People almost invariably arrive at their beliefs not on the basis of proof but on the basis of what they find attractive.

― Blaise Pascal

When we hear about supercomputing, the media focus, press release is always talking about massive calculations. The bigger is always better with as many zeros as possible with some sort of exotic name for the rate of computation, mega, tera, peta, eta, zeta,… Up and to the right! The implicit proposition is that bigger the calculation, the better the science. This is quite simply complete and utter bullshit. These big calculations providing the media footprint for supercomputing and winning prizes are simply stunts, or more generously technology demonstrations, and not actual science. Scientific computation is a much more involved and thoughtful activity involving lots of different calculations many at a vastly smaller scale. Rarely, if ever, do the massive calculations come as a package including the sorts of evidence science is based upon. Real science has error analysis, uncertainty estimates, and in this sense the massive calculations produce a disservice to computational science by skewing the picture of what science using computers should look like.

This post aims to correct this rather improper vision, and replace it with a discussion of what computational science should be.

With a substantial amount of focus on the drive toward the first exascale supercomputer, it is high time to remind everyone that a single massive calculation is a stunt meant to sell the purchase of said computers, and not science. This week the supercomputing community is meeting in Salt Lake City for a trade show masquerading as a scientific conference. It is simply another in a phalanx of echo chambers we seem to form with increasing regularity across every sector of society. I’m sure the cheerleaders for supercomputing will be crowing about the transformative power of these computers and the boon for science they represent. There will be celebrations of enormous calculations and pronouncements about their scientific value. There is a certain lack of political correctness to the truth about all this; it is mostly pure bullshit.

The entire enterprise pushing toward exascale is primarily a technology push program. It is a furious and futile attempt to stave off the death of Moore’s law. Moore’s law has provided an enormous gain in the power of computers for 50 years and enabled much of the transformative power of computing technology. The key point is that computers and software are just tools; they are incredibly useful tools, but tools nonetheless. Tools allow a human being to extend their own biological capabilities in a myriad of ways. Computers are marvelous at replicating and automating calculations and thought operations at speeds utterly impossible for humans. Everything useful done with these tools is utterly dependent on human beings to devise. My key critique about this approach to computing is the hollowing out of the investigation into devising better ways to use computers and focusing myopically on enhancing the speed of computation.

Truth is only relative to those that ignore hard evidence.

― A.E. Samaan

The core of my assertion that its mostly bullshit comes from looking at the scientific method and its application to these enormous calculations. The scientific method is fundamentally about understanding the World (and using this understanding via engineering). The World is observed either in its natural form, or through experiments devised to unveil difficult to see phenomena. We then produce explanations or theories to describe what we see, and allow us to predict what we haven’t see yet. The degree of comparison between the theory and the observations confirms our degree of understanding. There is always a gap between our theory and our observations, and each is imperfect in its own way. Observations are intrinsically prone to a variety of errors, and theory is always imperfect. The solutions to theoretical models are also imperfect especially when solved via computation. Understanding these imperfections and the nature of the comparisons between theory and observation is essential to a comprehension of the state of our science.

As I’ve stated before, the scientific method applied to scientific computing is embedded in the practice of verification and validation. Simply stated, a single massive calculation cannot be verified or validated (it could be, but not with current computational techniques and the development of such capability is a worthy research endeavor). The uncertainties in the solution and the model cannot be unveiled in a single calculation, and the comparison with observations cannot be put into a quantitative context. The proponents of our current approach to computing want you to believe that massive calculations have intrinsic scientific value. Why? Because they are so big, they have to be the truth. The problem with this thinking is that any single calculation does not contain steps necessary for determining the quality of the calculation, or putting any model comparison in context.

The context of any given calculation is determined by the structure of the errors associated with the computational modeling. For example it is important to understand the nature of any numerical errors, and producing an estimate of these errors. In some (many, most) cases a very good comparison between reality and a model is the result of calibration of uncertain model parameters. In many cases the choices for the modeling parameters are mesh dependent, which produces the uncomfortable outcome where a finer mesh produces a systematically worse comparison. This state of affairs is incredibly common, and generally an unadvertised feature.

An important meta-feature of the computing dialog is the skewing of computer size, design and abilities. For example, the term capability computer comes up where these computers can produce the largest calculations we see, the ones on press releases. These computers are generally the focus of all the attention and cost the most money. The dirty secret is that they are almost completely useless for science and engineering. They are technology demonstrations and little else. They do almost nothing of value to the myriad of programs reporting to use computations to do produce results. All of the utility to actual science and engineering come from the homely cousins of these supercomputers, the capacity computers. These computers are the workhorses of science and engineering because they are set up to do something useful. The capability computers are just show ponies, and perfect exemplars of the modern bullshit based science economy. I’m not OK with this; I’m here to do science and engineering. Are our so-called leaders OK with the focus of attention (and bulk of funding) being non-scientific, media-based, press release generators?

How would we do a better job with science and high performance computing?

The starting point is the full embrace of the scientific method. Taken at face value the observational or experimental community is expected to provide observational uncertainties with their data. These uncertainties should be de-convolved between errors/uncertainties in raw measurement and any variability in the phenomena. Those of us using such measurements for validating codes should demand that observations always come with these uncertainties. By the same token, computational simulations have uncertainties from a variety numerical errors and modeling choices and assumptions that should be demanded. Each of these error sources needs to be characterized to put any comparison with observations/experimental data into context. Without knowledge of these uncertainties on both sides of the scientific process, any comparison is completely untethered.

If nothing else, the uncertainty in any aspect of this process provides a degree of confidence and impact of comparative differences. If a comparison between a model and data is poor, but the data has large uncertainties, the comparison suddenly becomes more palatable. On the other hand small uncertainties with the data would imply that the model is potentially too incorrect. This conclusion would be made once the modeling uncertainty has been explored. One reasonable case would be the identification of large numerical errors in the model’s solution. This is the case where a refined calculation might be genuinely justified. If the bias with a coarse grid is sufficient, a finer grid calculation could be a reasonable way of getting more agreement. There are certainly cases where exascale computing is enabling for model solutions with small enough error to make models useful. This case is rarely made or justified in any massive calculation rather being asserted by authority.

On the other hand numerical error could be a small contributor to the disagreement. In this case, which is incredibly common, a finer mesh does little to rectify model error or uncertainty. The lack of quality comparison is dominated by modeling error, or uncertainty about the parameterization of the models. Worse yet, the models are poor representations of the physics of interest. If the model is a poor representation solving it very accurately is a genuinely wasteful exercise, at least if your goal is scientific in nature. If you’re interested in colorful graphics and a marketing exercise, computer power is your friend, but don’t confuse this with science (or at least good science). The worst case of this issue is a dominant model form error. This is the case where the model is simply wrong, and incapable of reproducing the data. Today many examples exist where models we know are wrong are beat to death with a supercomputer. This does little to advance science, which needs to work at producing a new model that ameliorates the deficiencies in the old model. Unfortunately our supercomputing programs are sapping the vitality from our modeling programs. Even worse, many people seem to confuse computing power as a remedy to model form error.

Equidistributed error is probably the best goal of modeling and simulation that is a balance of numerical and modeling error/uncertainty. This would be the case where the combination of modeling error and uncertainty with a numerical solution has the smallest value. The standard exascale computing driven model would have the numerical error driven to be nearly zero without regard for the modeling error. This ends up being a small numerical error by fiat or proof by authority, proof by overwhelming power. Practically, this is foolhardy and technically indefensible. The issue is the inability to effectively hunt down modeling uncertainties under these conditions, which is hamstrung by the massive calculations. The most common practice is to assess the modeling uncertainty via some sort of sampling approach. This requires many calculations because of the high-dimensional nature of the problem. Sampling converges very slowly with any mean value for the modeling being proportional to the inverse square root of the number of samples and the measure of the variance of the solution.

Thus a single calculation will have an undefined variance. With a single massive calculation you have no knowledge of the uncertainty either modeling or numerical (at least without have some sort of embedded uncertainty methodology). Without assessing the uncertainty of the calculation you don’t have a scientific or engineering activity. For driving down the inherent uncertainties especially where the modeling uncertainty dominates, you are aided by smaller calculations that can be executed over and over as to drive down the uncertainty. These calculations are always done on capacity computers and never on capability computers. In fact if you try to use a capability computer to do one of these studies, you will be punished and get kicked off. In other words the rules of use enforced via the queuing policies are anti-scientific.

The uncertainty structure can be approached at a high level, but to truly get to the bottom of the issue requires some technical depth. For example numerical error has many potential sources: discretization error (space, time, energy, … whatever we approximate in), linear algebra error, nonlinear solver error, round-off error, solution regularity and smoothness. Many classes of problems are not well posed and admit multiple physically valid solutions. In this case the whole concept of convergence under mesh refinement needs overhauling. Recently the concept of measure-valued (statistical) solutions has entered the fray. These are taxing on computer resources in the same manner as sampling approaches to uncertainty. Each of these sources requires specific and focused approaches to their estimation along with requisite fidelity.

Modeling uncertainty is similarly complex and elaborate. The hardest aspect to evaluate is the form of the physical model. In cases where multiple reasonable models exist, the issue is evaluating the model’s (or sub-model’s) influence on solutions. Models often have adjustable parameters that are unknown or subject to calibration. Most commonly the impact of these parameters and their values are investigated via sampling solutions, an expensive prospect. Similarly there are modeling issues that are purely random, or statistical in nature. The solution to the problem is simply not determinate. Again sampling the solution of a range of parameters that define such randomness is a common approach. All this sampling is very expensive and very difficult to accurately compute. All of our focus on exascale does little to enable good outcomes.

The last area of error is the experimental or observational error and uncertainty. This is important in defining the relative quality of modeling, and the sense and sensibility of using massive computing resources to solve models. We have several standard components in the structure of the error in experiments: the error in measuring a quantity, and then the variation in the actual measured quantity. In one case there is some intrinsic uncertainty in being able to measure something with complete precision. The second part of this is the variation of the actual value in the experiment. Turbulence is the archetype of this sort of phenomena. This uncertainty is intrinsically statistical, and the decomposition is essential to truly understand the nature of the world, and put modeling in proper and useful context.

The bottom line is that science and engineering is evidence. To do things correctly you need to operate on an evidentiary basis. More often than not, high performance computing avoids this key scientific approach. Instead we see the basic decision-making operating via assumption. The assumption is that a bigger, more expensive calculation is always better and always serves the scientific interest. This view is as common as it is naïve. There are many and perhaps most cases where the greatest service of science is many smaller calculations. This hinges upon the overall structure of uncertainty in the simulations and whether it is dominated by approximation error, modeling form or lack of knowledge, and even the observational quality available. These matters are subtle and complex, and we all know that today neither subtle, nor complex sells.

What can be asserted without evidence can also be dismissed without evidence.

― Christopher Hitchens

There is nothing more frightful than ignorance in action.

Our political climate and capability as a nation to engage each other in meaningful, respectful conversations has plummeted to dismal lows. The best description of our 2016 political campaign is a “rolling dumpster fire.” At the core of all of our dysfunction is a critical break from fact-based discussion, confronting objective reality and the ascendency of emotion and spin into the fact-vacuum and alternative reality. One might think that working at a scientific-engineering Laboratory would free me from this appalling trend, but the same dynamic is acutely felt there too. The elements undermining facts and reality in our public life are infesting my work. Many institutions are failing society and contributing to the slow-motion disaster we have seen unfolding. We need to face this issue head-on and rebuild our important institutions and restore our functioning society, democracy and governance.

A big part of the public divorce from facts is the lack of respect and admiration for expertise. Just as the experts and the elite have become suspicious and suspect in the broader public sphere, the same thing has happened in the conduct of science. In many ways the undermining of expertise in science is even worse and more corrosive. Increasingly, there is no tolerance or space for the intrusion of expertise into the conduct of scientific or engineering work. The way this tolerance manifests itself is subtle and poisonous. Expertise is tolerated and welcomed as long as it is confirmatory and positive. Expertise is not allowed to offer strong criticism or the slightest rebuke without regard for the shoddiness of work. If an expert does offer anything that seems critical or negative they can expect to be dismissed and never invited back to provide feedback again. Rather than welcome their service and attention, they are derided as troublemakers and malcontents. We see in every corner of the scientific and technical World a steady intrusion of mediocrity and outright bullshit into our discourse as a result.

Let’s give an example of how this plays out. I’ve seen this happen personally and witnessed it play out via external reviews I observed. I’ve been brought in

to to review technical work for a large important project. The expected outcome was a “rubber stamp” that said the work was excellent, and offered no serious objections. Basically the management wanted me to sign off on the work as being awesome. Instead, I found a number of profound weaknesses in the work, and pointed these out along with some suggested corrective actions. These observations were dismissed and never addressed by the team conducting the work. It became perfectly clear that no such critical feedback was welcome and I wouldn’t be invited back. Worse yet, I was punished for my trouble. I was sent a very clear and unequivocal message, “don’t ever be critical of our work.” 

This personal example of dysfunction is simply the tip of the iceberg for an adversarial attitude toward critical feedback. . We have external review committees visit and treated the same. Most seasoned reviewers know that this is not to be a critical review. It is a light touch and everyone expects to get a glowing report. Any real issues are addressed on the down low and even that is treated with kid gloves. If any reviewer has the audacity to raise an important issue they can expect not to be ever invited back. The end result is the increasingly meaningless nature of any review, and the hollowing out of expertise’s seal of approval. In the process experts and expertise become covered in the bullshit they pedal and become diminished in the end.

This dynamic in review is widespread and fuels the rise of bullshit in public life as well as science and engineering. This propensity to bullshit is driven by a system that cannot deal with conflict or critical feedback. Moreover the system is tilted toward a preconceived result, all is well and no changes are necessary. When this is not the case one is confronted with engaging in conflict against these expectations, or simply getting in line with the bullshit. More and more the bullshit is winning the day. I’ve been personally punished for not towing the line and making a stink. I’ve seen others punished too. It is very clear that failing to provide the desired result bullshit will be punished. The punishments for honesty means that bullshit is on the rise as nothing exists to produce a drive toward quality and results. In the end bullshit is a lot less effort and rewarded a lot more highly.

At the end of the day we can see that the system starts to seriously erode integrity at every level. This is exactly what we are witnessing society-wide. Institutions across the spectrum of public and private life are losing their integrity. Such erosion of integrity in an environment that cannot deal with critical feedback produces a negative loop that feeds upon itself. Bullshit begets more bullshit until the whole thing collapses. We may have just witnessed what the collapse of our political system looks like. We had an election that was almost completely bullshit start to finish. We have elected a completely and utterly incompetent bullshit artist president. Donald Trump was completely unfit to hold office, but he is a consummate con man and bullshit artist. In a sense he is the emblem of the age and the perfect exemplar of our addiction to bullshit over substance.

I personally see myself as a person of substance and integrity. It is increasingly difficult to square who I am with the system I am embedded in. I am not a bullshitter, when I produce bullshit people notice, and I am embarrassed. I am a straight shooter who is committed to progress and excellence. I have a broad set of expertise in science and engineering with a deep desire to contribute to meaningful things. This fundamental nature is increasingly at odds with how the World operates today. I feel a deep drive on the part of the workplace to squash everything positive I stand for. In the place of standing up for my basic nature as a scientific expert, a member of the elite, if you will, I am expected to tow the line and produce bullshit. This bullshit is there to avoid dealing with real issues head on and avoid conflict. The very nature of things stands in opposition to progress and quality, which are threatened by the current milieu.

This gets to the heart of the discussion about what we are losing in this dynamic. We are losing progress society wide. When we allow bullshit to creep into every judgment we make, progress is sacrificed. We bury immediate conflict for long-term decline and plant the seeds for far more deep, widespread and damaging conflict. Such horrible conflict may be unfolding right in front of us in the nature of the political process. By finding our problems and being critical we identify where progress can be made, where work can be done to make the World better. By bullshitting our way through things, the problems persist and fester and progress is sacrificed.

In the current environment where expertise is suspect we see wrong beliefs persist without any real resistance. Falsehoods and myths stand shoulder to shoulder with trust and get treated with equivalence. In this atmosphere the sort of political movements founded completely on absolute bullshit can thrive. Make no mistake, Donald Trump is a master bullshitter, and completely lacks all substance, yet in today’s World he has complete viability. All of us are responsible because we have allowed bullshit to stand on even footing with fact. We have allowed the mechanisms and institutions standing in the way of such bullshit to be weakened and infested with bullshit too. It is time to stand up for truth, integrity and expertise as a shield against this assault against society.

Everything present in the political rise of Donald Trump is playing out in the dynamic at my workplace. It is not as extreme and its presence is subtle, but it is there. We have allowed bullshit to become ubiquitous and accepted. We turn away from calling bullshit out and demanding that real integrity be applied to our work. In the process we implicitly aid and abed the forces in society undermining progress toward a better future. The result of this acceptance of bullshit can be seen in the reduced production of innovation, and breakthrough work, but most acutely in the decay of these institutions.

We have lost the ability to demand difficult decisions to solve seemingly intractable problems. When we do not operate on facts, we can turn away from difficulties and soothe ourselves with falsehoods. Instead of identifying problems and working toward progressive solutions, the problems are minimized and allowed to fester. This is true in the broader public sphere as well as in our scientific environment. I have been actively discouraged from pointing out problems or being critical. The result is stagnation and the steady persistence of problematic states. Instead of working to solve weaknesses, we are urged to accept them or explain them away. This will ultimately yield a catastrophic outcome. At the National level we may have just witnessed such a catastrophe play out in plain view.

In the workplace I feel the key question to ask is “If we don’t look for problems, how can we do important work?” Progress depends on finding weakness and attacking it. This is the principle that I focus on. Confidence comes from being sure you know where to look for problems and up to the challenge of solving them. Empty positivity is a sign of weakness. Yet this is exactly what I am being asked to do at work. The resulting bullshit is a sign of weakness and lack of confidence is being able to constructively solve problems. The need to be positive all the time and avoid criticism is weakness, lack of drive, and lack of conviction in the possibility of progress. We need to refresh out commitment to be constructively critical in the knowledge and belief that we are equal to the task of making the World better. This means stamping out bullshit wherever we see it. There is a lot to do because today we are drowning in it.

With the benefit of time i have a couple projections for the future:

1. The GOP and President Trump will do little or nothing to help the people that voted for them. The key to our democracy is whether they will take any responsibility. If history is our guide they will deflect the blame onto minorities, LBGT, women and
   

   U.S. Republican presidential candidate Donald Trump speaks at a rally in Columbus, Ohio, November 23, 2015. REUTERS/Jay LaPrete – RTX1VIY0

   everyone, but themselves. Will the people fall for the same con as they did when they elected these charlatans?

2. Things will be very dark and dismal for an extended time, and we will spiral toward violence. This may be violence directed by the new ruling class against “enemies of the state”. It also may be violence directed toward the ruling class. Mark my words blood will be shed by Americans at the hands of other Americans.
3. The only way out of this darkness is to work steadfastly to repair our institutions and figure out how to solve our problems in a collective manner for the benefit of all. I work for one of these institutions and we should be taking a long hard look at our role in the great unraveling we are in the midst of.

Facts are stubborn things; and whatever may be our wishes, our inclinations, or the dictates of our passion, they cannot alter the state of facts and evidence.

― John Adams

Footnote: I started writing this on Monday, and like almost everyone I thought the election would turn out differently. It was a genuinely shocking result that makes this topic all the more timely. Instead the results amplified the importance of this entire discussion immensely. The prospect of a President Trump fills me with dread because of the very issues discussed here. Trump exists in an alternative reality and his lack of presence in an objective reality will have real consequences. He is a reality TV star and professional buffoon. He is the most stunningly unqualified person to ever hold that office. I fear what is coming. I also feel the need to be resolved to pick up the pieces from the disaster that will likely unfold. We need to rebuild our institutions and reinstitute a knowledge/facts/reality based governance to guide society forward.

In the past quarter century the role of software in science has made a huge change in importance. I work in a computer research organization that employs many applied mathematicians. One would think that we have a little maelstrom of mathematical thought. Very little actual mathematics takes place with most of them writing software as their prime activity. A great deal of emphasis is placed on software as something to be preserved or invested in. This dynamic places a great deal of other forms of work on the backburner like mathematics (or modeling or algorithmic-methods investigation). The proper question to think about is whether the emphasis on software along with collateral decreases in focus on mathematics or physical modeling is a benefit to the conduct of science.

Doing mathematics should always mean finding patterns and crafting beautiful and meaningful explanations.

― Paul Lockhart

I’ll focus on my wife’s favorite question, “what is code?”(I put this up on a slide when she was in the audience, and she rolled her eyes at me and walked out). If we understand what exactly code is we can answer the question of whether it can be preserved and whether it is worthwhile to do.

The simplest answer to the question at hand is that code is a set of instructions that a computer can understand that provides a recipe provided by humans for conducting some calculations. These instructions could integrate a function, or a differential equations, sort some data out, filter an image, or millions of other things. In every case the instructions are devised by humans to do something, and carried out by a computer with greater automation and speed than humans can possibly manage. Without the guidance of humans, the computer is utterly useless, but with human guidance it is a transformative tool. We see modern society completely reshaped by the computer. Too often the focus of humans is on the tool and not the things that give it power, skillful human instructions devised by creative intellects. Dangerously, science is falling into this trap, and the misunderstanding of the true dynamic may have disastrous consequences for the state of progress. We must keep in mind the nature of computing and man’s key role in its utility.

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 manner of treating applied mathematics today serves as an instructive lesson in how out of balance the dynamic is today. Among the sciences mathematics may be the most purely thoughtful endeavor. Some have quipped that mathematics is the most cost efficient discipline, requiring nothing more than time, pen and paper. Often massive progress happens without pen and paper whole the mathematical mind ponders and schemes about theorem, proof and conceptual breakthroughs. Increasingly this idealized model is foreign to mathematicians and the desire for a more concrete product has taken hold. This is most keenly seen in the drive for software as a tangible end product.

Nothing is remotely wrong with creating working software to demonstrate a mathematical concept. Often mathematics is empowered by the tangible demonstration of the utility of the ideas expressed in code. The problem occurs when the code becomes the central activity and mathematics is subdued in priority. Increasingly, the essential aspects of mathematics are absent from the demands of the research being replaced by software. This software is viewed as an investment that must be transferred along to new generations of computers. The issue is that the porting of libraries of mathematical code has become the raison d’etre for research. This porting has swallowed innovation in mathematical ideas whole, and the balance in research is desperately lacking.

Instead of focusing on being mathematicians, we increasingly see software engineering and programming as the focal point for people’s work. Software engineering and maintenance of complex software is a worthy endeavor (more later), but our talented mathematicians should be discovering math, not porting code and finding bugs as their principle professional focus. The discovery of deep, innovative and exciting mathematics promises to provide far more benefit to the future of computing than any software instantiation. New mathematical ideas if focused upon and delivered will ultimately unleash far greater benefits in the long run. This is an obvious thing, yet focus is entirely away from this model. We are steadfastly turning our mathematicians into software engineers.

Let’s get to the crux of the problem with current thinking about software. Mathematical software is like a basic plumbing of lots of codes used for scientific activities, but this model is deeply flawed. It is not like infrastructure at all where the code would be repaired and services after it is built. This leads to the current maintainers of the code to not innovate or extend the intellectual ideas in software, which I would contend is necessary to intellectually own the software. Instead a mathematical body of code is more like an automobile. The auto must be fueled and services, but over time becomes old and outdated needing to be replaced. The classic car has a certain luster and beauty, but its efficiency and utility is far less than a new car. Any automobile can take you places, but eventually the old car cannot compete with the new car. This is how we should think about our mathematical software. It should be serviced and maintained by software professionals, but mathematicians should be working on a new model all the time.

For so much of what we do with computers mathematics forms the core and foundation of the capability. The lack of focus on the actual execution of mathematical research will have long lasting effects on our future. In essence we are living on the mathematical (and physics, engineering, …) research of the past without reinvesting in the next generation of breakthroughs. We are emptying the pipeline of discovery and impoverishing our future. In addition we are failing to take advantage of the skills, talents and imagination of the current generation of scientists. We are creating a deficit of possibility that will harm our future in ways we can scarcely imagine. The guilt lies in the failure of our leaders to have sufficient faith in the power of human thought and innovation to continue to march forward into the future in the manner we have in the
past. People if turned loose on challenging problems will solve them; we always have and past is prolog.

Progress is possible only if we train ourselves to think about programs without thinking of them as pieces of executable code.

― Edsger W. Dijkstra

The key to this notion is putting software in its proper place. Just as a computer itself, software is a tool. Software is an expression of intellect plain and simple. If the intellectual capital isn’t present the value of the software is diminished. Intellectual ownership is a big deal and the key to real value. Increasingly we are creating software where no one working on really owns the knowledge encoded. This is a massively dangerous trend. Unfortunately we are not funding the basic process where the ownership is obtained. Full ownership is established through the creative process, the ability to innovate and create new knowledge grants ownership. Without the creation of new knowledge the intellectual ownership is incomplete. An additional benefit of the ownership is new capability for mankind. The foundation of all of this is mathematical research.

Our foundation is crumbling beneath our feet from abject neglect. Again, like everything else today, the reason for this is a focus on money as the arbitrator of all that is good or bad. We simply do what we are paid to do, no more and no less. No one is paying for math, they are paying for software, it’s as simple as that.

Programs must be written for people to read, and only incidentally for machines to execute.

― Harold Abelson

Progress, productivity, quality, independence, value, … Compliance kills almost everything I prize about work as a scientist. Compliance is basically enslavement to mediocrity and subservience to authority unworthy of being followed.

In the republic of mediocrity, genius is dangerous.

― Robert G. Ingersoll

Earlier this week I had an interesting exchange on Twitter with my friend Karen and a current co-worker Si. It centered around the fond memories that Karen and I have about working at Los Alamos. The gist of the conversation was that the Los Alamos we worked at was wonderful, even awesome. To me the experience at Los Alamos from 1989-1999 was priceless and the result of an impressively generous and technically masterful organization. I noted that it isn’t the way it was and that fact is absolutely tragic. Si countered that it’s still full of good people that are great to interact with. All of this can be true and not the slightest bit contradictory. Si tends to be positive all the time, which can be a wonderful characteristic, but I know what Los Alamos used to mean, and it causes me a great deal of personal pain to see the magnitude of the decline and damage we have done to it. The changes at Los Alamos have been done in the name of compliance, to bring an unruly institution to heel and conform to imposed mediocrity.

How the hell did we come to this point?

In relative terms, Los Alamos is still a good place largely because of the echos of the same culture that Karen and I so greatly benefited from. Organizational culture is a deep well to draw from. It shapes so much of what we see from different institutions. At Los Alamos it has formed the underlying resistance to the imposition of the modern compliance culture. On the other hand, my current institution is tailor made to complete compliance, even subservience to the demands of our masters. When those masters have no interest in progress, quality, or productivity, the result in unremitting mediocrity. This is the core of the discussion, our master’s prime directive is compliance, which bluntly and specifically means “don’t ever fuck up!” In this context Los Alamos is the king of the fuck-ups, and others simply keep places nose clean thus succeeding in the eyes of the masters..

The second half of the argument comes down to recognizing that accomplishment and productivity is never a priority in the modern world. This is especially true once the institutions realized that they could bullshit their way through accomplishment without risking the core value of compliance. Thus doing anything real and difficult is detrimental because you can more easily BS your way to excellence and not run the risk of violating the demands of compliance. In large part compliance assures the most precious commodity in the modern research institution, funding. Lack of compliance is punished by lack of funding. Our chains are created out of money.

In the end they will lay their freedom at our feet and say to us, Make us your slaves, but feed us.

― Fyodor Dostoyevsky

A large part of the compliance is lack of resistance to intellectually poor programs. There was once a time when the Labs helped craft the programs that fund them. With each passing year this dynamic breaks down, and the intellectual core of crafting well-defined programs to accomplish important National goals wanes. Why engage in the hard work of providing feedback when it threatens the flow of money? Increasingly the only sign of success is the aggregate dollar figure flowing into a given institution or organization. Any actual quality or accomplishment is merely coincidental. Why focus on excellence or quality when it is so much easier to simply generate a press release that looks good.

We make our discoveries through our mistakes: we watch one another’s success: and where there is freedom to experiment there is hope to improve.

― Arthur Quiller-Couch

This entire compliance dynamic is at the core of so many aspects dragging us into the mire of mediocrity. Instead of working to produce a dynamic focused on excellence, progress and impact, we simply focus on following rules and bullshitting something that resembles an expected product. Managing a top rate scientific or engineering institution is difficult and requires tremendous focus on the things that matter. Every bit of our current focus is driving us away from the elements of success. Our masters are incapable
of supporting hard-nosed, critical peer reviews, allowing failure to positively arise from earnest efforts, empowering people to think independently, and rewarding efforts essential for progress. At the heart of everything is an environment that revolves around
fear, and control. We have this faulty belief that we can manage everything to avoid any bad things ever happening. In the end the only way to do this is stop all progress and make sure no one ever accomplishes anything substantial.

So in the end make sure you get those TPS reports in on time. That’s all that really matters.

Disobedience is the true foundation of liberty. The obedient must be slaves.

― Henry David Thoreau

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

Over the past few decades there has been a lot of sturm and drang around the prospect that computation changed science in some fundamental way. The proposition was that computation formed a new way of conducting scientific work to compliment theory, experiment/observation. In essence computation had become the third way for science. I don’t think this proposition stands the test of time and should be rejected. A more proper way to view computation is as a new tool that aids scientists. Traditional computational science is primarily a means of investigating theoretical models of the universe in ways that classical mathematics could not. Today this role is expanding to include augmentation of data acquisition, analysis, and exploration well beyond the capabilities of unaided humans. Computers make for better science, but recognizing that it does change science at all is important to make good decisions.

The key to my rejection of the premise that computation is a close examination of what science is. Science is a systematic endeavor to understand and organize knowledge of the universe in a testable framework. Standard computation is conducted in a systematic manner to conduct studies of the solution to theoretical equations, but the solutions always depend entirely on the theory. Computation also provides more general ways of testing theory and making predictions well beyond the approaches available prior to computation. Computation frees us of limitations for solving the equations comprising the theory, but nothing about the fundamental dynamic in play. The key point is that utilizing computation is as an enhanced tool set to conduct science in an otherwise standard way.

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

Why is this discussion worth having now?

Some of the best arguments for the current obsession with exascale computing are couched in advertising computing as a new way of doing science that somehow is game changing. It just isn’t a game changer; computation is an incredible tool that opens new options for progress. Looking at computing, as simply a really powerful tool that enhances standard science just doesn’t sound as good or as compelling for generating money. The problem is that computing is just that, a really useful and powerful tool, and little more. The proper context for computing carries with it important conclusions about how it should be used, and how it should not be used, neither is evident in today’s common rhetoric. As with any tool, computation must be used correctly to yield its full benefits.

This correct use and full benefit is the rub with current computing programs. The current programs focus almost no energy on doing computing correctly. None. They treat computing as a good unto itself rather than treating it as a deep, skillful endeavor that must be completely entrained within the broader scientific themes. Ultimately science is about knowledge and understanding of the World. This can only come from two places: the observation of reality, and theories to explain those observations. We judge theory by how well it predicts what we observe. Computation only serves as a vehicle for more effectively apply theoretical models and/or wrangling our observations practically. Models are still the wellspring of human thought. Computation does little to free us from the necessity for progress to be based on human creativity and inspiration.

Observations still require human ingenuity and innovation to be achieved. This can take the form of the mere inspiration of measuring or observing a certain factor in the World. Another form is the development of measurement devices that allow measurements. Here is a place where computation is playing a greater and greater role. In many cases computation allows the management of mountains of data that are unthinkably large by former standards. Another way of changing data that is either complementary or completely different is analysis. New methods are available to enhance diagnostics or see effects that were previously hidden or invisible. In essence the ability to drag signal from noise and make the unseeable, clear and crisp. All of these uses are profoundly important to science, but it is science that still operates as it did before. We just have better tools to apply to its conduct.

One of the big ways for computation to reflect the proper structure of science is verification and validation (V&V). In a nutshell V&V is the classical scientific method applied to computational modeling and simulation in a structured, disciplined manner. The high performance computing programs being rolled out today ignore verification and validation almost entirely. Science is supposed to arrive via computation as if by magic. If it is present it is an afterthought. The deeper and more pernicious danger is the belief by many that modeling and simulation can produce data of equal (or even greater) validity than nature itself. This is not a recipe for progress, but rather a recipe for disaster. We are priming ourselves for believing some rather dangerous fictions.

This is a healthy attitude expressed by Einstein. Replace theory with computation and ask the same question then inquire whether our attitudes toward models and simulation are equally healthy?

You make experiments and I make theories. Do you know the difference? A theory is something nobody believes, except the person who made it. An experiment is something everybody believes, except the person who made it.

― Albert Einstein

The archetype of this thought process is direct numerical simulation (DNS). DNS is most prominently associated with turbulence, but the mindset presents itself in many fields. The logic behind DNS is the following: if we solve the governing equations without any modeling in a very accurate manner, the solutions are essentially exact. These very accurate and detailed solutions are just as good as measurements of nature. Some would contend that the data from DNS is better because it doesn’t have any measurement error. Many modelers are eager to use DNS data to validate their models, and eagerly await more powerful computers to expand the grasp of DNS to more complex situations. This entire mindset is unscientific and prone to the creation of bullshit. A big part of the problem is a lack of V&V with DNS, but the core issue is deeper. The belief that the equations are exact, not simply accepted models from currently accepted theory.

Let me explain why I would condemn such a potentially useful and powerful activity so strongly. The problem with DNS used in this manner is that it does include a model of reality. Of course this ignores the fact that the equations themselves are a model of reality. The argument behind DNS is that the equations being solved are unquestioned. This lack of questioning is itself unscientific on the face of it, but let me go on. Others will argue that the equations being solved have been formally validated, thus their validity for modeling reality established. Again, this has some truth to it, but the validation is invariably for quantities that may be observed directly, and generally statistically. In this sense the data being used from DNS is validated by inference, but not directly. Using such unvalidated data for modeling is dangerous (it may be useful too, but needs to be taken with a big grain of salt). The use of DNS data needs to exercise caution and be applied in a circumspect manner, not in evidence today.

Perhaps one of the greatest issues with the application of DNS is its failure to utilize V&V systematically. The first leap of faith with DNS believes that no modeling is happening. The equations being solved are not exact, but rather models of reality. Next the error associated with the numerical integration of the equations is rarely (to never) quantified simply assumed to be negligibly small. Even if we were to accept DNS as equivalent to experimental data, the error needs to be defined as part of the data set (in essence the error bar). Other uncertainties almost required for any experimental dataset are also lacking with DNS. The treatment of data from DNS should be higher than any experimental data reflecting the caution such artificial information should be used with. Instead, the DNS computations are treated with less caution. In this way standard practice today veers all the way into cavalier.

The deepest issue with current programs pushing forward on the computing hardware is their balance. The practice of scientific computing requires the interaction and application of great swathes of scientific disciplines. Computing hardware is a small component in the overall scientific enterprise and among the aspect least responsible for the success. The single greatest element in the success of scientific computing is the nature of the models being solved. Nothing else we can focus on has anywhere close to this impact. To put this differently, if a model is incorrect no amount of computer speed, mesh resolution or numerical accuracy can rescue the solution. This is the statement of how scientific theory applies to computation. Even if the model is unyieldingly correct, then the method and approach to solving the model is the next largest aspect in terms of impact. The damning thing about exascale computing is the utter lack of emphasis on either of these activities. Moreover without the application of V&V in a structured, rigorous and systematic manner, these shortcomings will remain unexposed.

In summary, we are left to draw a couple of big conclusions: computation is not a new way to do science, but rather an enabling tool for doing standard science better. If we want to get the most out of computing requires a deep and balanced portfolio of scientific activities. The current drive for performance with computing hardware ignores the most important aspects of the portfolio, if science is indeed the objective. If we want to get the most science out of computation, a vigorous V&V program is one way to inject the scientific method into the work. V&V is the scientific method and gaps in V&V reflect gaps in scientific credibility. Simply recognizing how scientific progress occurs and following that recipe can achieve a similar effect. The lack of scientific vitality in current computing programs is utterly damning.

A computer lets you make more mistakes faster than any other invention with the possible exceptions of handguns and Tequila.

― Mitch Ratcliffe

The problem with incompetence is its inability to recognize itself.

― Orrin Woodward

My wife has a very distinct preference in late night TV shows. First, the show cannot be on late night TV, she is fast asleep by 9:30 most nights. Secondly, she is quite loyal. More than twenty years ago she was essentially forced to watch late night TV while breastfeeding our newborn daughter. Conan O’Brien kept her laughing and smiling through many late night feedings. He isn’t the best late night host, but he is almost certainly the silliest. His shtick is simply stupid with a certain sophisticated spin. One of the dumb bits on his current show is “Why China is kicking our ass”. It features Americans doing all sorts of thoughtless and idiotic things on video with the premise being that our stupidity is the root of any loss of American hegemony. As sad as this might inherently be, the principle is rather broadly applicable and generally right on the money. The loss of preeminence nationally is more due to shear hubris; manifest overconfidence and sprawling incompetence on the part of Americans than anything being done by our competitors.

The conventional view serves to protect us from the painful job of thinking.

― John Kenneth Galbraith

High performance computing is no different. By our chosen set of metrics, we are losing to the Chinese rather badly through a series of self-inflicted wounds instead of superior Chinese execution. Nonetheless, we are basically handing the crown of international achievement to them because we have become so incredibly incompetent at intellectual endeavors. Today, I’m going to unveil how we have thoughtlessly and idiotically run our high performance computing programs in a manner that undermines our success. My key point is that stopping the self-inflicted damage is the first step toward success. One must take careful note that the measure of superiority is based on a benchmark that has no practical value. Having metric of success with no practical value is a large part of the underlying problem.

Never attribute to malevolence what is merely due to incompetence

― Arthur C. Clarke

As a starting point I’ll state that current program kicking off, the Exascale Computing Project is a prime example of how we are completely screwing things up. It is basically a lexicon of ignorance and anti-intellectual thought paving the way to international mediocrity. The biggest issue is the lack of intellectual depth in the whole basis of the program, “The USA must have the faster computer”. The fastest computer does not mean anything unless we know how to use it. The fastest computer does not matter if it is fastest at doing meaningless things, or it isn’t fast doing things that are important. The fastest computer is simply a tool in a much larger “ecosystem” of computing. This fastest computer is the modern day equivalent of the “missile gap” from the cold war, which ended up being nothing but a political vehicle.

If part of this ecosystem is unhealthy, the power of the tool is undermined. The extent to which it is undermined should be a matter of vigorous debate. This current program is inadvertently designed to further unbalance an ecosystem that has been under duress for decades. We have been focused on computer hardware for the past quarter of a century while failing to invest in physics, engineering, modeling and mathematics all essential to the utility of the tool of computing. We have starved innovation in the use of computing and the set of most impactful aspects of the computing ecosystem. The result is an intellectually hollow and superficial program that will be a relatively poor investment in terms of benefit to society for dollar spent. In essence the soul of computing is being lost. Our quest for exascale computing belies a program that is utterly and unremittingly hardware focused. This hardware focus is myopic in the extreme and starves the ecosystem of major elements of its health. These elements are the tie to experiments, modeling, numerical methods and solution algorithms. The key to Chinese superiority, or not is whether they are making the same mistakes as we are making. If they are, their “victory” is hollow; if they aren’t their victory will be complete.

If you conform, you miss all of the adventures and stand against the progress of society.

― Debasish Mridha

Scientific computing has been a thing for about 70 years being born during World War 2. During that history there has been a constant push and pull of capability of computers, software, models, mathematics, engineering, method and physics. Experimental work has been essential to keep computations tethered to reality. An advance in one area would spur the advances in another in a flywheel of progress. A faster computer would make new problems previously seeming impossible to solve suddenly tractable. Mathematical rigor may suddenly give people faith in a method that previously seemed ad hoc and unreliable. Physics might ask new questions counter to previous knowledge, or experiments would confirm or invalidate model applicability. The ability to express ideas in software allows algorithms and models to be used that may have been too complex with older software systems. Innovative engineering provides new applications for computing that extend the scope and reach of computing to new areas of societal impact. Every single one of these elements is subdued in the present approach to HPC, and robs the ecosystem of vitality and power. We have learned these lessons in the recent past, yet swiftly forgotten them when composing this new program.

Control leads to compliance; autonomy leads to engagement.

― Daniel H. Pink

This alone could be a recipe for disaster, but it’s the tip of the iceberg. We have been mismanaging and undermining our scientific research in the USA for a generation both at research institutions like Labs and our universities. Our National Laboratories are mere shadows of their former selves. When I look at how I am managed the conclusion is obvious: I am well managed to be compliant to a set of conditions that have nothing to do with succeeding technically. Good management is applied to following rules and basically avoid any obvious “fuck ups”. Good management is not applied to successfully executing a scientific program. This being the prime directive today, the entire scientific enterprise is under siege. The assault on scientific competence is broad-based and pervasive as expertise is viewed with suspicion rather than respect. Part of this problem is the lack of intellectual stewardship reflected in numerous empty thoughtless programs. The second piece is the way we are managing science. A couple of easy things engrained into the way we do things that lead to systematic underachievement is inappropriately applied project planning and intrusive micromanagement into the scientific process. The issue isn’t management per se, but its utterly inappropriate application and priorities that are orthogonal to technical achievement.

One of the key elements in the downfall of American supremacy in HPC is the inability to tolerate failure as a natural outgrowth of any high-end endeavor. Our efforts are simply not allowed to fail at anything lest it be seen as a scandal or waste of money. In the process we deny ourselves the high-risk, but high-payoff activities that yield great leaps forward. Of course a deep-seated fear is at the root of the problem. As a direct result of this attitude, we end up not trying very hard. Failure is the best way to learn anything, and if you aren’t failing, your aren’t learning. Science is nothing more than a giant learning exercise. The lack of failure means that science simply doesn’t get done. All of this is obvious, yet our management of science has driven failure out. It is evident across a huge expanse of scientific endeavors, and HPC is no different. The death of failure is also the death of accomplishment. Correcting this problem alone would allow for significantly greater achievement, yet our current governance attitude seems utterly incapable of making progress here.

Tied like a noose around a neck is the problem of short-term focus. The short-term focus is the twin of the “don’t fail” attitude. We have to produce results and breakthroughs on a quarterly basis. We have virtually no idea where we are going beyond an annual basis, and the long-term plans continually shift with political whims. This short-term myopic view is being driven harder with each passing year. We effectively have no big long-term goals as a nation beyond simple survival. Its like we have forgotten to dream big and produce any sort of inspirational societal goals. Instead we create big soulless programs in the place of big goals. Exascale computing is perfect example. It is a goal without a real connection to anything societally important and is crafted solely for the purpose of getting money. It is absolutely vacuous and anti-intellectual at its core by viewing supercomputing as a hardware-centered enterprise. Then it is being managed like everything else with relentless short-term focus and failure avoidance. Unfortunately, even if it succeeds, we will continue our tumble into mediocrity.

This tumble into mediocrity is fueled by an increasingly compliance oriented attitude toward all work. Instead of working to conduct a balanced and impactful program to drive the capacity of computing to impact the real World, our programs simply comply with the intellectually empty directives from above. There is no debate about how the programs are executed because PI’s and Labs are just interested in getting money. The program is designed to be funded instead of succeed, and the Labs don’t act as honest brokers any longer being primarily interested in filling their own coffers. In other words, the program is designed as a marketing exercise, not a science program. Instead of a flywheel of innovative excellence and progress we produce a downward spiral of compliance driven mediocrity serving intellectually empty and unbalanced goals. If everyone gets their money and can successfully fill out their time sheets and gets a paycheck, it is a success.

At the end of the Cold War in the early 1990’s the USA’s Nuclear Weapons’ Labs were in danger of a funding free fall. Nuclear weapons’ testing ended in 1992 and the prospect of maintaining the nuclear weapons’ stockpile without testing, loomed large. A science-based stockpile stewardship (SBSS) program was devised to serve as a replacement, and HPC was one of the cornerstones of the program. SBSS provided a backstop against financial catastrophe at the Labs and provided long-term funding stability. This HPC element in SBSS was the ASCI program (which became the ASC program as it matured). The original ASCI program was relentlessly hardware focused with lots of computer science, along with activities to port older modeling and simulation codes to the new computers. This should seem very familiar to anyone looking at the new ECP program. The ASCI program is the model for the current exascale program. Within a few years it became clear that ASCI’s emphasis on hardware and computer science was inadequate to provide modeling and simulation support for SBSS with sufficient confidence. Important scientific elements were added to ASCI including algorithm and method development, verification and validation, and physics model development as well as stronger ties to experimental programs. These additions were absolutely essential for success of the program. That being said, these elements are all subcritical in terms of support, but they are much better than nothing.

If one looks at the ECP program the composition and emphasis looks just like the original ASCI program without the changes made shortly into its life. It is clear that the lessons learned by ASCI were ignored or forgotten by the new ECP program. It’s a reasonable conclusion that the main lesson taken from ASC program was how to get money by focusing on hardware. Two issues dominate the analysis of this connection:

1. none of the lessons learned by ASC necessary to conduct science have been learned by the exascale program. The exascale program is designed like the original ASCI program and fails to implement any of the programmatic modifications necessary for applied success. It is reasonable to conclude that the program has no serious expectation of applied scientific impact. Of course they won’t say this, but actions do speak louder than words!
2. The premise that exascale computing is necessary for science is an a priori assumption that has been challenged repeatedly (see JASONS reviews for example). The unfunded and neglected aspects of modeling, methods and algorithms all provide historically validated means to answer these challenges. Rather than address these challenges, they were rejected out of hand and never technically addressed. We simply see an attitude that bigger is better by definition and its been sold more as a patriotic call to arms than a balanced scientific endeavor. It remains true that faster computers are better, if you do everything right, we are not supporting the activities to do everything right (V&V, experimental connection and model development being primal in this regard).

Beyond the troubling lack of learning from past mistakes other issues remain. Perhaps the most obviously damning aspect of our current programs is their lack of connection to massive national goals. We simply don’t have any large national goals beyond being “great” or being “#1”. The HPC program is a perfect example. The whole program is tied to simply making sure that the USA is #1. In the past when computing came of age, the supercomputer was merely a tool that demonstrated utility in accomplishing something important to the nation or the world. It was not an end unto itself. This assured a definite balance in how the HPC was executed because the success was measured by HPC’s impact on a goal beyond itself. Today there is no goal beyond the HPC and the supercomputing as an activity suffers greatly. It has no measure of success outside itself. Any science done by supercomputer is largely for marketing, and press release. Quite often the results have little or no importance aside from the capacity to generate a flashy picture to impress people who know little or nothing about science.

Taken in sufficient isolation the objectives of the exascale program are laudable. An exascale computer is useful if it can be reasonably used. The issue is that such a computer does not live in isolation; it exists in a complex trade space where other options exist. My premise has never been that better or faster computer hardware is inherently bad. My premise is that the opportunity cost associated with such hardware is too high. The focus on the hardware is starving other activities essential for modeling and simulation success. The goal of producing an exascale computer is not an objective of opportunity, but rather a goal that we should actively divest ourselves of. Gains in supercomputing are overly expensive and work to hamper progress in related areas simply by the implicit tax produced by how difficult the new computers are to use. Improvements in real modeling and simulation capability would be far greater if we invested our efforts in different aspects of the ecosystem.

The key to holding a logical argument or debate is to allow oneself to understand the other person’s argument no matter how divergent their views may seem.

― Auliq Ice

God save me from idealists.

― Jim Butcher

Coming up with detailed mathematical analysis, much less the solution of (partial) differential equations is extremely difficult. In the effort to progress on this important, but critically difficult task various simplifications and idealizations can make all the difference between success and failure. This difficulty highlights the power and promise of numerical methods for solving such equations because simplifications and idealizations are not absolutely necessary for solution. Nonetheless, much of the faith in a numerical method is derived by congruence of the solution numerically with analytical solutions. This process is known as verification and paly an essential role in helping to provide evidence of the credibility of numerical simulations. Our faith in the ability of numerical simulations to solve difficult problems is thus grounded to some degree by the scope and span of our analytical knowledge. This tie is important to both recognize and carefully control because of analytical knowledge is necessarily limited in ways that numerical methods should not be.

In developing and testing computational methods, we spend a lot of time working on solving the ideal equations for a phenomenon. This is true in fluids, plasma, and many other fields. These ideal equations are usually something that comes from the age of classical physics and mathematics. Most commonly these ideal equations are associated with the names of greats of science, Newton, Euler, Poincare. This near obsession is one of the greatest dangers to progress I can think of. The focus on the ideal is the consequence of some almost religious devotion to classical ideas, and deeply flawed. By focusing on the classical ideal equations many of the important, critical and interesting aspects of reality escape attention. We remain anchored to the past in a way that undermines our ability to master reality with modernity.

Whenever a theory appears to you as the only possible one, take this as a sign that you have neither understood the theory nor the problem which it was intended to solve.

― Karl R. Popper

These ideal equations are starting points for investigations of the physical world, and arose in an environment where analytical work was the only avenue for understanding. Simplicity and stripping away the complexities of reality were the order of the day. Today we are freed to a very large extent from the confines of analytical study by the capacity to approximate solutions to equations. We are free to study the universe as it actually is, and produce a deep study of reality. The analytical methods and ideas still have utility for gaining confidence in these numerical methods, but their lack of grasp on describing reality should be realized. Our ability to study the reality should be celebrated and be the center of our focus. Our seeming devotion to the ideal simply distracts us and draws attention from understanding the real World.

The more pernicious and harmful aspect of ideality was a reverence for divinity insolutions. The ideal equations are supposed to represent the perfect, and in a sense the “hand of God” working in the cosmos. As such they represent the antithesis of moderni
ty. As such they represent the inappropriate injection of religiosity into the study of reality. For this reason alone the ideal equations should be deeply suspect at a philosophical level. These sort of religious ideas should not be polluting the unfettered investigation of reality. More than this we can see that the true engine of beauty in the cosmos is removed from these equations. So much of what is extraordinary about the universe is the messiness driven by the second law of thermodynamics. This law takes many forms and always removes the ideal from the equations, and injects the hard, yet beautiful face of reality.

A thing can be fine on paper but utterly crummy in the field.

― Robert A. Heinlein

Not only are these equations suspect for philosophical reasons, they are suspect for the imposed simplicity of the time they are taken from. In many respects the ideal equations miss most of fruits of the last Century of scientific progress. We have faithfully extended our grasp of reality to include more and more “dirty” features of the actual physical World. To a very great extent the continued ties to the ideal contribute to the lack of progress in some very important endeavors. Perhaps no case more amply demonstrates this handicapping of progress as well as turbulence. Our continued insistence that turbulence is tied to the ideal nature of incompressibility is becoming patently ridiculous. It highlights that important aspects of the ideal are synonymous with the unphysical.

I have spoken out about the issues with incompressibility several times in the past (https://williamjrider.wordpress.com/2014/03/07/the-clay-prize-and-the-reality-of-the-navier-stokes-equations/, https://williamjrider.wordpress.com/2015/03/06/science-requires-that-modeling-be-challenged/, https://williamjrider.wordpress.com/2016/04/08/the-singularity-abides/, https://williamjrider.wordpress.com/2016/04/15/the-essential-asymmetry-in-fluid-mechanics/, https://williamjrider.wordpress.com/2016/09/27/the-success-of-computing-depends-on-more-than-computers/). Here I will simply reiterate these points from the perspective of the concept of ideal equations. Incompressibility is a simple and utterly ideal in the sense that no nontrivial flow is exactly incompressible (). Real and nontrivial flow fields are only approximately incompressible. It is important to recognize that approximate and exactly incompressible are very different at their core. Exactly incompressible flows are fundamentally unphysical and unrealizable in the real world. Put differently, they are absolutely pathological.

An important thing to recognize in this discussion is the number of important aspects of reality that are sacrificed with incompressibility. The list is stunning and gives a hint of the depth of the loss. Gone is the second law of thermodynamics unless viscous effects are present. Gone is causality. Gone are important nonlinearities. This approximation is taken to the extreme of being an unphysical constraint that produces a deeply degenerate system of equations. Of greater consequence is the demolition of physics that may be at the heart of explaining turbulence itself. The essence of turbulence needs a singularity formation to make sense of observations. This is at the core of the Clay Prize, yet in the derivation of the incompressible equations, the natural nonlinear process for singularity formation is removed by fiat. Incompressibility creates a system of equations that is simple and yet only a shadow of the more general equations it claims to represent. I fear it is an albatross about the neck of fluid mechanics.

There are other idealities that need to be overturned. In many corners of fluid mechanics symmetries are assumed. Many scientists desire that they should be maintained under all sorts of circumstances. They rarely ask whether the symmetry is maintained in the face of perturbations from the symmetry that would reasonably be expected to exist in reality (in fact it is absolutely unreasonable to assume perfect symmetry). Some assumptions are reasonable in some situations where the flows are stable, but other cases would destroy these symmetries for any realistic flow. Pushing a numerical method to maintain symmetry under such circumstances where the instability would grow should be abhorrent and avoided. In the physical actual universe the destruction of symmetry is the normal evolution of a system and preservation is rarely observed. As such expectations of symmetry preservation in all cases define an unhealthy community norm.

A great example of this sort of dynamic occurs in modeling stars that end their lives in an explosion like type II supernovas. The classic picture was a static spherical star that burned elements in a series of concentric spheres or increasing mass as one got deeper into the star. Eventually the whole process becomes unstable as the nuclear reactions shift from exothermic to endothermic when iron is created. We observe explosions in such stars, but the idealized stars would not explode. Even if we forced the explosion, the evolution of the post-explosion could not match important observational evidence that implied deep mixing of heavy elements into the expanding envelope of the star.

It is a place where the idealized view stood in the way of progress of decades and the release of ideality allowed progress and understanding. Once these extreme symmetries were released and the star was allowed to rotate, have magnetic fields, and mix elements across the concentric spheres models and simulations started to match observations. We got exploding stars; we got the deep mixing necessary for both the explosion itself and the post explosion evolution. The simulations began to explain what we saw in nature. The process of these exploding stars is essential for the understanding of the universe because such stars are the birthplace of the matter that our World is built from. When things were more ideal the simulations failed to a very large extent.

This sort of issue appears over and over in science. Time and time again, the desire to study things in an ideal manner acts to impede the unveiling of reality. By now we should know better, but it is clear that we don’t. The idea of sustaining the ideal equations and evolution as the gold standard is quite strong. Another great example of this is the concept of kinetic energy conservation. Many flows and numerical methods are designed to exactly conserve kinetic energy. This only occurs in the most ideal of circumstances when flows have no natural dissipation (itself deeply unphysical) while retaining well-resolved smooth structure. So the properties are only seen in flows that are unphysical. Many believe that such flows should be exactly preserved as the foundation for numerical methods. This belief is somehow impervious to the observation that such flows are utterly unphysical and could never be observed in reality. It is difficult to square this belief system with the desire to model anything practical.

We need to recognize the essential tension between the need to test methods using the solution to idealized equations with the practical simulation of reality. We need to free ourselves of the limiting aspects of the mindset around the ideal equations. The important aspect of matching solutions to ideal equations must be acknowledged without imposing unphysical limits on the simulation. The imperative for numerical methods is modeling reality. To match aspects of the ideal equations solution many sacrifice physical aspect of numerical methods. Modeling reality should always be the preeminent concern for the equations and the methods for solution. Numerical methods unleash many of the constraints that analytical approaches abide by and these should be taken advantage of to a maximal degree.

Quite frequently, the way that numerical methods developers square their choices is an unfortunate separation of modeling from the numerical solutions. In some cases the choice that is followed is the philosophy where the ideal equations are solved along with the explicit modeling of any non-ideal physics. As such the numerical method is desired to be unwaveringly true to the ideal equations. Quite often the problem with this approach is that the non-ideal effects are necessary for the stability and quality of the solution. Moreover the coupling between the numerical solution and modeling is not clean, and the modeling can’t be ignored in the assessment of the numerical solution.

A great example of this dichotomy is turbulent fluid mechanics and it’s modeling. It is instructive to explore the issues surrounding the origin of the models with connections to purely numerical approaches. There is the classical thinking about modeling turbulence that basically comes down to solving the ideal equations as perfectly as possible, and modeling the entirety of turbulence with additional models added to the ideal equations. It is the standard approach and by comparison to many other areas of numerical simulation, a relative failure. Nonetheless this approach is followed with almost a religious fervor. I might surmise that the lack of progress in understanding turbulence is somewhat related to the combination of adherence to a faulty basic model (incompressibility) and the solution approach that supposes that all the non-ideal physics can be modeled explicitly.

It is instructive in closing to peer more keenly at the whole turbulence modeling problem. A simple, but very successful model for turbulence is the Smagorinsky model originally devised for climate and weather modeling, but forming the foundation for the practice of large eddy simulation (LES). What is under appreciated about the Smagorinsky model is its origins. This model was originally created as a way of stabilizing shock calculations by Robert Richtmyer and applied to an ideal differencing method devised by John Von Neumann. The ideal equation solution without Richtmyer’s viscosity was unstable and effectively useless. With the numerically stabilizing term added to the solution, the method was incredibly powerful and forms the basis of shock capturing. The same term was then added to weather modeling to stabilize those equations. It did just that and remarkably it suddenly transformed into a “model” for turbulence. In the process we lost the role it played for numerical stability, but also the strong and undeniable connection between the entropy generated by a shock and observed turbulence behavior. This connection was then systematically ignored because the unphysical incompressible equations we assume turbulence is governed by do not admit shocks. In this lack perspective we find the recipe for lack of progress. It is too powerful for a connection not to be present. Such connections creates issues that undermine some core convictions in the basic understanding of turbulence that seem too tightly held to allow the lack of progress to question.

We cannot become what we need by remaining what we are.

― John C. Maxwell

What’s measured improves

― Peter F. Drucker

In area of endeavor standards of excellence are important. Numerical methods are no different. Every area of study has a standard set of test problems that researchers can demonstrate and study their work on. These test problems end up being used not just to communicate work, but also test whether work has been reproduced successfully or compare methods. Where the standards are sharp and refined the testing of methods has a degree of precision and results in actionable consequences. Where the standards are weak, expert judgment reigns and progress is stymied. In shock physics, the Sod shock tube (Sod 1978) is such a standard test. The problem is effectively a “hello World” problem for the field, but suffers from weak standards of acceptance focused on expert opinion of what is good and bad without any unbiased quantitative standard being applied. Ultimately, this weakness in accepted standards contributes to stagnant progress we are witnessing in the field. It also allows a rather misguided focus and assessment of capability to persist unperturbed by results (standards and metrics can energize progress, https://williamjrider.wordpress.com/2016/08/22/progress-is-incremental-then-it-isnt/).

Sod’s shock tube is an example of a test problem being at the right time in the right place. It was published right at the nexus of progress in hyperbolic PDE’s, but before breakthroughs were well publicized. The article introduced a single problem applied to a large number of methods all of which performed poorly in one way or another. The methods were an amalgam of old and new methods demonstrating the general poor state of affairs for shock capturing methods in the late 1970’s. Since its publication is has become the opening ante for a method to demonstrate competence in computing shocks. The issues with this problem were highlighted in an earlier post, https://williamjrider.wordpress.com/2016/08/18/getting-real-about-computing-shock-waves-myth-versus-reality/, where a variety of mythological thoughts are applied to computing shocks.

This problem is a very idealized shock problem in one dimension that is amenable to semi-analytical solution. As a result an effectively exact solution may be obtained via solution of nonlinear equations. The evaluation of the exact solution appropriately for comparison with numerical solutions is itself slightly nontrivial. The analytical solution needs to be properly integrated over the mesh cells to represent the correct integrated control volume values (more over this integration needs to be done for the correct conserved quantities). Comparison is usually done via the primitive variables, which may be derived from the conserved variable using standard techniques (I wrote about this a little while ago https://williamjrider.wordpress.com/2016/08/08/the-benefits-of-using-primitive-variables/ ). A shock tube is the flow that is results when two semi-infinite slabs of gas at different state conditions are held separately. They are then allowed to interact and a self-similar flow is created. This flow can contain all the basic compressible flow structures, shocks, rarefactions, and contact discontinuities.

Specifically, Sod’s shock tube (https://en.wikipedia.org/wiki/Sod_shock_tube) has the following conditions: in a one dimensional domain filled with an ideal gamma law gas, , , the domain is divided into two equal regions; on , ; on , . The flow is described by the compressible Euler equations (conservation of mass, , momentum and energy ), and an equation of state, . At time zero the flow develops a self-similar structure with a right moving shock followed by a contact discontinuity, and a left moving rarefaction (expansion fan). This is the classical Riemann problem. The solution may be found through semi-analytical means solving a nonlinear equation defined by the Rankine-Hugoniot relations (see Gottlieb and Groth for a wonderful exposition on this solution via Newton’s method).

The crux of the big issue with how this problem is utilized is that the analytical solution is not used for more than display in plotting comparisons with numerical solutions. The quality of numerical solutions is then only assessed qualitatively. This is a huge problem that directly inhibits progress. This is a direct result of having no standard beyond expert judgment on the quality. It leads to the classic “hand waving” argument for the quality of solutions. Actual quantitative differences are not discussed as part of the accepted standard. The expert can deftly focus on the parts of the solution they want to and ignore the parts that might be less beneficial to their argument. Real problems can persist and effectively be ignored (such as the very dissipative nature of some very popular high-order methods). Under this lack of standard relatively poorly performing methods can retain a high level of esteem while better performing methods are effectively ignored.

With all these problems, why does this state of affairs persist year after year? The first thing to note is that the standard of expert judgment is really good for experts. The expert can rule by asserting their expertise creating a bit of a flywheel effect. For experts whose favored methods would be exposed by better standards, it allows their continued use with relative impunity. The experts are then gate keepers for publications and standards, which tends to further the persistence of this sad state of affairs. The lack of any standard simply energizes the status quo and drives progress into hiding.

The key thing that has allowed this absurdity to exist for so long is the loss of accuracy associated with discontinuous solutions. For nonlinear solutions of the compressible Euler equations, high order accuracy is lost in shock capturing. As a result the designed order of accuracy for a computational method cannot be measured with a shock tube solution. As a result, one of the primary aims of verification is not achieved using this problem. One must always remember that order of accuracy is the confluence of two aspects, the method and the problem. Those stars need to align for the order of accuracy to be delivered.

Order of accuracy is almost always shown in results for other problems where no discontinuity exists. Typically a mesh refinement study, error norms, order of accuracy is provided as a matter of course. The same data is (almost) never shown for Sod’s shock tube. For discontinuous solutions the order of accuracy is (less than one). Ideally, the nonlinear features of the solution (shocks and expansions) converge at first-order, and the linearly degenerate feature (shears and contacts) converge at less than first order based on the details of the method (see the paper by Aslam, Banks and Rider (me). The core of the acceptance of the practice of not showing the error or convergence for shocked problems is the lack of differentiation of methods due to similar convergence rates for all methods (if they converge!). The relative security offered by the Lax-Wendroff theorem further emboldens people to ignore things (the weak solution guaranteed by it has to be entropy satisfying to be the right one!).

This is because the primal point of verification cannot be satisfied, but other aspects are still worth (or even essential) to pursue. Verification is also all about error estimation, and when the aims of order verification cannot be achieved, this becomes a primary concern. What people do not report and the aspect that is missing from the literature is the relatively large differences in error levels from different methods, and the impact of these differences practically. For most practical problems, the design order of accuracy cannot be achieved. These problems almost invariably converge at the lower order, but the level of error from a numerical method is still important, and may vary greatly based on details. In fact, the details and error levels actually have greater bearing on the utility of the method and its efficacy pragmatically under these conditions.

For all these reasons the current standard and practice with shock capturing methods are doing a great disservice to the community. The current practice inhibits progress by hiding deep issues and failing to expose the true performance of methods. Interestingly the source of this issue extends back to the inception of the problem by Sod. I want to be clear that Sod wasn’t to blame because none of the methods available to him were acceptable, but within 5 years very good methods arose, but the manner of presentation chosen originally persisted. Sod on showed qualitative pictures of the solution at a single mesh resolution (100 cells), and relative run times for the solution. This manner of presentation has persisted to the modern day (nearly 40 years almost without deviation). One can travel through the archival literature and see this pattern repeated over and over in an (almost) unthinking manner. The bottom line is that it is well past time to do better and set about using a higher standard.

At a bare minimum we need to start reporting errors for these problems. This ought to not be enough, but it is an absolute minimum requirement. The problem is that the precise measurement of error is prone to vary due to details of implementation. This puts the onus on the full expression of the error measurement, itself an uncommon practice. It is uncommonly appreciated that the difference between different methods is actually substantial. For example in my own work with Jeff Greenough, the error level for the density in Sod’s problem between fifth order WENO, and a really good second-order MUSCL method is a factor of two in favor of the second-order method! (see Greenough and Rider 2004, the data is given in the tables below from the paper). This is exactly the sort of issue the experts are happy to resist exposing. Beyond this small step forward the application of mesh refinement with convergence testing should be standard practice. In reality we would be greatly served by looking at the rate of convergence to problems feature-by-feature. We could cut up the problem into regions and measure the error and rate of convergence separately for the shock, rarefaction and contact. This would provide a substantial amount of data that could be used to measure quality of solutions in detail and spur progress.

Two tables of data from Greenough and Rider 2004 displaying the density error for Sod’s problem (PLMDE = MUSCL).

We still use methods quite commonly that do not converge to the right solution for discontinuous problems (mostly in “production” codes). Without convergence testing this sort of pathology goes undetected. For a problem like Sod’s shock tube, it can still go undetected because the defect is relatively small. Usually it is only evident when the testing is on a more difficult problem with stronger shocks and rarefactions. Even then it is something that has to be looked for showing up as reduced convergence rates, or the presence of constant un-ordered error in the error structure, instead of the standard . This subtlety is usually lost in a field where people don’t convergence test at all unless they expect full order of accuracy for the problem.

Now that I’ve thrown a recipe for improvement out there to consider, I think it’s worthwhile to defend expert judgment just a bit. Expertise has its role to play in progress. There are aspects of science that are not prone to measurement, science is still a human activity with tastes and emotion. This can be a force of good and bad, the need for dispassionate measurement is there as a counter-weight to the worst instincts of mankind. Expertise can be used to express a purely qualitative assessment that can make the difference between something that is merely good and great. Expert judgment can see through complexity to remediate results into a form with greater meaning. Expertise is more of a tiebreaker than the deciding factor. The problem today is that current practice means all we have is expert judgment and this is a complete recipe for the status quo and an utter lack of meaningful progress.

The important outcome from this discussion is crafting a path forward that makes the best use of our resources. Apply appropriate and meaningful metrics to the performance of methods and algorithms to make progress or lack of it concrete. Reduce, but retain the use of expertise and apply it to the qualitative aspects of results. The key to doing better is striking an appropriate balance. We don’t have it now, but getting to an improved practice is actually easy. This path is only obstructed by the tendency of the experts to hold onto their stranglehold.

An expert is someone who knows some of the worst mistakes that can be made in his subject, and how to avoid them.

― Werner Heisenberg

Historical montage of Sod shock tube results from Sod 1978, Harten 1983, Huynh 1996, Jiang and Shu 1996.  First Sod’s result for perhaps the best performing method from his paper (just expert judgment on my part LOL).

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

Suresh, A., and H. T. Huynh. “Accurate monotonicity-preserving schemes with Runge–Kutta time stepping.” Journal of Computational Physics 136, no. 1 (1997): 83-99.

Jiang, Guang-Shan, and Chi-Wang Shu. “Efficient Implementation of Weighted ENO Schemes.” Journal of Computational Physics 126, no. 1 (1996): 202-228.

Sod, Gary A. “A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws.” Journal of computational physics 27, no. 1 (1978): 1-31.

Gottlieb, J. J., and C. P. T. Groth. “Assessment of Riemann solvers for unsteady one-dimensional inviscid flows of perfect gases.” Journal of Computational Physics 78, no. 2 (1988): 437-458.

Banks, Jeffrey W., T. Aslam, and W. J. Rider. “On sub-linear convergence for linearly degenerate waves in capturing schemes.” Journal of Computational Physics 227, no. 14 (2008): 6985-7002.

Greenough, J. A., and W. J. Rider. “A quantitative comparison of numerical methods for the compressible Euler equations: fifth-order WENO and piecewise-linear Godunov.” Journal of Computational Physics 196, no. 1 (2004): 259-281.

 The best dividends on the labor invested have invariably come from seeking more knowledge rather than more power.

– The Wright brothers

Some messages are so important that they need to be repeated over and over. This is one of those times. Computing is mostly not about computers. A computer is a tool, powerful, important and unrelentingly useful, but a tool. Using computing is a fundamentally human activity that uses a powerful tool to augment human’s capacity for calculation and monotony. Today we see attitudes expressing more interest in the computers themselves with little regard for how they are used. The computers are essential tools that enable a certain level of utility, but the holistic human activity is at the core of what they do. This holistic approach is exactly the spirit that has been utterly lost by the current high performance computer push. In a deep way the program lacks the appropriate humanity in its composition, which is absolutely necessary for progress.

Most clearly the computers themselves are not an end in themselves, but only useful insofar as they can provide benefit to the solution of problems for mankind. Taking human thinking and augmenting it derives the benefits for humanity. It is our imagination and inspiration automated as to enable solutions through primarily approximate means. The key to all of the true benefits of computing come from the fields of physics, engineering, medicine, biology, chemistry and mathematics. Subjects closer to the practice of computing do not necessarily push benefits forward to society at large. It is this break in the social contract that current high performance computing has entirely ignored. The societal end product is a mere after thought and little more than simply a marketing ploy for a seemingly unremitting focus on computer hardware.

Mathematics is the door and key to the sciences.

— Roger Bacon

This approach is destined to fail, or at the very most not reap the potential benefits the investment should yield. It is completely and utterly inconsistent with the venerated history of scientific computing. The key to the success and impact of scientific computing has been its ability to augment its foundational fields as a supplement to human’s innate intellect in an area that human’s ability is a bit diminished. While it supplements raw computational power, the impact of the field depends entirely on human’s natural talent as expressed in the base science and mathematics. One place of natural connection is the mathematical expression of the knowledge in basic science. Among the greatest sins of modern scientific computing is the diminished role of mathematics in the march toward progress.

Computing should never be an excuse to not think; the truth is that computing has become exactly that; it is an excuse to stop thinking, and simply automatically get “answers”. The importance of this connection cannot be underestimated. It is the complete and total foundation of computing. This is where the current programs become completely untethered from logic, common sense and the basic recipe of success. The mathematics program is virtually absent from the drive toward greater scientific computing. For example I work in an organization that is devoted to applied mathematics, yet virtually no mathematics actually takes place. Our applied mathematics programs have turned into software programs. Somehow the decision was made 20-30 years ago that software “weaponized” mathematics, and in the process the software became the entire enterprise, and the mathematics itself became lost, an afterthought of the process. Without the actual mathematical foundation for computing, important efficiencies, powerful insights and structural understanding is scarified.

The software has become the major product and end point of almost all research efforts in mathematics to the point of displacing actual math. The product of work needs to be expressed in software and the construction and maintenance of the software packages has become the major enterprise being conducted. In the process the centrality of the mathematical exploration and discovery has been submerged. Software is a difficult, valuable and important endeavor in itself, but distinct from mathematics. In many cases the software itself has become the raison d’être for math programs. In the process of the emphasis on the software instantiating mathematical ideas, the production and assault on mathematics has stalled. It has lost its centrality to the enterprise. This is horrible because there is so much yet to do.

Worse yet, the mathematical software is horribly expensive to maintain and loses its modernity with a frightful path. We hear calls to preserve the code base because it was so expensive. A preserved code base loses its value more surely than a car depreciates. The software is only as good as the intellect of the people maintaining it. In the process we lose intellectual ownership of the code. This is beyond the horrible accumulation of technical debt in the software, which erodes its value like mold or dry rot. None of these problems are the worst of the myriad of issues around this emphasis; the worst issue is the opportunity cost of turning our mathematicians into software engineers and removing the attention from some of our most pressing issues.

A single discovery of a new concept, principle, algorithms or technique can render one of these software packages completely obsolete. We seem to be in an era where we believe that more computer power is all that is needed to bring reality to heel. These discoveries can allow results and efficiencies that were completely unthinkable to be achieved. Discoveries make the impossible, possible, and we are denying ourselves the possibility of these results through our inept management of mathematics proper role in scientific computing. What might be some of the important topics in need of refined and focused mathematical thinking?

The work of Peter Lax and others has brought great mathematical understanding, discipline and order to the world of shock physics. Amazingly this has all happened in one dimension plus time. In two or three dimensions where the real World happens, we know far less. As a result our knowledge and mastery over the equations of (compressible) fluid dynamics is limited and incomplete. Bringing order and understanding to the real World of fluids could have a massive impact on our ability to solve realistic problems. Today we largely exist on the faith that our limited one-dimensional knowledge gives us the key to multi-dimensional real World problems. A program to expand our knowledge and fill these gaps in knowledge would be a boon to analytical and numerical methods seeding a new renaissance for scientific computing, physics and engineering.

One of the key things to understanding the power of computing is the comprehension that the ability compute belies a deep understanding that enables analytical, physical and domain specific knowledge. A problem intimately related to the multi-dimensional issues with compressible fluids is the topic of one of the Clay prizes. This is a million dollar prize for proving the existence of solutions to the Navier-Stokes equations. There is a deep problem with the way this problem is posed that may make its solution both impossible and practically useless. The equations posed in the problem statement are fundamentally wrong. They are physically wrong, not mathematically although this wrongness has consequences. In a very deep practical way fluids are never truly incompressible; incompressible is an approximation, but not a fact. This makes the equations have an intrinsically elliptic character (because incompressibility implies infinite sound speeds, and lack of thermodynamic character).

Physically the infinite sound speeds remove causality from the equations, and the removal of thermodynamics takes them further outside the realm of reality. This also creates immense mathematical difficulties that make these equations almost intractable. So this problem touted as the route to mathematically contribute to understanding turbulence may be a waste of time for that endeavor as well. Again, we need a concerted effort to put this part of the mathematical physics World into better order. The benefits to computation through some order would be virtually boundless.

This gets at one of the greatest remaining unsolved problems in physics, turbulence. The ability to solve problems depends critically upon models and the mathematics that makes such models tractable or not. The existence theory problems for the incompressible Navier-Stokes equations are essential for turbulence. For a century it has largely been assumed that the Navier-Stokes equations describe turbulent flow with an acute focus on incompressibility. More modern understanding should have highlighted that the very mechanism we depend upon for creating the sort of singularities turbulence observations imply has been removed in the process of the choice of incompressibility. The irony is absolutely tragic. Turbulence brings almost an endless amount of difficulty to its study whether experimental, theoretical, or computational. In every case the depth of the necessary contributions by mathematics is vast. It seems somewhat likely that we have compounded the difficulty of turbulence by choosing a model with terrible properties. If so, it is likely that the problem remains unsolved, not due to its difficulty, but rather our blindness to the shortcomings, and the almost religious faith many have followed in attacking turbulence with such a model.

Before I close I’ll touch on a few more areas where some progress could either bring great order to a disordered, but important area, or potentially unleash new approaches to problem solving. An area in need of fresh ideas, connections and better understanding is mechanics. This is a classical field with a rich and storied past, but suffering from a dire lack of connection between the classical mathematical rigor and the modern numerical world. Perhaps in no way is this more evident in the prevalent use of hypo-elastic models where hyper-elasticity would be far better. The hypo-elastic legacy comes from the simplicity of its numerical solution being the basis of methods and codes used around the World. It also only applies to very small incremental deformations. For the applications being studied, it should is invalid. In spite of this famous shortcoming, hypo-elasticity rules supreme, and hyper-elasticity sits in an almost purely academic role. Progress is needed here and mathematical rigor is part of the solution.

A couple of areas of classical numerical methods are in dire need of breakthroughs with the current technology simply being accepted as good enough. A key one is the solution of sparse linear systems of equations. The current methods are relatively fragile and it’s been 30-40 years since we had a big improvement. Furthermore these successes are somewhat hollowed by the lack of a robust solution path. Right now the gold standard of scaling comes from multigrid, invented in the mid-1970’s to mid-1980’s. Robust solvers use some sort of banded method with quadratic scaling or pre-conditioned Krylov methods (which are less reliable). This area needs new ideas and a fresh perspective in the worst way. The second classical area of investigation that has stalled is high-order methods. I’ve written about this a lot. Needless to say we need a combination of new ideas, and a somewhat more honest and pragmatic assessment of what is needed in practical terms. We have to thread the needle of accuracy, efficiency and robustness in both cases. Again without mathematics holding us to the level of rigor it demands progress seems unlikely.

Lastly we have broad swaths of application and innovation waiting to be discovered. We need to work to make optimization something that yields real results on a regular basis. The problem in making this work is similar to the problem with high-order methods; we need to combine the best technology with an unerring focus on the practical and pragmatic. Optimization today only applies to problems that are far too idealized. Other methodologies are laying in wait of great impact among these the generalization of statistical methods. There is an immense need for better and more robust statistical methods in a variety of fields (turbulence being a prime example). We need to unleash the forces of innovation to reshape how we apply statistics.

When you change the way you look at things, the things you look at change.

― Max Planck

The depth of the problem for mathematics does seem to be slightly self-imposed. In a drive for mathematical rigor and professional virtue in applied mathematics, the field has lost a great deal of connection to physics and engineering. If one looks to the past for guidance, the obvious truth is that the ties between physics, engineering and mathematics have been quite fruitful. There needs to be healthy dynamics of push and pull between these areas of emphasis. The worlds of physics and engineering need to seek mathematical rigor as a part of solidifying advances. Mathematics needs to seek inspiration from physics and engineering. Sometimes we need the pragmatic success in the ad hoc “seat of the pants” approach to provide the impetus for mathematical investigation. Finding out that something works tends to be a powerful driver to understanding why something works. For example the field of compressed sensing arose from a practical and pragmatic regularization method that worked without theoretical support. Far too much emphasis is placed on software and far too little on mathematical discovery and deep understand. We need a lot more discovery and understanding today, perhaps no place more than scientific computing!

Mathematics is as much an aspect of culture as it is a collection of algorithms.

—  Carl Boyer

Note: Sometimes my post is simply a way of working on narrative elements for a talk. I have a talk on Exascale computing and (applied) mathematics next Monday at the University of New Mexico. This post is serving to help collect my thoughts in advance.

Above all, don’t lie to yourself. The man who lies to himself and listens to his own lie comes to a point that he cannot distinguish the truth within him, or around him, and so loses all respect for himself and for others. And having no respect he ceases to love.

― Fyodor Dostoyevsky

Over the past handful of years the capacity to maintain professional success and personal integrity has become increasingly strained. Simultaneously the whole concept of integrity within the public life has similarly become strained. I’ve been startled by the striking symmetry between my private, professional and public life as the incredible and terrifying shit show of the 2016 American Presidential election. It all seems to be coming together in a massive orgy of angst, lack of honesty and fundamental integrity across the full spectrum of life. As an active adult within this society I feel the forces tugging away at me, and I want to recoil from the carnage I see. A lot of days it seems safer to simply stay at home and hunker down and let this storm pass. It seems to be present at every level of life involving what is open and obvious to what is private and hidden.

How do we deal with all of the conflict, tension and danger when the rules of the game seem to have been thrown out? Is this confluence of effects common to everyone helping to explain the World, or is it personal? I think it’s worth pondering the breadth and scope of the challenges as we head forward toward a hopefully more optimistic future and 2017.

I’ve highlighted the concept of integrity as the focal point and the thing at more imminent risk. This is an expansion of my previous discussion of peer review that revolves around the same axis. It gets to the ability to care with some depth about the work I do, and whether that concern is actually appreciated. What does personal integrity mean to me? Like most things it is complex and combines multiple aspects of the content of my daily life. The greatest element of personal angst deals with the truth and any willingness for truths to be articulated openly. Research and progress depends upon good ideas being applied to areas of opportunity. Another less charitable way of saying the same thing is that progress depends on finding important, valuable problems and driving solutions. A second piece of integrity is hard work, persistence, and focus on the important valuable problems linked to great National or World concerns. Lastly, a powerful aspect of integrity is commitment to self. This includes a focus on self-improvement, and commitment to a full and well-rounded life. Every single bit of this is in rather precarious and constant tension, which is fine. What isn’t fine is the intrusion of outright bullshit into the mix undermining integrity at every turn.

When people don’t express themselves, they die one piece at a time.

― Laurie Halse Anderson

The core of the issue attacking integrity is the power and prevalence of bullshit in professional and public life. At a professional level bullshit has become the prevalent means of communication of results. Why create real work when fake work can be spun into results of equal and greater value. In fact bullshit is better because it can be whatever it needs to be for success. People continually produce results of minimal value that get marketed as breakthroughs. The lack of integrity at the level of leadership simply takes this bullshit and passes it along. Eventually the bullshit gets to people who are incapable of recognizing the difference. Ultimately, the result of the acceptance of bullshit produces a lowering of standards, and undermines the reality of progress. Bullshit is the death of integrity in the professional world.

There exists an appalling symmetry within the broadest public sphere. We are witnessing a political movement of disturbing power founded on bullshit. We see outright lies produced every day and never actively challenged. This bullshit may actually elect a completely unqualified and dangerous person President of the United States. Why work with facts or truth when bullshit is so incredibly effective? When we look more deeply at this problem we start to see that our political dysfunction is built upon a virtual mountain of bullshit. We see reality television, Facebook, Fox News, CNN, online dating, and a host of other modern things all-operating within a vibrant and growing bullshit economy. Taken in this broad context, the dominance of bullshit in my professional life and the potential election of Donald Trump are closely connected.

A big part of the acceptance of bullshit as the medium of universal discourse is related to fear. Increasingly our professional and public lives are ruled by irrational fears. Fear of failure professionally is rampant. Fear of terrorism is also rampant. Fear of immigrants is yet another common fear tied to terrorism, racism and economic stress. Fear is a powerful emotion that overrules most rational responses to problems. It leads to people shrinking away from the sort of professional risk that research depends upon. Fear is also one of the most powerful tools of political despots. In each of these cases bullshit can be used to either quell or amplify fears. In the technical World bullshit can produce seeming success without regard to actual technical accomplishment. The acceptance of bullshit in place of actual results quells the risk of failure. In the political World bullshit can produce fear where little or none is warranted. We are seeing trillions of dollars and millions of votes being generated by fear mongering bullshit. Even worse it has paved the way for greater purveyors of bullshit like Trump. We may well see bullshit providing the vehicle to elect an unqualified con man as the President.

Here are two major paths to take in life, follow and be rewarded by the power structure, or confront power with the reality of their failing. So I’m confronted with guidance “don’t be a troublemaker” versus show integrity “speak truth to power”. Which is it? Which of these paths do our institutions support today? That’s easy. Be quiet and go quietly through life, and don’t make waves. Better yet, join the bullshit economy and contribute to its vibrant growth as our greatest export.

If you can’t speak truths and provide honest assessments in today’s World, or call out lies in public, can you have personal integrity? When actual bullshit has become the path to professional success, how does one come back? How much of yourself gets left behind in the process? And how does one keep ones self from being a mere shadow of one’s true self? I’ve been struggling with these questions at work with ever-greater regularity. My personal devotion to progress and research is continually undermined by bullshit replacing progress. Today it is easier to make shit up and pass it off as being completely equivalent to the result of honest good work. Moreover bullshit doesn’t have the downside of potentially not working, its “success” is virtually guaranteed. Just tell the people above you in the food chain what they want to hear, you’ll be rewarded in spades! The beauty of it is the made up shit can conform to whatever narrative you desire, and completely fit whatever your message is. In such a world real problems can simply be ignored when the path to progress is problematic for those in power.

These same issues happen to energize a political dialog that becomes a shouting match. There is no truth and the whole thing degenerates into a shouting match. Beyond the shouting match you get the ability to ignore real problems that are inconvenient. A perfect example is climate change. For many traditional businesses climate change is really an inconvenient truth. When bullshit rules, it can be publically ignored without real risk to political success. We are seeing this play out right in front of all of us. We are in danger of losing all connection to facts, truth and science as guiding forces for determining optimal solutions to our very real problems. The road to this end is paved by allowing bullshit to be viewed as equivalent to solid facts.

Is there a path forward? Perhaps things simply need to devolve into the natural outcome from the current path. Nothing short of catastrophe will stop this orgy of bullshit choking public life. One might hope that we have the collective wisdom to avoid a calamity, one might hope.

One of the greatest regrets in life is being what others would want you to be, rather than being yourself.

― Shannon L. Alder