There is only one valid definition of business purpose: to create a customer.

― Peter F. Drucker

In today’s world money is the prime mover in almost every decision made. Money is the raison d’etre in how we are managed, and what we perceive as correct. It has become a surrogate for what is morally correct, and technically proper. Fundamentally money is just a tool. Instead of balance and some sort of holistic attitude toward what we do, money ends up being the core of meaning. The signaling from society is clear and unambiguous; money is what matters. In the process of making this tool the center of focus, we lose the focus on reality. In terms of business and those who make decisions for business, money is all that matters. If it puts more money in the pockets of those in power (i.e., the stockholder), it is by current definition a good decision. The flow and availability of money is maximized by a short business cycle, and an utter lack of long-term perspective. In the work that I do, we define the correctness of our work by whether money is allocated for it. This attitude has led us toward some really disturbing outcomes.

What gets measured gets improved.

—Peter Drucker

Stepping away from the big picture of science for a moment is instructive in seeing how money distorts things. Medicine and medical care is a good example of the sort of abominable things that money does for decision making. The United States spends an immense amount of its aggregate wealth on health care, yet the outcomes for Americans are poor. For people with lots of resources (i.e., money) the health care is better than anywhere in the World. For the common person the health care approaches second World standards, and for the poor third World standards. The outcomes for our citizens follow these outcomes in terms of life expectancy. The reasons for our terrible health care system are clear, as the day is long, money. More specifically, the medical system is tied to profit motive rather than responsibility and ethics resulting in outcomes being directly linked to people’s ability to pay. The moral and ethical dimension of health care in the United States is indefensible and appalling. It is because money is the prime mover for decisions. Worse yet, the substandard medical care for most of our citizens is a drain on society, produces awful results, but provides a vast well of money for the rich and wealthy to leech off of.

Money is a tool. Period. Computers are tools too. When tools become reasons and central organizing principles we are bound to create problems. I’ve written volumes on the issues created by the lack of perspective on computers as tools as opposed to ends unto themselves. Money is similar in character. In my world these two issues are intimately linked, but the problems with money are broader. Money’s role as a tool is a surrogate for value and worth, and can be exchanged for other things of value. Money’s meaning is connected to the real world things it can be exchanged for. We have increasingly lost this sense and put ourselves in a position where value and money have become independent of each other. This independence is truly a crisis and leads to severe misallocation of resources. At work, the attitude is increasingly “do what you are paid to do” “the customer is always right” “we are doing what we get funded to do”. The law, training and all manner of organizational tools, enforces all of this. This shadowed by business where the ability to make money justifies anything. We trade, destroy and carve up businesses so that stockholders can make money. All sense of morality, justice, and long-term consequence is scarified if money can be extracted from the system. Today’s stock market is built to create wealth in this manner, and legally enforced. The true meaning of the stock market is a way of creating resources for businesses to invest and grow. This purpose has been completely lost today, and the entire apparatus is in place to generate wealth. This wealth generation is done without regard for the health of the business. Increasingly we have used business as the model for managing everything. To its disservice, science has followed suit and lost the sense of long-term investment by putting business practice into use to manage research. In many respects the core religion in the United States is money and profit with its unquestioned supremacy as an organizing and managing principle.

So much of what we call management consists of making it difficult for people to work.

—Peter Drucker

For example I noted how horribly we are managing a certain program at work, how poorly the work is suited toward the espoused outcomes. The response to this is always, “this is the program we could get funded.” Instead of doing what has value or what is needed, we construct programs to get money. Increasingly, the way we are managed pushes a deep level of accountability to the money instead of value and purpose. The workplace messaging is “only work on what you are paid to do.” Everything we do is based on the customer who is writing the checks. The vacuous and shallow end results of this management philosophy are clear. Instead of doing the best thing possible for real world outcomes, we propose what people want to hear and what is easily funded. Purpose, value and principles are all sacrificed for money. The biggest loss is the inability to deal with difficult issues or get to the heart of anything subtle. The money is increasingly uncoordinated and nothing is tied to large objectives. In the trenches people simply work on the thing they are being paid by and learn to not ask difficult questions or think in the long term. The customer cares nothing about the career development or expertise of those they fund. In the process of money first our career development and National scientific research is plummeting and in free fall whether we look at National Labs or Universities.

There is nothing quite so useless as doing with great efficiency something that should not be done at all.

—Peter Drucker

At the heart of the matter is difficulty with long-term value. The impact of short-term thinking is clear in business. Short term drives are great for making money for stockholders, the more activity in the stock market, the better. The long term health of business is always lost to the possibility of making more money in the now. By the same token, the short-term thinking is terrible for value to society and leads to many businesses simply being chewed up and spit out. Unfortunately our society has adopted the short term thinking for everything including science. All activities are measured quarterly (or even monthly) against the funded plans. Organizations are driving everyone to abide by this short-term thinking. No one can use their judgment or knowledge gained to change this for values that transcend money. The result is a complete loss of long-term perspective in decision-making. We have lost the ability to care for the health and growth of careers. The defined financial path has become the only arbiter of right and wrong. All of our judgment is based on money, if its funded, it is right, if it isn’t funded its wrong. More and more all the long-term interests aren’t funded, so our future whither right in front of us. The only ones benefiting from the short-term thinking are a small number of the wealthiest people in society. Most people and society itself are left behind, but forced to serve their own demise.

Long-range planning does not deal with the future decisions, but with the future of present decisions.

—Peter Drucker

Doing something better is relatively easy to devise, but seemingly impossible to implement in the near term. Large parts of the problem are laws that favor short-term interests and profit taking over long-term investment. These laws are entirely created to maximize the personal wealth creation. Instead laws are needed to maximize the societal creation of wealth, which is invariably long-term in perspective. We could bias the system in favor of long-term investment. Part of the answer is the tax system. Currently the system of taxation is completely oriented toward short-term and wealth creation for individuals. The attitude today is that if you can make lots of money; it is correct. This perspective needs to change to something more nuanced. We need to push a balance of this idea with value, impact and the long-term perspective. Ultimately this will require people in power to sacrifice wealth now, for more wealth in the future. People need to receive a significant benefit for putting off short-term profit to take the long-term perspective. We need to overhaul how science is done. The notably long-term investment is research must be recovered and freed from the business ideas that are destroying the ability of science to create value. The idea that business practices today are correct is utterly perverse and damaging.

Rank does not confer privilege or give power. It imposes responsibility.

― Peter F. Drucker

The problem with making these changes is primarily those who benefit from the current system. A small number of the most powerful and wealthy in society are significantly advantaged. They will work steadfastly to keep the current system in place because it benefits them. Everyone else can be damned and in many cases the powerful care little about society at large (some wealthy people seem to have adopted a more generous attitude, Bill Gates, Warren Buffet come to mind). Money having value over real world things is to their advantage. Creating a system that benefits all of society hurts them. This is true in the short term, but in the longer term it creates less overall wealth. We need a realization of the long-term effects of current attitudes and policies as a loss to everyone. A piece of this puzzle is a greater degree of responsibility for the future on the part of the rich and powerful. Our leaders need to work for the benefit of everyone, not for their accumulation of more wealth and power. Until this fact becomes more evident to the population as a whole we can expect the wealthy and powerful to continue to favor a system that benefits himself or herself to exclusion of everyone else.

Doing the right thing is more important than doing the thing right.

—Peter Drucker

Part of the overall puzzle is overcoming the infatuation with using business models to manage everything including science. It isn’t necessarily incompatible with the best interests of science, but today’s business practices are utterly orthogonal to good science.

Management is doing things right. Leadership is doing the right things.

—Peter Drucker

Only those who dare to fail greatly can ever achieve greatly.

― Robert F. Kennedy

In modeling and simulation numerical error is an extremely important yet generally unsatisfactorily understood thing. For general nonlinear problems dominating the use and utility of high performance computing, the state of affairs is quite incomplete. It has a central role in modeling and simulation making our gaps in theory, knowledge and practice rather unsettling. Theory is strong for linear problems where solutions are well behaved and smooth (i.e., continuously differentiable, or a least many derivatives exist). Almost every problem of substance driving National investments in computing is nonlinear and rough. Thus, we have theory that largely guides practice by faith rather than rigor. We would be well served by a concerted effort to develop theoretical tools better suited to our reality.

Sometimes a clearly defined error is the only way to discover the truth

― Benjamin Wiker

We have a fundamental existence theory for convergent solutions defined by Lax’s early work (the fundamental theorem of numerical analysis). It is quite limited, rigorously applying to linear differential equations, yet defining basic approaches to numerical approximations for models that are almost invariably nonlinear. The theorem states that when a stable approximation is consistent (approximates the differential equation properly), it will converge to the correct solution. By convergent we mean that the solution approaches the exact solution is the manner of approximation grows closer to a continuum, which is associated with small discrete steps/mesh and more computational resource. This theorem provides the basis and ultimate drive for faster, more capable computing. We apply it most of the time where it is invalid. We would be greatly served by having a theory that is freed of these limits. Today we just cobble together a set of theories, heuristics and lessons into best practices and we stumble forward.

Part of making use of this fundamental theorem is producing a consistent approximation to the model of choice. The tool for accomplishing this is a thing like Taylor series, polynomials and finite elements. All of these methods depend to some degree on solutions being well behaved and nice. Most of our simulations are neither well behaved nor nice. We assume an idealized nice solution then approximate using some neighborhood of discrete values. Sometimes this is done using finite differences, or cutting the world into little control volumes (equivalent in simple cases), or creating finite elements and using variational calculus to make approximations. In all cases the underlying presumption is smooth, nice solutions while most of the utility of approximations violates these assumptions. Reality is rarely well behaved or nice, so we have a problem. Our practice has done reasonably well and taken us far, but a better more targeted and useful theory might truly unleash innovation and far greater utility.

The aim of science is not to open the door to infinite wisdom, but to set a limit to infinite error.

― Bertolt Brecht

We don’t really know what happens when the theory falls apart, and simply rely upon bootstrapping ourselves forward. We have gotten very far with very limited theory, and simply moving forward largely on faith. We do have some limited theoretical tools, like conservation principles (Lax-Wendroff’s theorem), and entropy solutions (converging toward solutions associated with viscous regularization consistent with the second law of thermodynamics). The thing we miss is general understanding of what is guiding accuracy and defining error in these cases. We cannot design methods specifically to produce accurate solution in these circumstances and we are guided by heuristics and experience rather than rigorous theory. A more rigorous theoretical construct would provide a springboard for productive innovation. Let’s look at a few of the tools available today to put things in focus.

One of the first things one encounters in putting together discrete approximations in realistic circumstances is a choice. For nonlinear features leading to general and rough solutions, one can decide to track features in the solution explicitly. The archetype of this is shock tracking where the discrete evolution of a shock wave is defined explicitly in the approximation. In essence the shock wave (or whatever wave is tracked) becomes an internal boundary condition allowing regular methods to be used everywhere else. This typically involves the direct solution of the Rankine-Hugoniot relations (i.e. the shock jump conditions, algebraic relations holding at a discontinuous wave). The problems with this approach are extreme, including unbounded complexity if all waves are tracked, or with solution geometry in multiple dimensions. This choice has been with us since the dawn of computation including the very first calculations at Los Alamos that used this technique, but it rapidly becomes untenable.

To address the practical aspects of computation shock capturing methods were developed. Shock capturing implicitly computes the shock wave on a background grid through detecting its presence and adding a physically motivated dissipation to stabilize its evolution. This concept has made virtually all of computational science possible. Even when tracking methods are utilized the explosion of complexity is tamed by resorting to shock capturing away from the dominant features being tracked. The origin of the concept came from Von Neumann in 1944, but lacked a critical element for success, dissipation or stabilization. Richtmyer added this critical element with artificial viscosity in 1948 while working at Los Alamos on problems whose complexity was advancing beyond the capacity of shock tracking to deal with. Together Von Neumann’s finite differencing scheme and Richtmyer’s viscosity enabled shock capturing. It was a proof of principle and its functionality was an essential springboard for others to have faith in computational science.

What one recognizes is that when dealing with shock wave physics must be added to the discrete representation. This happens explicitly in tracking where the shock itself becomes as discrete element or implicit with shock capturing where the approximation is adapted using the physics of shocks. Of course, shock capturing is useful for more than just shocks. It can be used to stabilize the computation of any feature. The overall methodology has some additional benefits not immediately recognized by its originators. For computing turbulence without fully resolving features shock capturing methods are essential (i.e., not DNS, but DNS can be criticized in its practice). Large eddy simulation was born out of adding the original Richtmyer-Von Neumann viscosity to weather modeling, and resulted in the creation of the Smagorinsky eddy viscosity. Other shock capturing methods developed for general purposes have provided the means for implicit Large Eddy Simulation. These methods all have the same origin, and rely upon the basic principles of shock capturing. The fact that all of this has the same origin almost certainly has a deep meaning that is lost in most of today’s dialog. We would be well served by aggressively exploring these connections in an open-minded and innovative fashion.

One of the key things about all of this capability is the realization of how heuristic it is at its core. Far too much of what we currently do in computational science is based upon heuristics, and experience gained largely through trial and error. Far too little is based upon rigorous theory. The advancement of our current approaches through theory would be a great service to the advancement of the field. Almost none of the current efforts are remotely associated with advancing theory. If one gets down to brass tacks about the whole drive for exascale, we see that it is predicated on the concept of convergence whose theoretical support is extrapolated from circumstances that don’t apply. We are really on thin ice, and stunningly unaware of the issues. This lack of awareness then translates to lack of action, lack of priority, lack of emphasis and ultimately lack of money. In today’s world if no one pays for it, it doesn’t happen. Today’s science programs are designed to be funded, rather than designed to advance science. No one speaks out about how poorly thought through our science programs are; they simply are grateful for the funding.

When I was a kid, they had a saying, ‘to err is human but to really fuck it up takes a computer.’

― Benjamin R. Smith

There are a host of technologies and efforts flowing out from our current efforts that could all benefit from advances in theory for numerical approximation. In addition to the development of larger computers, we see the application of adaptive mesh refinement (AMR) to define enhanced resolution. AMR is even more highly bootstrapped and leveraged in terms of theory. By the same token, AMR’s success is predicated on best practices and experience from a wealth of applications. AMR is an exciting technology that produces stunning results. Better and more appropriate theory can turn these results from the flashy graphics AMR produces to justifiable credible results. A big part of moving forward is putting verification and validation into practice. Both activities are highly dependent on theory that is generally weak or non-existent. Our ability to rigorously apply modeling and simulation to important societal problems is being held back by our theoretical failings.

Another area with critical importance and utter lack of support is subgrid closure modeling especially where it depends on the mesh scale itself. The general thinking about closure modeling is completely haphazard and heuristic. The combination of numerical modeling and closure at the mesh scale is poorly thought out, and generally lacking any theoretical support. Usually the closure models are tied directly to the mesh scale, yet numerical methods rarely produce good solutions on the smallest mesh, but rather over a number of mesh cells (or elements). We rarely think about we defined or resolved solution structures and how it connects to modeling. Instead models are thought of solely geometrically in terms of scale and tied to the mesh scale. As a result we don’t have consistency between our mesh, numerical solution and the resolution-fidelity of the numerical method. Often this leaves the modeling in the code as being completely mesh-dependent, and produces no chance of mesh independence.

A big issue is a swath of computational science where theory is utterly inadequate much of it involving chaotic solutions where there is extreme dependence on initial conditions. Turbulence is the classical problem most closely related to this issue. Our current theory and rigorous understand is vastly inadequate to spur progress. In most cases we are let down by both the physics modeling, mathematical and numerical theory. In every case we have weak to non-existent rigor leading to heuristic filled models and numerical solvers. Extensions of any of this work are severely hampered by the lack of theory (think higher order accuracy, uncertainty quantification, optimization,…). We don’t know how any of this converges, we just act like it does and use it to justify most of our high performance computing investments. All of our efforts would be massively assisted by almost any progress theoretically. Most of the science we care about is chaotic at a very basic level and lots of interesting things are utterly dependent on understanding this better. The amount of focus on this matter is frightfully low.

My overall view is that the lack of investment and attention to our theoretical shortcomings is a significant burden. The flipside is the loss of a massive opportunity to make some incredible advances. Instead of solving a whole new class of problems powered by deeper understanding of physics and mathematics, we are laboring under vast gaps. This lowers the effectiveness of everything we do, and every dollar we spend. While a focus on advancing theory and understanding is quite risky, the benefits are extreme. If we are not prepared to fail, we will not succeed.

Success is not built on success. Not great success. Great success is built on failure, frustration, even catastrophe.

— Sumner Redstone

Lax, Peter D., and Robert D. Richtmyer. “Survey of the stability of linear finite difference equations.” Communications on pure and applied mathematics 9, no. 2 (1956): 267-293.

Von Neumann, John. “Proposal and analysis of a new numerical method for the treatment of hydrodynamical shock problems.” The collected works of John von Neumann 6 (1944).

Richtmyer, R. D. “Proposed numerical method for calculation of shocks.” LANL Report, LA 671 (1948): 1-18.

VonNeumann, John, and Robert D. Richtmyer. “A method for the numerical calculation of hydrodynamic shocks.” Journal of applied physics 21, no. 3 (1950): 232-237.

Mattsson, Ann E., and William J. Rider. “Artificial viscosity: back to the basics.” International Journal for Numerical Methods in Fluids 77, no. 7 (2015): 400-417.

Richtmyer, Robert D., and Keith W. Morton. “Difference methods for initial-value problems.” Malabar, Fla.: Krieger Publishing Co.,| c1994, 2nd ed. (1994).

Smagorinsky, Joseph. “General circulation experiments with the primitive equations: I. The basic experiment.” Monthly weather review 91, no. 3 (1963): 99-164.

Smagorjnsky, Joseph. “The beginnings of numerical weather prediction and general circulation modeling: early recollections.” Advances in Geophysics 25 (1983): 3-37.

Boris, J. P., F. F. Grinstein, E. S. Oran, and R. L. Kolbe. “New insights into large eddy simulation.” Fluid dynamics research 10, no. 4-6 (1992): 199-228.

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

Margolin, Len G., and William J. Rider. “A rationale for implicit turbulence modelling.” International Journal for Numerical Methods in Fluids 39, no. 9 (2002): 821-841.

We are losing the ability to understand anything that’s even vaguely complex.

― Chuck Klosterman

I get asked, “what do you do?” quite often in conversation, and I realize the truth needs to be packaged carefully for most people. One of my issues is that advertise what I do on my body with some incredibly nerdy tattoos including an equation that describes one form of the second law of thermodynamics. What I do is complex and highly technical full of incredible subtlety. Even when talking with someone from a nearby technical background the subtlety of approximating physical laws numerically in a manner suitable for computing can be daunting. For someone without a technical background it is positively alien. This character comes to play rather acutely in the design and construction of research programs where complex, technical and subtle does not sell. This is especially true in today’s world where expertise and knowledge is regarded as suspicious, dangerous and threatening to so many. In today’s world one of the biggest insults to hurl at some one is to accuse them of being one of the “elite”. Increasingly it is clear that this isn’t just an American issue, but Worldwide in its scope. It is a clear and present threat to a better future.

I’ve written often about the sorry state of high performance computing. Our computing programs are blunt and naïve constructed to squeeze money out of funding agencies and legislatures rather then get the job done. The brutal simplicity of the arguments used to support funding is breathtaking. Rather than construct programs to be effective and efficient getting the best from every dollar spent, we construct programs to be marketed at the lowest common denominator. For this reason something subtle, complex and technical like numerical approximation gets no play. In today’s world subtlety is utterly objectionable and a complete buzz kill. We don’t care that it’s the right thing to do, or that it is massively greater in return than simply building giant monstrosities of computing. It would take an expert from the numerical elite to explain it, and those people are untrustworthy nerds, so we will simply get the money to waste on the monstrosities instead. So here I am, an expert and one of the elite using my knowledge and experience to make recommendations on how to be more effective and efficient. You’ve been warned.

Truth is much too complicated to allow anything but approximations.

— John Von Neumann

If we want to succeed at remaining a high performance computing superpower, we need change our approach and fast. Part of what is needed is a greater focus on numerical approximation. This is part of deep need to refocus on the more valuable aspects of the scientific computing ecosystem. The first thing to recognize is that our current hardware first focus is oriented on the least valuable part of the ecosystem, the computer itself. A computer is necessary, but horribly insufficient for high performance computing supremacy. The real value for scientific computing is the opposite end of the spectrum where work is grounded in physics, engineering and applied mathematics.

Although this may seem a paradox, all exact science is dominated by the idea of approximation.

— Bertrand Russell

I’ve made this argument before and it is instructive to unpack it. The model solved via simulation is the single most important aspect of the simulation. If the model is flawed, no amount of raw computer speed, numerical accuracy, or efficient computer code can rescue the solution and make it better. The model must be changed, improved, or corrected to produce better answers. If a model is correct the accuracy, robustness, fidelity and efficiency of its numerical solution is essential. Everything upstream of the numerical solution aimed toward the computer hardware is less important. We can move down the chain of activities all of which are necessary seeing the same effect, the further you get from the model of reality, the less efficient the measures are. This whole thing is referred to an ecosystem these days and every bit of it needs to be in place. What also needs to be in place is a sense of the value of each activity, and priority placed toward those that have the greatest impact, or the greatest opportunity. Instead of doing this today, we are focused on the thing with least impact, farthest from reality and starving the most valuable parts of the ecosystem. One might argue that the hardware is a subject of opportunity, but the truth is the opposite. The environment for improving the performance of hardware is at a historical nadir; Moore’s law is dead, dead, dead. Our focus on hardware is throwing money at an opportunity that has passed into history.

I’m a physicist, and we have something called Moore’s Law, which says computer power doubles every 18 months. So every Christmas, we more or less assume that our toys and appliances are more or less twice as powerful as the previous Christmas.

— Michio Kaku

At some point, Moore’s law will break down.

— Seth Lloyd

There is one word to describe this strategy, stupid!

At the core of the argument is a strategy that favors brute force over subtleties understood mainly by experts (or the elite!). Today the brute force argument always takes the lead over anything that might require some level of explanation. In modeling and simulation the esoteric activities such as the actual modeling and its numerical solution are quite subtle and technical in detail compared to the raw computing power that can be understood with ease by the layperson. This is the reason the computing power gets the lead in the program, not because of its efficacy in improving the bottom line. As a result our high performance-computing world is dominated by meaningless discussions of computing power defined by a meaningless benchmark. The political dynamics is basically a modern day “missile gap” like we had during the Cold War. It has exactly as much virtue as the original “missile gap”; it is a pure marketing and political tool with absolutely no technical or strategic validity aside from its ability to free up funding.

Each piece, or part, of the whole of nature is always merely an approximation to the complete truth, or the complete truth so far as we know it. In fact, everything we know is only some kind of approximation because we know that we do not know all the laws as yet.

— Richard P. Feynman

Once you have an entire program founded on bullshit arguments, it is hard to work your way back to technical brilliance. It is easier to double down on the bullshit and simply define everything in terms of the original fallacies. A big part of the problem is the application of modern verification and validation to the process. Both verification and validation are modern practices to accumulate evidence on the accuracy, correctness and fidelity of computational simulations. Validation is the comparison of simulation with experiments and in this comparison the relative correctness of models is determined. Verification determines the correctness and accuracy of the numerical solution of the model. Together the two activities should help energize high quality work. In reality most programs consider them to be nuisances and box checking exercises to be finished and ignored as soon as possible. Programs like to say they are doing V&V, but don’t want to emphasize or pay for doing it well. V&V is a mark of quality, but the programs want its approval rather than attend to its result. Even worse, if the results are poor or indicate problems, they are likely to be ignored or dismissed as being inconvenient. Programs get away with this because the practice of V&V is technical and subtle and in the modern world highly susceptible to bullshit.

Far better an approximate answer to the right question, which is often vague, than an exact answer to the wrong question, which can always be made precise.

— John W. Tukey

Numerical methods for solving models are even more technical and subtle. As such they are the focus of suspicion and ignorance. For high performance computing today they are considered to be yesterday’s work and largely a finished, completed product now simply needing a bigger computer to do better. In a sense this notion is correct, the bigger computer will produce a better result. The issue is that using the computer power, as the route to improvement is inefficient under the best of circumstances. We are not living under of the best of circumstances! Things are far from efficient, as we have been losing the share of computer power advances useful for modeling and simulation for decades now. Let us be clear, when we receive an ever-smaller proportion of the maximum computing power as each year passes. Thirty years ago we would commonly get 10, 20 or even 50 percent of the peak performance of the cutting edge supercomputers. Today even one percent of the peak performance is exceptional, and most codes doing real application work are significantly less than that. Worse yet, this dismal performance is getting worse with every passing year. This is one element of the autopsy of Moore’s law that we have been avoiding while its corpse rots before us.

So we are prioritizing improvement in an area where the payoffs are fleeting and suboptimal. Even these improvements are harder and harder to achieve as computers become ever more parallel and memory access costs become ever more extreme. Simultaneously we are starving more efficient means of improvement of resources and emphasis. Numerical methods and algorithms are two key areas not getting any significant attention or priority. Moreover support for these areas is actually diminishing so that support for the inefficient hardware path can be increased. Let’s not mince words; we are emphasizing a crude naïve and inefficient route to improvement at the cost of a complex and subtle route that is far more efficient and effective.

Numerical approximations and algorithms are complex and highly technical things poorly understood by non-experts even if they are scientists. The relative merits of one method or algorithm compared to another is difficult to articulate. The merits and comparison is highly technical and subtle. Since creating new methods and algorithms makes progress, this means improvements are hard to explain and articulate to non-experts. In some cases both methods and algorithms can produce breakthrough results and produce huge speed-ups. These cases are easy to explain. More generally a new method or algorithm produces subtle improvements like more robustness or flexibility or accuracy than the older options. Most of these changes are not obvious, but making this progress over time leads to enormous improvements that swamp the progress made by faster computers.

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

The huge breakthroughs are far and few between but provide much greater value than any hardware over similar periods of time. To get these huge breakthroughs requires continual investment in research for extended periods of time. For much of the time the research is mostly a failure producing small or non-existent improvements, until they don’t. Without the continual investment, the failure and the expertise failure produces, the breakthroughs will not happen. They are mostly serendipitous and the end product of many unsuccessful ideas. Today the failures and lack of progress is not supported; we exist in a system where insufficient trust exists to support the sort of failure needed for progress. The result is the addiction to Moore’s law and its seemingly guaranteed payoff because it frees us from subtlety.

Often a sign of expertise is noticing what doesn’t happen.

― Malcolm Gladwell

A huge aspect of expertise is the taste for subtlety. Expertise is built upon mistakes and failure just as basic learning is. Without the trust to allow people to gloriously make professional mistakes and fail in the pursuit of knowledge, we cannot develop expertise or progress. All of this lands heavily on the most effective and difficult aspects of scientific computing, the modeling and solution of the models numerically. Progress on these aspects is both highly rewarding in terms of improvement, and very risky being prone to failure. To compound matters progress is often highly subjective itself needing great expertise to explain and be understood. In an environment where the elite are suspect and expertise is not trusted such work is unsupported. This is exactly what we see, the most important and effective aspects of high performance computing are being starved in favor of brutish and naïve aspects, which sell well. The price we pay for our lack of trust is an enormous waste of time, money and effort.

Wise people understand the need to consult experts; only fools are confident they know everything.

― Ken Poirot

Again, I’ll note that we still have so much to do. Numerical approximations for existing models are inadequate and desperately in need of improvement. We are burdened by theory that is insufficient and heavily challenged by our models. Our models are all flawed and the proper conduct of science should energize them to improve.

…all models are approximations. Essentially, all models are wrong, but some are useful. However, the approximate nature of the model must always be borne in mind… [Co-author with Norman R. Draper]

— George E.P. Box

The fundamental law of computer science: As machines become more powerful, the efficiency of algorithms grows more important, not less.

― Nick Trefethen

Modern modeling and simulation is viewed as a transformative technology for science and engineering. Invariably the utility of modeling and simulation is grounded on the solution of models via numerical approximations. The fact that numerical approximation is the key to unlocking its potential seems largely lost in the modern perspective, and engaged in any increasingly naïve manner. For example much of the dialog around high performance computing is predicated on the notion of convergence. In principle, the more computing power one applies to solving a problem, the better the solution. This is applied axiomatically and relies upon a deep mathematical result in numerical approximation. This heritage and emphasis is not considered in the conversation to the detriment of its intellectual depth.

Where all think alike there is little danger of innovation.

― Edward Abbey

At this point, the mathematics and specifics of numerical approximation is then systematically ignored by the dialog. The impact of this willful ignorance is felt across the modeling and simulation world, a general lack of progress and emphasis on numerical approximation is evident. We have produced a situation where the most valuable aspect of numerical modeling is not getting focused attention. People are behaving as if the major problems are all solved and not worthy of attention or resources. The nature of the numerical approximation is the second most important and impactful aspect of modeling and simulation work. Virtually all the emphasis today is on the computers themselves based on the assumption of their utility in producing better answers. The most important aspect is the modeling itself; the nature and fidelity of the models define the power of the whole process. Once a model has been defined, the numerical solution of the model is the second most important aspect. The nature of this numerical solution is most dependent on the approximation methodology rather than the power of the computer.

The uncreative mind can spot wrong answers, but it takes a very creative mind to spot wrong questions.

― Anthony Jay

People act as if the numerical error is so small as not to be important on one hand, while encouraging great focus on computing power where the implicit reasoning for the computing power is founded on reducing numerical error. To make matters worse with this corrupt logic, the most effective way to reduce numerical error is being starved for attention and resources having little or no priority. The truth is that numerical errors are still too large, and increasing computing power is lousy way and inefficient to make them smaller. We are committed to a low-risk path that is also highly inefficient because the argument is accessible to the most naïve people in the room.

What is important is seldom urgent and what is urgent is seldom important.

― Dwight D. Eisenhower

Another way of getting to the heart of the issue is the efficacy of using gains in computer power to get better solutions. Increases in computing power are a terrible way to produce better results; it is woefully inefficient. One simply needs to examine the rate of solution improvement based on scaling arguments. First, we need to recognize that practical problems converge quite slowly in terms of the application of enhanced computational resources. For almost any problem of true real world applicability, high-order convergence (higher than first-order) is never seen. Generally we might expect solutions to improve at first-order with the inverse of mesh size. If we look at three dimensional, time dependent problems and we want to halve the numerical error, we need to apply at least 16 times the computing power. Usually convergence rates are less than first order, so the situation is actually even worse. As a result we are investing an immense amount in progressing in an incredibly inefficient manner, and starving more efficient means of progress. To put more teeth on the impact of current programs, the exascale initiative wants to compute things fifty times better, which will only result is reducing errors by slightly more than one half. So we will spend huge effort and billions of dollars in making numerical errors smaller by half. What an utterly shitty return on investment! This is doubly shitty when you realize that so much more could be done to improve matter by other means.

The first thing we need to recognize for progress is relative efficacy of different modes of investment. The most effective way to progress in modeling and simulation are better models. Better models require work on theory and experiment with deeply innovative thinking based on inspiration and evidence of limitations of current theory and modeling. For existing and any new models the next step is solving the models numerically. This involves detailed and innovative numerical approximations of the models. The power of modeling and simulation with computers is predicated on the ability to solve complex models that cannot be understood analytically (or analytically without severe restrictions or assumptions). The fidelity of the numerical approximations is the single most effective way to improve results once modeling errors have been addressed. Numerical approximations can make a huge difference in the accuracy of simulations far more effectively than computer power.

Don’t tell me about your effort. Show me your results.

― Tim Fargo

So why are we so hell bent on investing in a more inefficient manner of progressing? Our mindless addiction to Moore’s law providing improvements in computing power over the last fifty years for what in effect has been free for the modeling and simulation community.

Our modeling and simulation programs are addicted to Moore’s law as surely as a crackhead is addicted to crack. Moore’s law has provided a means to progress without planning or intervention for decades, time passes and capability grows almost if by magic. The problem we have is that Moore’s law is dead, and rather than moving on, the modeling and simulation community is attempting to raise the dead. By this analogy, the exascale program is basically designed to create zombie computers that completely suck to use. They are not built to get results or do science, they are built to get exascale performance on some sort of bullshit benchmark.

This gets to the core of the issue, our appetite for risk and failure. Improving numerical approximations is risky and depends on breakthroughs and innovative thinking. Moore’s law has sheltered the modeling and simulation community from risk and failure in computing hardware for a very long time. If you want innovation you need to accept risk and failure; innovation without risk and failure simply does not happen. We are intolerant of risk and failure as a society, and this intolerance dooms innovation literally strangling it to death in its crib. Moore’s law allowed progress without risk, as if it came for free. The exascale program will be the funeral pyre for Moore’s law and we are threatening the future of modeling and simulation with our unhealthy addiction to it.

If failure is not an option, then neither is success.

― Seth Godin

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

― Paulo Coelho

The key thing to realize about this discussion is that improving numerical approximations is risky and highly prone to failure. You can invest in improving numerical approximations for a very long time without any seeming progress until one gets a quantum leap in performance. The issue in the modern world is the lack of predictability to such improvements. Breakthroughs cannot be predicted and cannot be relied upon to happen on a regular schedule. The breakthrough requires innovative thinking and a lot of trial and error. The ultimate quantum leap in performance is founded on many failures and false starts. If these failures are engaged in a mode where we continually learn and adapt our approach, we eventually solve problems. The problem is that it must be approached as an article of faith, and cannot be planned. Today’s management environment is completely intolerant of such things, and demands continual results. The result is squalid incrementalism and an utter lack of innovative leaps forward.

Civilizations… cannot flourish if they are beset with troublesome infections of mistaken beliefs.

― Harry G. Frankfurt

What is the payoff for methods improvement?

If we improve a method we can achieve significantly better results without a finer computational mesh. This results in a large saving in computational cost as long as the improved method isn’t too expensive. As I mentioned before one needs 16 times the computational resources to knock error down by half for a 3-D time dependent calculation. If I produce a method with half the error, it can be more efficient if it is less than 16 times as expensive. In other word, the method can use 16 times the computational resource and still be more efficient. This is a lot of headroom to work with!

The most dangerous ideas are not those that challenge the status quo. The most dangerous ideas are those so embedded in the status quo, so wrapped in a cloud of inevitability, that we forget they are ideas at all.

― Jacob M. Appel

For some cases the payoff is far more extreme than these simple arguments. The archetype of this extreme payoff is the difference between first and second order monotone schemes. For general fluid flows, second-order monotone schemes produce results that are almost infinitely more accurate than first-order. The reason for this stunning claim are acute differences in the results comes from the impact of the form of the truncation error expressed via the modified equations (the equations solved more accurately by the numerical methods). For first-order methods there is a large viscous effect that makes all flows laminar. Second-order methods are necessary for simulating high Reynolds number turbulent flows because their dissipation doesn’t interfere directly with the fundamental physics.

As technology advances, the ingenious ideas that make progress possible vanish into the inner workings of our machines, where only experts may be aware of their existence. Numerical algorithms, being exceptionally uninteresting and incomprehensible to the public, vanish exceptionally fast.

― Nick Trefethen

We don’t generally have good tools for numerical error approximation in non-standard (or unresolved) cases. One digestion of one of the key problems is found in Banks, Aslam, Rider where sub-first-order convergence is described and analyzed for solutions of a discontinuous problem for the one-way wave equation. The key result in this paper is the nature of mesh convergence for discontinuous or non-differentiable solutions. In this case we see sub-linear fractional order convergence. The key result is a general relationship between the convergence rate and the formal order of accuracy for the method, , which is . This comes from the analysis of the solution to the modified equation including the leading order truncation error. For nonlinear discontinuous solutions, the observed result is first-order where one establishes a balance between the regularization and the self-steepening in shock waves. At present there is no theory of what this looks like theoretically. Seemingly this system of equations could be analyzed as we did for the linear equations. Perhaps this might provide guidance for numerical method development. It would seemingly be worthy progress if we could analyze such systems more theoretically providing a way to understand actual accuracy.

Another key limitation of existing theory is chaotic solutions classically associated with turbulent or turbulent-like flows. These solutions are extremely (perhaps even infinitely) sensitive to initial conditions. It is impossible to get convergence results for point values, and the only convergence is for integral measures. These measures are generally convergent very slowly and they are highly mesh-dependent. This issue is huge in high performance computing. One area of study is measure-valued solutions where convergence is examined statistically. This is a completely reasonable approach for convergence of general solutions to hyperbolic PDE’s.

The much less well-appreciated aspect comes with the practice of direct numerical simulation of turbulence (DNS really of anything). One might think that having a DNS would mean that the solution is completely resolved and highly accurate. They are not! Indeed they are not highly convergent even for integral measures. Generally speaking, one gets first-order accuracy or less under mesh refinement. The problem is the highly sensitive nature of the solutions and the scaling of the mesh with the Kolmogorov scale, which is a mean squared measure of the turbulence scale. Clearly there are effects that come from scales that are much smaller than the Kolmogorov scale associated with highly intermittent behavior. To fully resolve such flows would require the scale of turbulence to be described by the maximum norm of the velocity gradient instead of the RMS.

If you want something new, you have to stop doing something old

― Peter F. Drucker

When we get to the real foundational aspects of numerical error and limitations, we come to the fundamental theorem of numerical analysis. For PDEs it only applies to linear equations and basically states that consistency and stability is equivalent to convergence. Everything is tied to this. Consistency means you are solving the equations in a valid and correct approximation, stability is getting a result that doesn’t blow up. What is missing is the theoretical application to more general nonlinear equations along with deeper relationships to accuracy, consistency and stability. This theorem was derived back in the early 1950’s and we probably need something more, but there is no effort or emphasis on this today. We need great effort and immensely talented people to progress. While I’m convinced that we have no limit on talent today, we lack effort and perhaps don’t develop or encourage the talent to develop appropriately.

Beyond the issues with hardware emphasis, today’s focus on software is almost equally harmful to progress. Our programs are working steadfastly on maintaining large volumes of source code full of the ideas of the past. Instead of building on the theory, methods, algorithms and idea of the past, we are simply worshiping them. This is the construction of a false ideology. We would do far greater homage to the work of the past if we were building on that work. The theory is not done by a long shot. Our current attitudes toward high performance computing are a travesty, and embodied in a national program that makes the situation worse only to serve the interests of the willfully naive. We are undermining the very foundation upon which the utility of computing is built. We are going to end up wasting a lot of money and getting very little value for it.

We now live in a world where counter-intuitive bullshitting is valorized, where the pose of argument is more important than the actual pursuit of truth, where clever answers take precedence over profound questions.

― Ta-Nahisi Coates

Margolin, Len G., and William J. Rider. “A rationale for implicit turbulence modelling.” International Journal for Numerical Methods in Fluids 39, no. 9 (2002): 821-841.

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

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

Fjordholm, Ulrik S., Roger Käppeli, Siddhartha Mishra, and Eitan Tadmor. “Construction of approximate entropy measure-valued solutions for hyperbolic systems of conservation laws.” Foundations of Computational Mathematics (2015): 1-65.

Lax, Peter D., and Robert D. Richtmyer. “Survey of the stability of linear finite difference equations.” Communications on pure and applied mathematics 9, no. 2 (1956): 267-293.