, PPMLR and PPMDE. The LR stands for Lagrange-Remap where the solution is solved on a Lagrangian grid and then remapped back to the original grid for an utterly Eulerian solution. Both the Lagrangian and Eulerian grids are unevenly spaced, and this results in far more elaborate formulas. As a result it is hard to recognize the simpler core method lurking inside the pages of the paper. The DE stands for direct Eulerian, which can be very simple for the basic discretization. Unfortunately, the complication for the DE flavor of PPM comes with the Riemann solver, which is far more complex in the Eulerian frame. The Largangian frame Riemann solver is very simple and easy to evaluate numerically. Not so for the Eulerian version, which has many special cases and requires some exceedingly complex evaluations of the analytical structure of the Riemann solution. Advances that occurred later greatly simplified and clarified this presentation. This is a specific difficulty of being an early adopter of methods, the clarity of presentation and understanding is dimmed by purely narrative effects. Many of these shortcomings have been addressed in the recent literature discussed below.

The development of the PPM gas dynamics scheme grew out of earlier work in the mid 1970s with Bram van Leer on the MUSCL scheme. The work of Godunov inspired essential aspects of MUSCL.

– Paul R. Woodward

The paper had a host of interesting and important sub-techniques for solving hyperbolic PDEs. Many of these “bells” and “whistles” are not part of the repertoire for most methods today. The field actually suffers from some extent by not adopting most of these strategies for attacking difficult problems. It is useful to list the special approaches along with a description and context that might make them easier to adopt more broadly (https://williamjrider.wordpress.com/2016/06/14/an-essential-foundation-for-progress/, https://williamjrider.wordpress.com/2017/06/30/tricks-of-the-trade-making-a-method-robust/, https://williamjrider.wordpress.com/2016/08/08/the-benefits-of-using-primitive-variables/). The paper is written in such a way that these algorithms seem specifically tailored to PPM, but they are far broader in utility. Generalizing their use more broadly would serve the quality of numerical solutions immensely. To a large extent Phil Colella extended many of these techniques to piecewise linear methods that form the standard approach in production codes today.

• Shock flattening – Shocks are known to be horrifically nonlinear and difficult both forgiving and brutal. This technique acknowledges this issue by blending a bit of safe first order method with the nonlinearly adaptive high-order methods when a strong shock is encountered. The principle is to use a bit more first-order when the shock is strong because oscillations can escape. For weak shocks this is unnecessary. Rather than penalize the solution everywhere the method is made locally more dissipative where the danger is the greatest.
• Contact steepening – contact discontinuities will smear out without limit if dissipation is applied to them. In other words, errors made in their solution are with you forever. To keep this from happening, the amount of dissipation applied at these waves is minimized. This sort of technique must be applied with great caution because at a shock wave this is exceedingly dangerous. Additionally, the method to limit the dissipation can produce a very good interface tracking method that is far simpler than the elaborate methodology using interface geometry. It is a useful pragmatic way to move interfaces with little dissipation along with relative simplicity. This basic approach is the actual interface tracking method in many production codes today although few use methods as elaborate or as high quality as that used in the original PPM.
• Extra dissipation – Monotonicity preservation and Riemann solvers are two elaborate ways of producing dissipation while achieving high quality. For very nonlinear problems this is not enough. The paper describes several ways of adding a little bit more, one of these is the shock flattening, and another is an artificial viscosity. Rather than use the classical Von Neumann-Richtmyer approach (that really is more like the Riemann solver), they add a small amount of viscosity using a technique developed by Lapidus appropriate for conservation form solvers. There are other techniques such as grid-jiggling that only really work with PPMLR and may not have any broader utility. Nonetheless, there may be aspects of the thought process that may be useful.
• High-order edges – One of PPM’s greatest virtues is the use of formally higher order principles in the method. Classic PPM uses fourth-order approximations for its edge values. As a result, as the Courant number goes to zero, the method becomes formally fourth-order accurate. This is a really powerful aspect of the method. It is also one of the clear points where the method can be generalized. We can use whatever high-order edge value we like for PPM. One of the maxims to take from this approach is the power of including very high-order discretizations even with otherwise lower order approximation methods. The impact of the high-order is profoundly positive.
• Steepened edge values – For horrible nonlinear problems, the simple use of high-order differencing is not advisable. The nature of the high-order approximation can be decomposed into several pieces, and the approximation can be built more carefully and appropriately for complex problems. In this way, the high order edge values are a bit hierarchical. This is partially elaboration, but also reflects a commitment to quality that is imminently laudable.

Generalized Monotonicity – PPM uses a parabola and as a result the limiters so well-known don’t work to provide monotone results. As a result, the limiter for PPM takes two steps instead of the single step needed for a linear profile. I don’t like the original presentation in the paper and recast the limiter into an equivalent algorithm that uses two applications of the median function per edge. The first step makes sure the edge value being used is bounded by the cell averages adjacent to it. The second step asks whether the parabola is monotone in the cell and limits it to one that is by construction should it not be (https://williamjrider.wordpress.com/2016/06/07/the-marvelous-magical-median/, https://williamjrider.wordpress.com/2016/06/22/a-path-to-better-limiters/ https://williamjrider.wordpress.com/2015/08/06/a-simple-general-purpose-limiter/, https://williamjrider.wordpress.com/2014/01/11/practical-nonlinear-stability-considerations/, https://williamjrider.wordpress.com/2015/08/07/edge-or-face-values-are-the-path-to-method-variety-and-performance/ ).

Before launching into a systematic description of the PPM algorithm, it is worthwhile to first explain the goals and constraints that have influenced its design. These are:

1. Directional operator splitting.
2. Robustness for problems involving very strong shocks.
3. Contact discontinuity steepening.
4. Fundamental data in the form of cell averages only.
5. Minimal dissipation.
6. Numerical errors nevertheless dominated by dissipation, as opposed to dispersion.
7. Preservation of signals, if possible, even if their shapes are modified, so long as they travel at roughly the right speeds.

8. Minimal degradation of accuracy as the Courant number decreases toward 0.

– Paul R. Woodward

Over time PPM has mostly been interpreted monolithically as opposed to some basic principles. PPM is really a wonderful foundation with the paper only providing a single instantiation of a panoply of powerful methods. This aspect has come to the fore more recently, but would have served the community better far earlier. Some of these comments are the gift of 2020 hindsight. A great deal of the pedagogical clarity with regard to Godunov-type methods is the result of its success, and only came to common use in the late 1980’s, if not the 1990’s. For example, the language to describe Riemann solvers with clarity and refinement hadn’t been developed by 1984. Nevertheless, the monolithic implementation of PPM has been a workhorse method for computational science. Through Paul Woodward’s efforts it is often the first real method to be applied to brand new supercomputers, and generates the first scientific results of note on them.

The paper served as a companion to the adjacent paper that reviewed the performance of numerical methods for strong shocks. The review was as needed as it was controversial. The field of numerical methods for shock waves as set to explode into importance and creative energy. The authors Phil Colella and Paul Woodward would both play key roles in the coming revolution in methods. Woodward had already made a huge difference by spending time in Europe with Bram van Leer. Paul helped Bram with implementing advanced numerical methods using methodologies Paul learned at the Livermore Labs. Bram exposed Paul to his revolutionary ideas about numerical methods chronicled in Bram’s famous series of papers (https://williamjrider.wordpress.com/2014/01/11/designing-new-schemes-based-on-van-leers-ideas/, https://williamjrider.wordpress.com/2014/01/06/van-leers-1977-paper-paper-iv-in-the-quest-for-the-ultimate/, https://williamjrider.wordpress.com/2014/01/05/review-of-the-analysis-of-van-leers-six-schemes/). The ideas therein were immensely influential in changing how hyperbolic equations were solved.

One of the great successes in numerical methods for hyperbolic conservation laws has been the use of nonlinear hybridization techniques, known as limiters, to maintain positivity and monotonicity in the presence of discontinuities and underresolved gradients.

– Michael Sekora and Phil Collela

Bram’s ideas created a genuine successor to Godunov’s method. The methods he created were novel in producing a nonlinearly adaptive numerical method where the method would adapt locally to the nature of the solution. This overcame the limitations of Godunov’s theorem regarding the accuracy of numerical methods for hyperbolic equations. Bram’s ideas were geometric in nature, and reflected the approach of the physicist. Paul being a physicist gravitated into the same view, and added a genuine does of pragmatism. Bram also wasn’t the first person to overcome Godunov’s theorem. He may have actually been the third (or fourth). The first is most likely to have been Jay Boris who invented the flux-corrected transport (FCT) method in 1971. In addition, Kolgan in the Soviet Union and Ami Harten might lay claims to overcoming Godunov’s barrier theorem. Some of these different methods played a role in the comparison in the review article by Woodward and Colella. In the light of history many of the differences in the results were more due to the approaches to systems of equations and related difficulties than the nonlinearly adaptive principles in the methods.

The strong, fluid-dynamic shock problem had become the number one computational roadblock by the fall of 1970 so I was urged to concentrate on the problem full time, finally developing the FCT convection algorithm in the winter.

– Jay Boris

In totality, the methods developed by three or four men in the early 1970’s set the stage for revolutionary gains in method performance. At the time of the developments, the differences in the methods were fiercely debated and hotly contested. The reviews of the papers were contentious and resulted in bitter feelings. Looking back with the virtues of time and perspective several things stand out. All the methods represented a quantum leap in performance, and behavior over the methods available prior.  The competition and ideas so hotly contested probably helped to spur developments, but ultimately became counter-productive as the field matured. It seems clear that the time was ripe for the breakthrough. There was a combination of computers, mathematics and applications that seeded the developments. For the same basic idea to arise independently in a short period of time means the ideas were dangling just out of reach. The foundations for the breakthrough were common and well-known.

Paul Woodward is an astrophysicist, and PPM found its most common and greatest use in his field. For a long time the nature of PPM’s description meant that the exact versions of the method described in the canonical 1984 paper were the exact method used in other codes. Part of this results from PPM being a highly tuned, high-performance method with a delicate balance between high-resolution methodology and various safety measures needed for difficult highly nonlinear problems. In a manner of speaking it is a recipe that produces really great results. Imagine PPM as something akin to the Toll House chocolate chip cookie recipe. The cookies you get by following the package exactly are really, really good. At the same time, you can modify the recipe to produce something even better while staying true to the basic framework. The basic cookies will get you far, but with some modification you might just win contests or simply impress your friends. PPM is just like that.

At this point I’ve said quite little about the method itself. The core of the method is a parabolic representation of the solution locally in a cell. The method is totally one-dimensional in nature. This parabola is determined by the integral average in a cell and the point values of the quantity at the edge of the cell. What is not so widely appreciated is the connection of PPM to the fifth scheme in Van Leer’s 1977 paper. This method is interesting because the method evolves both cell averages like any finite volume code, and the point values at the cell boundary. It is compact and quite supremely accurate compared with other third-order methods. The PPM is a way of getting some of the nice properties of this method from a finite volume scheme. Rather than evolve the point values on the edge, they are recovered from the finite volumes.

Rather than belabor the technical details of PPM, I’ll point to the recent trends that have extended the method beyond its classical form. One of the original authors has used the parabola to represent valid extrema in the solution rather than damping them by forcing monotonicity. I did the same thing in my own work largely paralleling Phil’s work. In addition, the change in the high-order edge reconstruction has been recognized and implemented to good effect by both Phil, Paul, myself and others. The connection of Riemann solvers has also been generalized. All of this reflects the true power of the method when projected onto the vast body of work that arose after the publication of this paper.  Even today PPM remains one of the very best methods in existence especially with the modifications recently introduced.

Personally, I’ve come to know both Phil and Paul personally and professionally. In the numerical solution of hyperbolic PDEs both men have played a significant personal role and witnessed history being made. They helped make CFD what it is today. It’s always an interesting experience to read someone’s work then come to know the person. A big part of a deeper appreciation is finding out the underlying truths of the paper. You start to realize that the written, published record is a poor reflection of the real story. Some of this comes through the hard work of reading and re-reading a paper, then deriving everything in it for yourself. A deeper appreciation came from expressing the same method in my own language and mathematics. Finally taking each of these expressions into conversations with the authors who clarified most of the remaining questions. The academic literature is a scrubbed and largely white-washed reflection of reality. What we are allowed to read and see is not the truth, but an agreed upon distortion.

When the numerics fails, substitute the physics.

– Steve Zalesak

the scientists who use such algorithms must have both input to and knowledge of their design. There may come a day when we no longer hold to this view, when the design of such algorithms can be left to expert numerical analysts alone, but that day has not yet arrived.

– Steve Zalesak

Woodward, Paul, and Phillip Colella. “The numerical simulation of two-dimensional fluid flow with strong shocks.” Journal of computational physics 54, no. 1 (1984): 115-173.

Carpenter Jr, Richard L., Kelvin K. Droegemeier, Paul R. Woodward, and Carl E. Hane. “Application of the piecewise parabolic method (PPM) to meteorological modeling.” Monthly Weather Review 118, no. 3 (1990): 586-612.

Woodward, Paul R. “Piecewise-parabolic methods for astrophysical fluid dynamics.” In Astrophysical Radiation Hydrodynamics, pp. 245-326. Springer Netherlands, 1986.

Godunov, S. K. “A finite difference method for the computation of discontinuous solutions of the equations of fluid dynamics.” Sbornik: Mathematics 47, no. 8-9 (1959): 357-393.

Plewa, Tomasz, and Ewald Mueller. “The consistent multi-fluid advection method.” arXiv preprint astro-ph/9807241 (1998).

Van Leer, Bram. “Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method.” Journal of computational Physics 32, no. 1 (1979): 101-136.

Van Leer, Bram. “Towards the ultimate conservative difference scheme. IV. A new approach to numerical convection.” Journal of computational physics 23, no. 3 (1977): 276-299.

Bell, John B., Phillip Colella, and John A. Trangenstein. “Higher order Godunov methods for general systems of hyperbolic conservation laws.” Journal of Computational Physics 82, no. 2 (1989): 362-397.

Grinstein, Fernando F., Len G. Margolin, and William J. Rider, eds. Implicit large eddy simulation: computing turbulent fluid dynamics. Cambridge university press, 2007.

Rider, William J., Jeffrey A. Greenough, and James R. Kamm. “Accurate monotonicity-and extrema-preserving methods through adaptive nonlinear hybridizations.” Journal of Computational Physics 225, no. 2 (2007): 1827-1848.

Rider, William J. “Reconsidering remap methods.” International Journal for Numerical Methods in Fluids 76, no. 9 (2014): 587-610.

Kolgan, V. P. “Application of the principle of minimum values of the derivative to the construction of finite-difference schemes for calculating discontinuous gasdynamics solutions.” TsAGI, Uchenye Zapiski 3, no. 6 (1972): 68-77.

J. P. Boris “A Fluid Transport Algorithm That Works,” Proceedings of the seminar course on computing as a language of physics, 2-20 August 1971, InternationalCentre for Theoretical Physics, Triest, Italy.