Over the past couple of weeks I’ve experienced something very irritating time and time again. Each time I’ve been left more frustrated and angry than before. It has been a continual source of disappointment. I went into a room expecting to learn something and left knowing less than when I entered. What is it? “The finite element method”

“If you can’t explain it to a six year old, you don’t understand it yourself.” ― Albert Einstein

In short, the answer to my title is nothing at all and everything. Nothing is technically wrong with the finite element method, absolutely nothing at all. Given that nothing is wrong with it there is a lot wrong with what it does to the practice of mathematics and scientific computing. More specifically there isn’t a thing wrong with the method except how people use it, which is too damn abstractly. Much of the time the method is explained in a deep code undecipherable to anyone except a small cadre of researchers working in the field. Explaining finite elements to a six year old is a long suit, but a respectable goal. Too often you can’t explain what you’re doing to a 46 year old with a PhD unless they are part of the collective of PhD’s working directly in the field and have received the magic decoder ring during their graduate education.

A common occurrence for someone to begin their research career with papers that clearly state what they are doing, and as the researcher becomes successful, all clarity leaves their writing. I saw a talk at a meeting where a researcher who used to write clearly had simultaneously obscured their presentation while pivoting toward research on easier problems. This is utter madness! The mathematics of finite element research tends to take a method that works well on hard problems, and analyze them on simpler problems while making the whole thing less clear. One of the key reasons to work on simpler problems is to clarify, not complicate. Too often the exact opposite is done.

Sometimes this blog is about working stuff out that bugs me in a hopefully articulate way. I’ve spent most of the last month going to scientific meetings and seeing a lot of technical talks and one of the things that bugs me the most are finite element methods (FEM). More specifically the way FEM is presented. There really isn’t a lot wrong with FEM per se, it’s a fine methodology that might even be optimal for some problems. I can’t really say because its proponents so often do such an abysmal job of explaining what they are doing and why. That is the crux of the matter.

Scientific talks on the finite element method tend to be completely opaque and I walk out of them knowing less than I walked in. The talks are often given in a manner that seems to intentionally obscure the topic with the seeming objective of making the speaker seem much smarter than they actually are. I’m not fooled. The effect they have gotten is to piss me off, and cause me to think less of them. Presenting a simple problem in an intentionally abstract and obtuse way is simply a disservice to science. It serves no purpose, but to make the simple grandiose and distant. It ultimately hurts the field, deeply.

The point of a talk is to teach, explain and learn not to make the speaker seem really smart. Most FEM talks are about making the speaker seem smart instead of explaining why something works. The reality is that the simple clear explanation is actually the hallmark of intellectual virtue. Simplicity is a virtue that seems to be completely off the map with FEM, FEM is about making the simple, complex instead. To make matter more infuriating, much of the current research on FEM is focused on attacking the least important and most trivial mathematical problems instead of the difficult problems that are pacing computational science. Computational science is being paced today by issues such as multiphysics (where multiple physical effects interact to define a problem) particularly involving transport equations (defined by hyperbolic PDE’s). In addition uncertainty quantification along with verification and validation is extremely important.

Instead FEM research is increasingly focused on elliptic PDE’s, which are probably the easiest thing to solve in the PDE world. In other words, if you can solve an elliptic PDE well I know very little about the ability of a methodology’s capacity to attack the really hard important problems. It is nice, but not very interesting (the very definition of necessary and insufficient). Frankly the desire and interest in taking a method designed for solving hyperbolic PDE’s such as discontinuous Galerkin and applying it to elliptic PDE’s is worthwhile, but should not receive anywhere near the attention I see. It is not important enough to get the copious attention it is getting.

The effect is that we are focused on the areas of less importance, which has the impact of taking the methodology backwards. The research dollars are focused on less important problems instead of more important ones. Difficult important problems should be the focus of research, not the kind of “Mickey Mouse” stuff I’ve seen the whole month. On top of Mickey Mouse problems, the talks make the topic as complex as possible, and seem to be focused on trying not to explain anything in simple terms.

“Simplicity is a great virtue but it requires hard work to achieve it and education to appreciate it. And to make matters worse: complexity sells better.” ― Edgar Wybe Dijkstra

I think Dijkstra was talking about something entirely different, but the point is similar, the complexity sells and that is why it is trotted out time and time again. While it sells, it also destroys the sort of understanding that allows ideas to be extended and modified to solve new problems. The complexity tends to box ideas in rather than making them more general and less specific. There is a lot at stake beyond style, the efficacy of science is impacted by a false lack of simplicity. Ultimately it is the lack of simplicity that works against FEM, not the method itself. This is a direct failure of the practice of FEM rather than the ideas embedded within.

The people who tend to work on FEM tend to significantly overelaborate things. I’m quite close to 100% convinced that the overelaboration is completely unnecessary, and it actually serves a supremely negative purpose in the broader practice of science. One of the end products is short-changing the FEM. In a nutshell, people can solve harder problems with finite volume methods (FVM) than FEM. The quest for seemingly rigorous mathematics has created a tendency to work toward problems with well-developed math. Instead we need to be inventing math to attack important problems even if the rigor is missing. Additionally, researchers over time have been far more innovative with FVM than FVM.

The FEM folks usually trot out that bullshit quip that FEM is exactly like FVM with the properly chosen test function. OK, fair enough, FEM is equivalent to FVM, but this fails to explain the generic lack of innovation in numerical methods arising from the FEM community. In the long run it is the innovations that determine the true power of a method, not the elaborate theories surrounding relatively trivial problems. These elaborations actually undermine methods and lead to a cult of complexity that so often defines the practice.

Where FEM excels is the abstraction of geometry from the method and ability to include geometric detail in the simulation within a unified framework. This is extremely useful and explains the popularity of FEM for engineering analysis where geometric detail is important, or assumed to be important. Quite often the innovative methodology is shoehorned into FEM having been invented and perfected in the finite volume (or finite difference) world. Frequently the innovative devices have to be severely modified to fit into the FEM’s dictums. These modifications usually diminish the overall effectiveness of the innovations relative to their finite volume or difference forbearers. These innovative devices are necessary to solve the hard multiphysics problems often governed by highly nonlinear hyperbolic (conservation or evolution) equations. I personally would be more convinced by FEM if some of the innovation happened within the FEM framework instead of continually being imported.

Perhaps most distressingly FEM allows one to engage in mathematical masturbation. I say this with complete sincerity because the development of methods in FVM is far more procreative where methods are actually born of the activity. Too often FEM leads to mathematical fantasy that have no useful end product aside from lots of self-referential papers in journals, and opaque talks at meetings such as those I’ve witnessed in the last month. For example computational fluid dynamics (CFD) is dominated by FVM methods. CFD solvers are predominantly FVM not FEM largely for the very reason that innovative methods are derived first and used best in FVM. Without the innovative methods CFD would not be able to solve many of its most important and challenging problems today.

Mathematically speaking, I think the issue comes down to regularity. For highly regular and well-behaved problems FEM works very well, and it’s better than FVM. In a sense FEM often doubles down on regularity with test functions. When the solution is highly regular this yields benefits. The issue is that highly regular problems actually define the easier and less challenging problems to be solved, not the hard technology-pacing ones. FVM on the other hand hedges its’ bets. Discontinuous Galerkin (DG) is a particular example. It is a really interesting method because it sits between FEM and FVM. The DG community puts a lot of effort in making it a FEM method with all the attendant disadvantages of assumed regularity.  This is the heart of the maddening case of taking a method so well suited to very hard problems and studying in incessantly on very easy problems with no apparent gain in utility. It seems to me that DG methods have actually gone backwards in the last decade due to this practice.

In a sense the divide is defined by whether you don’t assume regularity and add it back, or you assume it is there and take measures to deal with it when it’s not there. Another good example comes from the use of FEM for hyperbolic PDE’s where conservation form is important. Conservation is essential, and the weak form of the PDE should give conservation naturally. Instead with the most common Galerkin FEM if one isn’t careful the implementation can destroy conservation. This should not happen, conservation should be a constraint, an invariant that comes for free. It does with FVM, it doesn’t with FEM, and that is a problem. Simple mistakes should not cause conservation errors. In FVM this would have been structurally impossible because of how it was coded. The conservation form would have been built in. In FEM the conservation is a specially property, which is odd for something built on the weak form of the PDE. This goes directly to the continuous basis selected in the construction of the scheme.

Another place where the FEM community falls short is stability and accuracy analysis. With all the mathematical brouhaha surrounding the method one might think that stability and accuracy analysis would be ever-present in FEM practice. Quite the contrary is true. Code and solution verification are common and well practiced in the FVM world and almost invisible in FEM. It makes no sense. A large part of the reason is the abstract mathematical focus of FEM instead of the practical approach of FVM. At the practical end where engineering and science are being accomplished with the aid of scientific computing, the mathematical energy seems to yield very little. It is utterly baffling.

“Simplicity is the ultimate sophistication.” ― Leonardo da Vinci

The issue is where the math community spends its time; do they focus on proving things for easy problems, or expand the techniques to handle hard problems? Right now, it seems to focus on making the problem easier and proving things rather than expanding the techniques available and create structures that would work on the harder problems.  The difference is rather extreme. The goal should be to solve the hard problems we are offered, not transform the hard problems into easy problems with existing math. If the math needed for the hard problems aren’t there we need to invent it and start extending ourselves to provide the rigor we want to see. Too often the opposite path is chosen.

A big issue is the importance or prevalence of problems for which strong convergence can be expected. How much of the work in the world is focused where this doesn’t or can’t happen. How much is? Where is the money or importance?

A think a much better path for FEM in the future is to focus on first making the style and focus of presentation simple and pedagogical. Secondarily the focus should be pushed toward solving harder problems that pace computational science rather than toys that are amenable to well-defined mathematical analysis. The advantages of FEM are clear, the hardest this we have to do is make the method clear, comprehensible and extensible.

I fully expect to catch grief over what I’m saying. Instead I’d like to spur those working on FEM to both attack harder problems, and make their explanation of what they are doing simple. The result will be a better methodology that more people understand. Maybe then the FEM will start to be the source of more innovative numerical methods. Everyone will benefit from this small, but important change in perspective.

“Any darn fool can make something complex; it takes a genius to make something simple.” ― Pete Seeger

I spent the week at a relatively massive conference (3000 attendees) in Barcelona Spain, the World Congress on Computational Mechanics. The meeting was large enough that I was constantly missing talks that I wanted to see because other talks were even more interesting. Originally I wanted to give four talks, but the organizers allowed only one so I was attending more talks and giving far less. Nonetheless such meetings are great opportunities to learn about what is going on around the World, get lots of new ideas, meet old friends and make new ones. It is exactly what I wrote about a few weeks ago, giving a talk is second, third or even fourth on the list of reasons to attend such a meeting.

The span and scope of the Congress is truly impressive. Computational modeling has become a pervasive aspect of modern science and engineering. The array of application is vast and impressively international in flavor. While all of this is impressive, such venues offer an opportunity to take stock of where I am, and where the United States and the rest of the World stand. All of this provides a tremendously valuable opportunity to gain much needed perspective.

An honest assessment is complex. On the one hand, the basic technical and scientific progress is immense, on the other hand there are concerns lurking around every corner. While the United States probably remain in a preeminent state for computational science and engineering, the case against this is getting stronger every day. Europe and Asia are catching up quickly if not having overtaken the USA in many subfields. Across the board there are signs of problems and stagnation in the field. It would seem clear that people know this and it isn’t clear whether there is any action to address problems. Among these issues is the increased use of a narrow set of software tools either commercial or open source computational tools with a requisite lack of knowledge and expertise in the core methods and algorithms used inside them. In addition, the nature of professional education and the state of professionalism is under assault by societal forces,

Despite the massive size of the meeting there are substantial signs that support for research in the field is declining in size and changing in character. It was extremely difficult to see “big” things happening in the field. The question is whether this is the sign of a mature field where slow progress is happening, or the broad lack of support for truly game-changing work. It could also be a sign that the creative energy in science has moved to other areas that are “hotter” such biology, medicine, materials, … There was a notable lack of exciting keynote lectures at the meeting. There didn’t seem to be any “buzz” with any of them. This was perhaps the single most disappointing aspect of the conference.

A couple of things are clear in the United States and Europe the research environment is in crisis under assault from short-term thinking, funding shortfalls (after making funding the end-all and be-all), and educational malaise. For example, I was horrified that Europeans are looking to the USA for guidance on improving their education. This comes on top of my increasing concern about the nature of professional development at the sort of Labs where I work, and the general lack of educational vitality at universities. More and more it is clear that the chief measure of academic success for professors in monetary. The claims of research quality are measured in dollars and the publish or perish mentality that has ravaged the scientific literature. It is a system in dire need of focused reform and should not be the blueprint for anything but failure. The monetary drive comes from the lack of support that education is receiving from the government, which has driven tuition higher at a stunning pace. At the same time the monetary objective of research funding is hollowing out the educational focus universities should possess. The research itself has a short-term focus, and the lack of emphasis or priority for developing people be they students or professionals shares the short sighted outcome. We are draining our system of the vital engine of innovation that has been the key to our recent economic successes.

Another clear trend that resonates with my attendance at the SIAM annual meeting a few weeks ago is the increasing divide between applied mathematic (or theoretical mechanics) and applications. The disparity in focus between the theoretically minded scientists and the application-focused scientist-engineer is growing to the detriment of the community. The application side of things is increasingly using commercial codes that tend to reflect a deep stagnation in capability (aside from the user interface). The theoretical side is focused on idealized problems stripped of real features that complicate the results making for lots of results that no one on the applied side cares about or can use. The divide is only growing with fewer and fewer reaching across the chasm to connect theory to application.

The push from applications has in the past spurred the theoretical side to advance by attacking more difficult problems. Those days appear to be gone. I might blame the prevalence of the sort of short-term thinking investing other areas for this. Both sides of this divide seem to be driven to take few chances and place their efforts into the safe and sure category of work. The theoretical side is working on problems where results can surely be produced (with the requisite publications). By the same token the applied side uses tried and true methods to get some results without having to wait or hope for a breakthrough. The result is a deep sense of abandonment of progress on many fronts.

The increasing dominance of a small number of codes either commercial or open source would be another deep concern. Part of the problem is a reality (or perception) of extensive costs associated with the development of software. People choose to use these off-the-shelf systems because they cannot afford to build their own. On the other hand, by making these choices they and their students or staff are denied the hands on knowledge of the methodology that leads to deep expertise. This is all part of this short-term focus that is bleeding the entire community of deep expertise development necessary for excellence. The same attitudes and approach happen at large laboratories that should seemingly not have the sort of financial and time pressures operating in academia. This whole issue is exacerbated by the theoretical versus applied divide. So far we haven’t made scientific and engineering software modular or componentized. Further the leading edge efforts with “modules” often are so divorced from real problems that they can’t really be relied upon for hard-core applications. Again we have problems with adapting to the modern world confounded with the short-term focus, and success measures that do not measure success.

Perhaps what I’m seeing is a veritable mid-life crisis. The field of computational science and engineering has become mature. It is remarkably broad and making inroads into new areas and considered a full partner with traditional activities in most high-tech industries. At the same time there is a stunning lack of self-awareness, and a loss of knowledge and perspective on the history of the past fifty to seventy years that led to this point. Larger societal pressures and trends are pushing the field in directions that are counter-productive and work actively to undermine the potential of the future. All of this is happening at the same time that computer hardware is either undergoing a crisis or phase transition to a different state. Together we are entering an exciting, but dangerous time that will require great wisdom to navigate. I truly fear that the necessary wisdom while available will not be called upon. If we continue to choose the shortsighted path and avoid doing some difficult things, the outcome could be quite damaging.

A couple of notes about the venue should be made. Barcelona is a truly beautiful city with wonderful weather, people, architecture food, mass transit, I really enjoyed the visit, and there is plenty to comment on. Too few Americans have visited other countries to put their own country in perspective. After a short time you start to hone in on the differences between where you visit and where you live. Coming from America and hearing about the Spanish economy I expected far more homelessness and obvious poverty. I saw very little of either societal ill during my visit. If this is what economic disaster looks like, then it’s hard to see it as an actual disaster. Frankly, the USA looks much worse by comparison with a supposedly recovering economy. There are private security guards everywhere. The amount of security and the meeting was actually a bit distressing. In contrast to this in a week, at a hotel across the street from the hospital, I heard exactly one siren, amazing. As usual getting away from my standard environment is thought provoking, which is always a great thing.

Last week I showed how the accuracy, stability and general properties of an ODE integrator might be studied with the aid of Mathematica. This week I will do the same for a partial differential equations solution. Again, I will provide the commands used in Mathematica to conduct the analysis reported at the end of the post.

It is good to start as simple as possible. That was the reason for retreading the whole ODE stability analysis last week. Now we can steadily go forward toward looking at something a bit harder partial differential equations, starting with a first-order method for a first-order hyperbolic equation, the linear advection equation,

, where the subscript denotes differentiation with respect to the variable. This equation is about as simple as PDEs get, but it is notoriously difficult to solve numerically.

Before getting to the analysis we can state a few properties of the equation. The exact solution is outrageously simple, . This means that the temporal solution is simply defined by the initial condition translated by the velocity (which is one in this case) and time. Nothing changes it simply moves in space. This is a very simple form of space-time self-similarity. If we are solving this equation numerically, any change in the waveform is an error. We can also note that the integral of the value is preserved (of course) making this a “conservation law”. Later when you’d like to solve harder problems this property is exceedingly important.

Now we can proceed to the analysis. The basic process is to replace the function with an analytical representation and similar to ODEs we use the complex exponential (Fourier transform), , where is the grid index of our discretized function, and is the angle parameterizing frequency of the waveform. The analysis then proceeds much as in the style as the ODE work from last week, one substitutes this function into the numerical scheme and works out the modification of the waveform by the numerical method. We then take this modification to be the symbol of the operator . In this form we have divided the symbol into two effects its amplitude and its modulation of the waveform or phase. Finishing our conceptual toolbox is the expression of the exact solution as .

We are now ready to apply the analysis technique to the scheme. We can start off with something horribly simple like first-order upwind. The numerical method is easy to write down as  where is the Courant or CFL number and is the upwind edge value. The CFL number is the similarity variable (dimensionless) of greatest important for numerical schemes for hyperbolic PDEs. Now we plug our Fourier function into the grid values in the scheme and evaluate for a single grid point . Without showing the trivial algebraic steps this gives . We can make the substitution of the trigonometric functions for the complex exponential, $\exp\left(-\imath \theta\right) = \cos\left(\theta\right) – \imath \sin\left(\theta\right)$.

Now it is time to use these relations to provide the properties of the numerical scheme. We will divide these effects into two categories, changes in the amplification of the function that will define stability, $\latex \left| A \right|$ and the phase error . The exact solution has amplitude of one, and a phase of . Once we have separated the symbol into its pieces we can then examine the formal truncation error of the method (as is equivalent to ) in a straightforward manner.

 

We can also expand these in a Taylor series to get a result for the truncation error. For the amplitude we get the following . The phase error can be similarly treated, . Please note that the phase error is actually one order higher than I’ve written because of its definition where I have divided

through by . The last bit of analysis we conduct is to make an estimate of the rate of convergence as a function of the mesh spacing and CFL number. Given the symbol we can compute the error . We then compute the error with a refined grid by a factor of two and note that it must applied twice to get the solution to the same point in time. The error for the refined calculation is , which is squared to account for two time steps being taken to get to the same simulation time, .Given these errors the local rate of convergence is simple, . We can then plot the function where we see that the convergence rate deviates significantly from one (the expected value) for finite values of and .

We can now apply the same machinery to more complex schemes. Our first example is the time-space coupled version of Fromm’s scheme, which is a second-order method. Conducting the analysis is largely a function of writing the numerical scheme in Mathematica much in the same fashion we would use to write the method into a computer code.

The first version of Fromm’s scheme uses a combined space time differencing introduced by Lax-Wendroff implemented using a methodology similar to Richtmyer’s two-step scheme, which makes the steps clear. First, define a cell-centered slope  and then use this to define a edge-centered, time-centered value, . This choice has a “build-in” upwind bias. If the velocity in the equation were oriented oppositely, this choice would be instead (). Now we can write the update for the cell-centered variables as , substitute in the Fourier transform and apply all the same rules as for the first-order upwind method.

Just note that in the Mathematica the slope, and edge variables are defined as general functions of the mesh index and the substitution is accomplished without any pain. This property is essential for analyzing complicated methods that effectively have very large or complex stencils.

The results then follow as before. We can plot the amplitude and phase error easily and the first thing we should notice is the radical improvement over the first-order method, particularly the amplification error at large wavenumbers (i.e., the grid scale). We can go further and use the Taylor series expansion to express the formal accuracy for the amplification and phase error. The amplification error is two orders higher than upwind and is . The phase error is smaller than the upwind scheme, but the same order, . This is the leading order error in Fromm’s scheme.

We can finish by plotting the convergence rate as a function of finite time step and wavenumber. Unlike the upwind scheme as the wavenumber approaches one the rate of convergence is larger than the formal order of accuracy.

The Mathematica commands used to conduct the analysis above:

(* 1st order 1-D *)

U[j_] := T[j]

U1[j_] := U[j] – v (U[j] – U[j – 1])

sym = 1/2 U[0] + 1/2 U1[0] – v/2 (U1[0] – U1[-1]);

T[j_] := Exp[I j t]

Series[sym – Exp[-I v t], {t, 0, 5}]

Simplify[sym]

Sym[v_, t_] := 1/2 E^(-2 I t) (-2 E^(I t) (-1 + v) v + v^2 + E^(2 I t) (2 – 2 v + v^2))

rg1 = Simplify[ComplexExpand[Re[sym]]];

ig1 = Simplify[ComplexExpand[Im[sym]]];

amp1 = Simplify[Sqrt[rg1^2 + ig1^2]];

phase1 = Simplify[ ArcTan[-ig1/rg1]/(v t)];

Series[amp1, {t, 0, 5}]

Series[phase1, {t, 0, 5}]

Plot3D[amp1, {t, 0, Pi}, {v, 0, 1}, AxesLabel -> {t, v}, LabelStyle -> Directive[18, Bold, Black]]

Export[“amp-1.jpg”, %]

ContourPlot[amp1, {t, 0, Pi}, {v, 0, 1}, PlotPoints -> 250, Contours -> {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.99, 1}, ContourShading -> False, Axes -> {True, True}, AxesLabel -> {t, v}, ContourLabels -> All, LabelStyle -> Directive[18, Bold, Black]]

Export[“amp-1-cont.jpg”, %]

Plot3D[phase1, {t, 0, Pi}, {v, 0, 1}, AxesLabel -> {t, v}, LabelStyle -> Directive[18, Bold, Black]]

Export[“phase-1.jpg”, %]

ContourPlot[phase1, {t, 0, Pi}, {v, 0, 1}, PlotPoints -> 250, Contours -> {-1, -0.5, -0.25, 0, 0.25, 0.5, 0.75, 0.9, 0.99, 1, 1.25, 1.5}, ContourShading -> False, Axes -> {True, True}, AxesLabel -> {t, v}, ContourLabels -> All, LabelStyle -> Directive[18, Bold, Black]]

Export[“phase-1-cont.jpg”, %]

err = Sym[v, t] – Exp[-I v t];

err2 = Sym[v, t/2]^2 – Exp[-I v t];

Plot3D[Log[Abs[err]/Abs[err2]]/Log[2], {t, 0.01, Pi/2}, {v, 0, 1}, AxesLabel -> {t, v}, LabelStyle -> Directive[18, Bold, Black]]

Export[“conv-rate-1.jpg”, %]

ContourPlot[Log[Abs[err]/Abs[err2]]/Log[2], {t, 0, Pi}, {v, 0, 1}, PlotPoints -> 250, Contours -> {0.5, 0.75, 0.9, 0.95, 0.99}, ContourShading -> False, Axes -> {True, True}, AxesLabel -> {t, v}, ContourLabels -> All, LabelStyle -> Directive[18, Bold, Black]]

Plot3D[Abs[sym/Exp[-I v t]], {t, 0, Pi}, {v, 0, 5}]

ContourPlot[ If[Abs[sym/Exp[-I v t]] <= 1, Abs[sym/Exp[-I v t]], 0], {t, 0, Pi}, {v, 0, 5}, PlotPoints -> 250, Contours -> {0.1, 0.25, 0.5, 0.75, 0.9, 1}, ContourShading -> False, Axes -> {True, True}, AxesLabel -> {t, v}, ContourLabels -> All, LabelStyle -> Directive[18, Bold, Black]]

errs = Sym[v, t/2]^2 – Sym[v, t];

errs2 = Sym[v, t/4]^4 – Sym[v, t/2]^2;

Plot3D[Log[Abs[errs]/Abs[errs2]]/Log[2], {t, 0.01, Pi/2}, {v, 0, 1}]

errt = Sym[v/2, t]^2 – Sym[v, t];

errt2 = Sym[v/4, t]^4 – Sym[v/2, t]^2;

Plot3D[Log[Abs[errt]/Abs[errt2]]/Log[2], {t, 0, Pi/2}, {v, 0, 1}]

(* classic fromm *)

U[j_] := T[j]

S[j_] := 1/2 (U[j + 1] – U[j – 1])

Ue[j_] := U[j] + 1/2 S[j] (1 – v)

sym2 = U[0] – v (Ue[0] – Ue[-1]);

T[j_] := Exp[I j t]

Series[sym2 – Exp[-I v t], {t, 0, 5}];

Simplify[Normal[%]];

Collect[Expand[Normal[%]], t]

Simplify[sym2]

Sym2[v_, t_] := 1/32 E^(-4 I t) (v^2 – 10 E^(I t) v^2 + E^(6 I t) v^2 + 2 E^(5 I t) v (-4 + 3 v) – 4 E^(3 I t) v (-10 + 7 v) + E^(2 I t) v (-8 + 31 v) – E^(4 I t) (-32 + 24 v + v^2))

rg2 = Simplify[ComplexExpand[Re[sym2]]];

ig2 = Simplify[ComplexExpand[Im[sym2]]];

amp2 = Simplify[Sqrt[rg2^2 + ig2^2]];

phase2 = Simplify[ ArcTan[-ig2/rg2]/(v t)];

Series[amp2, {t, 0, 5}]

Series[phase2, {t, 0, 5}]

Plot3D[amp2, {t, 0, Pi}, {v, 0, 1}, AxesLabel -> {t, v}, LabelStyle -> Directive[18, Bold, Black]]

Export[“amp-2.jpg”, %]

ContourPlot[amp2, {t, 0, Pi}, {v, 0, 1}, PlotPoints -> 250, Contours -> {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.99, 1}, ContourShading -> False, Axes -> {True, True}, AxesLabel -> {t, v}, ContourLabels -> All, LabelStyle -> Directive[18, Bold, Black]]

Export[“amp-2-cont.jpg”, %]

Plot3D[phase2, {t, 0, Pi}, {v, 0, 1}, AxesLabel -> {t, v}, LabelStyle -> Directive[18, Bold, Black]]

Export[“phase-2.jpg”, %]

ContourPlot[phase2, {t, 0, Pi}, {v, 0, 1}, PlotPoints -> 250, Contours -> {-1, -0.5, -0.25, 0, 0.25, 0.5, 0.75, 0.9, 0.99, 1, 1.25, 1.5}, ContourShading -> False, Axes -> {True, True}, AxesLabel -> {t, v}, ContourLabels -> All, LabelStyle -> Directive[18, Bold, Black]]

Export[“phase-2-cont.jpg”, %]

err = Sym2[v, t] – Exp[-I v t];

err2 = Sym2[v, t/2]^2 – Exp[-I v t];

Plot3D[Log[Abs[err]/Abs[err2]]/Log[2], {t, 0.01, Pi/2}, {v, 0, 1}, AxesLabel -> {t, v}, LabelStyle -> Directive[18, Bold, Black]]

Export[“conv-rate-2.jpg”, %]

ContourPlot[ Log[Abs[err]/Abs[err2]]/Log[2], {t, 0.01, Pi}, {v, 0.01, 1}, PlotPoints -> 250, Contours -> {1, 1.5, 1.75, 1.9, 2, 2.1, 2.2, 2.3, 2.4}, ContourShading -> False, Axes -> {True, True}, AxesLabel -> {t, v}, ContourLabels -> All, LabelStyle -> Directive[18, Bold, Black]]

Export[“conv-rate-2-cont.jpg”, %]

ContourPlot[ If[Abs[sym2/Exp[-I v t]] <= 1, Abs[sym2/Exp[-I v t]], 0], {t, 0, Pi}, {v, 0, 1}, PlotPoints -> 250, Contours -> {0.1, 0.25, 0.5, 0.75, 0.9, 1}, ContourShading -> False, Axes -> {True, True}, AxesLabel -> {t, v}, ContourLabels -> All, LabelStyle -> Directive[18, Bold, Black]]

errs = Sym2[v, t] – Sym2[v, t/2]^2;

errs2 = Sym2[v, t/4]^4 – Sym2[v, t/2]^2;

Plot3D[Log[Abs[errs]/Abs[errs2]]/Log[2], {t, 0.01, Pi/2}, {v, 0, 1}, AxesLabel -> {t, v}, LabelStyle -> Directive[18, Bold, Black]]

errt = Sym2[v, t] – Sym2[v/2, t]^2;

errt2 = Sym2[v/4, t]^4 – Sym2[v/2, t]^2;

Plot3D[Log[Abs[errt]/Abs[errt2]]/Log[2], {t, 0.0, Pi/2}, {v, 0, 0.1}]

(* 2nd order Fromm – RK *)

U[j_] := T[j]

S[j_] := 1/2 (U[j + 1] – U[j – 1])

Ue[j_] := U[j] + 1/2 S[j]

U1[j_] := U[j] – v (Ue[j] – Ue[j – 1])

S1[j_] := 1/2 (U1[j + 1] – U1[j – 1])

Ue1[j_] := U1[j] + 1/2 S1[j]

sym2 = 1/2 U[0] + 1/2 U1[0] – v/2 (Ue1[0] – Ue1[-1]);

T[j_] := Exp[I j t]

Series[sym2 – Exp[-I v t], {t, 0, 5}];

Simplify[Normal[%]];

Collect[Expand[Normal[%]], t]

Simplify[sym2]

Sym2[v_, t_] := 1/32 E^(-4 I t) (v^2 – 10 E^(I t) v^2 + E^(6 I t) v^2 + 2 E^(5 I t) v (-4 + 3 v) – 4 E^(3 I t) v (-10 + 7 v) + E^(2 I t) v (-8 + 31 v) – E^(4 I t) (-32 + 24 v + v^2))

rg2 = Simplify[ComplexExpand[Re[sym2]]];

ig2 = Simplify[ComplexExpand[Im[sym2]]];

amp2 = Simplify[Sqrt[rg2^2 + ig2^2]];

phase2 = Simplify[ ArcTan[-ig2/rg2]/(v t)];

Series[amp2, {t, 0, 5}]

Series[phase2, {t, 0, 5}]

Plot3D[amp2, {t, 0, Pi}, {v, 0, 1}, AxesLabel -> {t, v}, LabelStyle -> Directive[18, Bold, Black]]

Export[“amp-2rk.jpg”, %]

ContourPlot[amp2, {t, 0, Pi}, {v, 0, 1}, PlotPoints -> 250, Contours -> {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.99, 1}, ContourShading -> False, Axes -> {True, True}, AxesLabel -> {t, v}, ContourLabels -> All, LabelStyle -> Directive[18, Bold, Black]]

Export[“amp-2rk-cont.jpg”, %]

Plot3D[phase2, {t, 0, Pi}, {v, 0, 1}, AxesLabel -> {t, v}, LabelStyle -> Directive[18, Bold, Black]]

Export[“phase-2rk.jpg”, %]

ContourPlot[phase2, {t, 0, Pi}, {v, 0, 1}, PlotPoints -> 250, Contours -> {-1, -0.5, -0.25, 0, 0.25, 0.5, 0.75, 0.9, 0.99, 1, 1.25, 1.5}, ContourShading -> False, Axes -> {True, True}, AxesLabel -> {t, v}, ContourLabels -> All, LabelStyle -> Directive[18, Bold, Black]]

Export[“phase-2rk-cont.jpg”, %]

err = Sym2[v, t] – Exp[-I v t];

err2 = Sym2[v, t/2]^2 – Exp[-I v t];

Plot3D[Log[Abs[err]/Abs[err2]]/Log[2], {t, 0.01, Pi/2}, {v, 0, 1}, AxesLabel -> {t, v}, LabelStyle -> Directive[18, Bold, Black]]

Export[“conv-rate-2rk.jpg”, %]

ContourPlot[ Log[Abs[err]/Abs[err2]]/Log[2], {t, 0.01, Pi}, {v, 0.01, 1}, PlotPoints -> 250, Contours -> {1, 1.5, 1.75, 1.9, 2, 2.1, 2.2, 2.3, 2.4}, ContourShading -> False, Axes -> {True, True}, AxesLabel -> {t, v}, ContourLabels -> All, LabelStyle -> Directive[18, Bold, Black]]

Export[“conv-rate-2rk-cont.jpg”, %]

ContourPlot[ If[Abs[sym2/Exp[-I v t]] <= 1, Abs[sym2/Exp[-I v t]], 0], {t, 0, Pi}, {v, 0, 1.5}, PlotPoints -> 250, Contours -> {0.1, 0.25, 0.5, 0.75, 0.9, 1}, ContourShading -> False, Axes -> {True, True}, AxesLabel -> {t, v}, ContourLabels -> All, LabelStyle -> Directive[18, Bold, Black]]

errs = Sym2[v, t] – Sym2[v, t/2]^2;

errs2 = Sym2[v, t/4]^4 – Sym2[v, t/2]^2;

Plot3D[Log[Abs[errs]/Abs[errs2]]/Log[2], {t, 0.01, Pi/2}, {v, 0, 1}]

errt = Sym2[v, t] – Sym2[v/2, t]^2;

errt2 = Sym2[v/4, t]^4 – Sym2[v/2, t]^2;

Plot3D[Log[Abs[errt]/Abs[errt2]]/Log[2], {t, 0.0, Pi/2}, {v, 0, 0.1}]

In order to avoid going on another (epic) rant this week, I’ll change gears and touch upon a classic technique for analyzing the stability of numerical methods along with extensions or the traditional approach.

Before diving into partial differential equations, I thought it would be beneficial to analyze the stability of ordinary differential equation integrators first. This provides the basis of the approach. Next I will show how the analysis proceeds for an important second-order method for linear advection. I will close with providing the analysis of second-order discontinuous Galerkin methods, which introduce an important wrinkle on the schemes. I will close by producing the Mathematica commands used to give the results.

It is always good to have references that can be read for detail and explanation than I will give a few seminal ones here:

* Ascher, Uri M., and Linda R. Petzold. Computer methods for ordinary differential equations and differential-algebraic equations. Vol. 61. Siam, 1998.

* Durran, Dale R. Numerical methods for wave equations in geophysical fluid dynamics. No. 32. Springer, 1999.

* LeVeque, Randall J., and Randall J. Le Veque. Numerical methods for conservation laws. Vol. 132. Basel: Birkhäuser, 1992.

* LeVeque, Randall J. Finite volume methods for hyperbolic problems. Vol. 31. Cambridge university press, 2002.

* Strikwerda, John C. Finite difference schemes and partial differential equations. Siam, 2004.

Let’s jump into the analysis of ODE solvers by looking at a fairly simple method, the forward Euler method. We can write the solver for a simple ODE, , as simply . We take the right hand side , and do some algebra. We have several principal goals, establish conditions for stability, accuracy, and overall behavior of the method.

For stability we determine how much the value of the solution is amplified by the action of the integration scheme, , we remove the variable and call the result the “symbol’’ of the integrator, , we then solve for then take its magnitude, , being less than one for stability. We can write down this answer explicitly . We can also plot this result easily (see the commands I used in Mathematica at the end of the post).  On all the plots the horizontal axis is the real values and the vertical axis is the imaginary values .

 

This plot just includes the values where they amplitude of the symbol is less than or equal to one.

Next we look at accuracy using a Taylor series expansion. The Taylor series is simple given the analytical solution to the ODE, , and the Taylor series expansion is classical, . We simply subtract this Taylor series from the symbol of the operator and look at the remainder, $\latex E= \frac{1}{2}(\lambda \Delta t)^2+ O(\Delta t^3)$, where the time has been replaced by the time step size.

The last couple of twists can add some significant texture to the behavior of the integration scheme. We can plot the “order stars’’ which shows whether the numerical scheme changes the amplitude of the answer more or less than the exact operator. These are call stars because they start to show star-like shapes for higher order methods (mostly starting at third- and higher order accuracy). He is the plot for forward Euler.

The last thing we will examine for the forward Euler scheme is the order of accuracy you should see during a time step refinement study as part of a verification exercise. Usually this is thought of as being the same as the order of the numerical scheme, but for a finite time step size the result deviates from the analytical order substantially. Computing this is really quite simple, one simply compute the symbol of the operator for half the time step size , for two time steps (so that the time ends up at the same place as you get for a single step with . This is simply the square of the operator at the smaller time step size. To get the order of accuracy you take the operators and subtract the exact solution, take the absolute value of the result, then compute the order of accuracy like usual verification,

We can plot the result easily with Mathematica. It is notable how different from the asymptotic value of one the results are for reasonable, but finite values of . As the operator becomes unstable, the convergence rate actually becomes very large. This is a word of warning to the practitioner that very high rates of convergence can actually be a very bad sign for a calculation.

Look for

We can now examine a second-order method with relative ease.  Doing the analysis is akin to writing a computer code albeit symbolically.  The second-order is using a predictor-corrector format where a half step is taken using forward Euler and this result is used to advance the solution the full step. This is an improved forward Euler method.  It is explicit in that the solution can be evaluated solely in terms of the initial data. The scheme is the following: for the predictor, and .  The symbol is computed as before giving .  Getting the full properties of the method now just requires “turning the crank’’ as we did for the forward Euler scheme.

The truncation error has gained an order of accuracy and now is .

The stability plot is more complex giving a larger region for the stability particularly along the imaginary axis.

The order star looks much more like a star.

Finally the convergence rate plot is much less pathalogical although some of the same conclusions can be drawn from the behavior where the scheme is unstable (giving the very large convergance rates).

We will finish this week’s post by turning our attention to a second order implicit scheme, the backwards differentiation formula (BDF).  Everything will follow from the previous two example, but the scheme adds an important twist (or two).  The first twist is that the method is implicit, meaning that the left and right hand sides of the method are coupled, and the second is that the method depends on three time levels of data, not two as the first couple of methods.

The update for the method is written , and the amplification is now a quadratic equation,  with two roots. One of these roots will have a Taylor series expansion that demonstrates second-order accuracy for the scheme, the other will not. The inaccurate root must still be stable for the scheme to be stable. The accurate root is

with a error of .

The second inaccurate root is also called spurious and has the form

.

The stability of the scheme requires taking the maximum of the magnitude of both roots.

Using the accurate root we can examine the order star, and the rate of convergence of the method as before.

 

Next week we will look at a simple partial differential equation analysis, which adds new wrinkles.

While this sort of analysis can be done by hand, the greatest utility can be achieved by using symbolic or numerical packages such as Mathematica. Below I’ve included the Mathematica code used for the analyses given above.

Soln = Collect[Expand[Normal[Series[Exp[h L], {h, 0, 6}]]], h]

(* Forward Euler *)

a =.; b =.

A = 1 + h L

Aab2 = (1 + h L/2 )^2

Collect[Expand[Normal[Series[A, {h, 0, 6}]]], h];

Soln – %

L = ( a + b I)/h;

ContourPlot[If[Abs[A] < 1, -Abs[A], 1], {a, -3, 1}, {b, -2, 2},
PlotPoints -> 250,
Contours -> {-0.1, -0.2, -0.3, -0.4, -0.5, -0.6, -0.7, -0.8, -0.9},
ContourShading -> False, Axes -> {True, True},
AxesLabel -> {a dt, b dt}, LabelStyle -> Directive[18, Bold, Black]]

Export[“forwardEuler.jpg”, %]

“forwardEuler.jpg”

ContourPlot[
If[Abs[A]/Abs[Exp[a + b I]] < 1, -Abs[A]/Abs[Exp[a + b I]],
1], {a, -3, 2}, {b, -2, 2}, PlotPoints -> 250,
Contours -> {-0.1, -0.2, -0.3, -0.4, -0.5, -0.6, -0.7, -0.8, -0.9},
ContourShading -> False, Axes -> {True, True},
AxesLabel -> {a dt, b dt}, LabelStyle -> Directive[18, Bold, Black]]

Export[“forwardEuler-star.jpg”, %]

“forwardEuler-star.jpg”

Plot3D[Log[Abs[A – Exp[a + b I]]/Abs[Aab2 – Exp[a + b I]]]/
Log[2], {a, -3, 3}, {b, -2, 2}, PlotPoints -> 100,
AxesLabel -> {a dt, b dt, n},
LabelStyle -> Directive[18, Bold, Black], PlotRange -> All]

Export[“forwardEuler-order.jpg”, %]

“forwardEuler-order.jpg”

ContourPlot[
Log[Abs[A – Exp[a + b I]]/Abs[Aab2 – Exp[a + b I]]]/Log[2], {a, -3,
3}, {b, -2, 2}, PlotPoints -> 250,
Contours -> {0, 0.25, 0.5, 0.75, 0.9, 1, 1.1, 1.25, 1.5, 2, 3, 4, 5,
10}, ContourShading -> False, ContourLabels -> All,
Axes -> {True, True}, AxesLabel -> {a dt, b dt},
LabelStyle -> Directive[18, Bold, Black], PlotRange -> All]

Export[“forwardEuler-order-contour.jpg”, %]

“forwardEuler-order-contour.jpg”

ContourPlot[
If[Abs[A] < 1, Abs[A – Exp[a + b I]]/Abs[Aab2 – Exp[a + b I]],
0], {a, -3, 2}, {b, -2, 2}, PlotPoints -> 250,
Contours -> {2, 2.5, 3, 3.5, 4}, ContourShading -> False,
Axes -> {False, True}]

Plot3D[Abs[A – Exp[a + b I]]/Abs[Aab2 – Exp[a + b I]], {a, -3,
3}, {b, -2, 2}, PlotPoints -> 100]

ContourPlot[
If[Abs[A]/Abs[Exp[a + b I]] < 1, -Abs[A]/Abs[Exp[a + b I]],
1], {a, -3, 2}, {b, -2, 2}, PlotPoints -> 250,
Contours -> {-0.1, -0.2, -0.3, -0.4, -0.5, -0.6, -0.7, -0.8, -0.9},
ContourShading -> False, Axes -> {False, True}]

(* RK 2 *)

L =.

A1 = 1 + 1/2 h L

A = 1 + h L A1

A12 = 1 + 1/4 h L;

Aab2 = (1 + h L A12/2 )^2

Collect[Expand[Normal[Series[A, {h, 0, 6}]]], h] – Soln

L = ( a + b I)/h;

ContourPlot[If[Abs[A] < 1, -Abs[A], 1], {a, -3, 1}, {b, -2, 2},
PlotPoints -> 250,
Contours -> {-0.1, -0.2, -0.3, -0.4, -0.5, -0.6, -0.7, -0.8, -0.9},
ContourShading -> False, Axes -> {True, True},
AxesLabel -> {a dt, b dt}, LabelStyle -> Directive[18, Bold, Black]]

Export[“rk2.jpg”, %]

ContourPlot[
If[Abs[A]/Abs[Exp[a + b I]] < 1, -Abs[A]/Abs[Exp[a + b I]],
1], {a, -3, 2}, {b, -2, 2}, PlotPoints -> 250,
Contours -> {-0.1, -0.2, -0.3, -0.4, -0.5, -0.6, -0.7, -0.8, -0.9},
ContourShading -> False, Axes -> {True, True},
AxesLabel -> {a dt, b dt}, LabelStyle -> Directive[18, Bold, Black]]

Export[“rk2-star.jpg”, %]

Plot3D[Log[Abs[A – Exp[a + b I]]/Abs[Aab2 – Exp[a + b I]]]/
Log[2], {a, -3, 3}, {b, -2, 2}, PlotPoints -> 100,
AxesLabel -> {a dt, b dt, n},
LabelStyle -> Directive[18, Bold, Black], PlotRange -> All]

Export[“rk2-order.jpg”, %]

ContourPlot[
Log[Abs[A – Exp[a + b I]]/Abs[Aab2 – Exp[a + b I]]]/Log[2], {a, -3,
3}, {b, -2, 2}, PlotPoints -> 250,
Contours -> {0, 0.25, 0.5, 0.75, 0.9, 1, 1.1, 1.25, 1.5, 2, 3, 4, 5,
10}, ContourShading -> False, ContourLabels -> All,
Axes -> {True, True}, AxesLabel -> {a dt, b dt},
LabelStyle -> Directive[18, Bold, Black], PlotRange -> All]

Export[“rk2-order-contour.jpg”, %]

(* BDF 2 *)

A =.

L =.

Solve[3/2 A^2 – 2 A + 1/2 == h A^2 L, A]

A1 = (-2 – Sqrt[1 + 2 h L])/(-3 + 2 h L); A2 = (-2 + Sqrt[
1 + 2 h L])/(-3 + 2 h L);

Solve[3/2 A^2 – 2 A + 1/2 == 1/2 h A^2 L, A]

Aab2 = ((-2 – Sqrt[1 + h L])/(-3 + h L) )^2;

Collect[Expand[Normal[Series[A1, {h, 0, 6}]]], h] – Soln

Collect[Expand[Normal[Series[A2, {h, 0, 6}]]], h] – Soln

L = ( a + b I)/h;

ContourPlot[
If[Max[Abs[A1], Abs[A2]] < 1, -Max[Abs[A1], Abs[A2]], 1], {a, -8,
6}, {b, -5, 5}, PlotPoints -> 250,
Contours -> {-0.1, -0.2, -0.3, -0.4, -0.5, -0.6, -0.7, -0.8, -0.9},
ContourShading -> False, Axes -> {True, True},
AxesLabel -> {a dt, b dt}, LabelStyle -> Directive[18, Bold, Black]]

Export[“bdf2.jpg”, %]

ContourPlot[

If[Abs[A1]/Abs[Exp[a + b I]] < 1, -Abs[A1]/Abs[Exp[a + b I]],
1], {a, -3, 2}, {b, -2, 2}, PlotPoints -> 250,
Contours -> {-0.1, -0.2, -0.3, -0.4, -0.5, -0.6, -0.7, -0.8, -0.9},
ContourShading -> False, Axes -> {True, True},
AxesLabel -> {a dt, b dt}, LabelStyle -> Directive[18, Bold, Black]]

Export[“bdf2-star.jpg”, %]

Plot3D[Log[Abs[A1 – Exp[a + b I]]/Abs[Aab2 – Exp[a + b I]]]/
Log[2], {a, -8, 6}, {b, -5, 5}, PlotPoints -> 100,
AxesLabel -> {a dt, b dt, n},
LabelStyle -> Directive[18, Bold, Black]]

Export[“bdf2-order.jpg”, %]

ContourPlot[
Log[Abs[A1 – Exp[a + b I]]/Abs[Aab2 – Exp[a + b I]]]/Log[2], {a, -8,
6}, {b, -5, 5}, PlotPoints -> 250,
Contours -> {-1, 0, 0.5, 0.9, 1, 1.1, 1.5, 2, 5, 10},
ContourShading -> False, ContourLabels -> All, Axes -> {True, True},
AxesLabel -> {a dt, b dt}, LabelStyle -> Directive[18, Bold, Black],
PlotRange -> All]

Export[“bdf2-order-contour.jpg”, %]

 

Donald Knuth -“If you find that you’re spending almost all your time on theory, start turning some attention to practical things; it will improve your theories. If you find that you’re spending almost all your time on practice, start turning some attention to theoretical things; it will improve your practice.”

This week I visited Chicago for the 2014 SIAM Annual Meeting (Society for Industrial and Applied Mathematics). It was held at the Palmer House, which is absolutely stunning venue swimming in old-fashioned style and grandeur. It is right around the corner from Millennium Park, which is one of the greatest Urban green spaces in existence, which itself is across the street from the Art Institute. What an inspiring setting to hold a meeting. Chicago itself is one of the great American cities with a vibrant downtown and numerous World-class sites.

The meeting included a lot of powerful content and persuasive applications of applied mathematics. Still some of the necessary gravity for the work seems to be missing from the overall dialog with most of the research missing the cutting edge of reality. There just seems to be a general lack of vitality and importance to the overall scientific enterprise, and applied mathematics is suffering likewise. This isn’t merely the issue of funding, which is relatively dismal, but overall direction and priority. In total, we aren’t asking nearly enough from science, and mathematics is no different. The fear of failure is keeping us from collectively attacking society’s most important problems. The distressing part of all of this is the importance and power of applied mathematics and the rigor it brings to science as a whole. We desperately need some vision moving forward.

The importance of applied mathematics to the general scientific enterprise should not be in doubt, but it is. I sense a malaise in the entire scientific field stemming from the overall lack of long-term perspective for the Nation as a whole. Is the lack of vitality specific to this field, or a general description of research?

I think it is useful to examine how applied mathematics can be an important force for order, confidence and rigor in science. Indeed applied mathematics can be a powerful force to aid the practice of science. For example there is the compelling example of compressed sensing (told in the Wired article http://www.wired.com/2010/02/ff_algorithm/). The notion that the L1 norm had magical properties to help unveil the underlying sparsity in objects was an old observation, but not until mathematical rigor was put in place to underpin this observation did the practice take off. There is no doubt that the entire field exploded in interest when the work of Candes, Tao and Donoho put a rigorous face on the magical practice of regularizing a problem with the L1 norm. It shouldn’t be under-estimated that the idea came at the right time; this is a time when we are swimming in data from an increasing array of sources, and compressed sensing conceptually provides a powerful tool for dealing with this. At the same time, the lack of rigor limited the interest in the technique prior to 2004 or 2005.

One of the more persuasive cases where applied mathematics has provided a killer theory is the work of Peter Lax on hyperbolic conservation laws. He laid the groundwork for stunning progress in modeling and simulating with confidence and rigor. There are other examples such as the mathematical order and confidence of the total variation diminishing theory of Harten to power the penetration of high-resolution methods into broad usage for solving hyperbolic PDEs. Another example is the relative power and confidence brought to the solution of ordinary differential equations, or numerical linear algebra by the mathematical rigor underlying the development of software. These are examples where the presence of applied mathematics makes a consequential and significant difference in the delivery of results with confidence and rigor. Each of these is an example of how mathematics can unleash a capability in truly “game-changing” ways. A real concern is why this isn’t happening more broadly or in targeted manner.

I started the week with a tweet of Richard Hamming’s famous quote – “The purpose of computing is insight, not numbers.” During one of the highlight talks of the meeting we received a modification of that maxim by the lecturer Leslie Greengard,

“The purpose of computing is to get the right answer”.

A deeper question with an uncertainty quantification spin would be “which right answer?” My tweet in response to Greengard then said

“The purpose of computing is to solve more problems than you create.”

This entire dialog is the real topic of the post. Another important take was by Joseph Teran on scientific computing in special effects for movies. Part of what sat wrong with me was the notion that looking right becomes equivalent to being right. On the other hand the perception and vision of something like turbulent fluid flow shouldn’t be underestimated. If it looks right there is probably something systematic lying beneath the superficial layer of entertainment. The fact that the standard for turbulence modeling for science and movies might be so very different should be startling. Ideally the two shouldn’t be that far apart. Do special effects have something to teach us? Or something worthy of explanation? I think these questions make mathematicians very uncomfortable.

If it makes you uncomfortable, it might be a good or important thing to ask. That uncomfortable question might have a deep answer that is worth attacking. I might prefer to project this entire dialog into the broader space of business practice and advice. This might seem counter-intuitive, but the broader societal milieu today is driven by business.

“Don’t find customers for your products, find products for your customers.” ― Seth Godin

One of the biggest problems in the area where I work is the maturity of the field. People simply don’t think about what the entire enterprise is for. Computational simulation and modeling is about using a powerful tool to solve problems. The computer allows certain problem solving approaches to be used that aren’t possible with out it, but the problem solving is the central aspect. I believe that the fundamental utility of modeling and simulation is being systematically taken for granted. The centrality of the problem being solved has been lost and replaced by simpler, but far less noble pursuits. The pursuit of computational power has become a fanatical desire that has swallowed the original intent. Those engaging in this pursuit have generally good intentions, but lack the well-rounded perspective on how to achieve success. For example, the computer is only one small piece of the toolbox and to use a mathematical term, necessary, but gloriously insufficient.

Currently the public policy is predicated upon the notion that a bigger faster computer provides an unambiguously better solution. Closely related to this notion is a technical term in computational modeling and mathematics known as convergence. The model converges or approaches a solution as more computational resource is applied. If you do everything right this will happen, but as problems become more complex you have to do a lot of things right. The problem is that we don’t have the required physical or mathematical knowledge to have the expectation of this in many cases. These are the very cases that we use to justify the purchase of new computers.

The guarantee of convergence ought to be at the very heart of where applied mathematics is striving; yet the community as a whole seems to be shying away from the really difficult questions. Today too much applied mathematics focuses upon simple model equations that are well behaved mathematically, but only capture cartoon aspects of the real problems facing society. Over the past several decades focused efforts on attacking these real problems have retreated. This retreat is part of the overall base of fear of failure in research. Despite the importance of these systems, we are not pushing the boundaries of knowledge to envelop them with better understanding. Instead we spend effort redoubling our efforts to understand simple model equations. This lack of focus on real problems is one of the deepest and most troubling aspects of the current applied mathematics community.

We have evolved to the point in computational modeling and simulation where today we don’t actually solve problems any more. We have developed useful abstractions that have taken the place of the actual problem solving. In a deep sense we now solve cartoonish versions of actual problems. These cartoons allow the different sub-fields to work independently of one another. For example, the latest and greatest computers require super high-resolution 3-D (or 7-D) solutions to the model problems. Actual problem solving rarely (never) works this way. If the problem can be solved in a lower-dimensional manner, it is better. Actual problem solving always starts simple and builds its way up. We start in one dimension and gain experience, run lots of problems, add lots of physics to determine what needs to be included in the model. The mantra of the modern day is to short-circuit this entire approach and jump to add in all the physics, and all the dimensionality, and all the resolution. It is the recipe for disaster, and that disaster is looming before us.

The reason for this is a distinct lack of balance in how we are pursing the objective of better modeling and simulation. To truly achieve progress we need a return to a balanced problem solving perspective. While this requires attention to computing, it also requires physical theory and experiment, deep engineering, computer science, software engineering, mathematics, and physiology. Right now, aside from computers themselves and computer science, the endeavor is woefully out of balance. We have made experiments almost impossible to conduct, and starved the theoretical aspects of science in both physics and mathematics.

Take our computer codes as an objective example of what is occurring. The modeling and simulation is no better than the physical theory and the mathematical approximations used. In many cases these ideas are now two or three decades old. In a number of cases the theory gives absolutely no expectation of convergence as the computational resource is increased. The entire enterprise is predicated on this assumption, yet it has no foundation in theory! The divorce between what the codes do and what the applied mathematicians at SIAM do is growing. The best mathematics is more and more irrelevant to the codes being run on the fastest computers. Where excellent new mathematical approximations exist they cannot be applied to the old codes because of the fundamental incompatibility of the theories. Despite these issues little or no effort exists to rectify this terrible situation.

Why?

Part of the reason is our fixation on short-term goals, and inability to invest in long-term ends. This is true in science, mathematics, business, roads, bridges, schools, universities, …

Long-term thinking has gone the way of the dinosaur. It died in the 1970’s. I came across a discussion of one of the key ideas of our time, the perspective that business management is all about maximizing shareholder value. It was introduced in 1976 by Nobel Prize-winning economist, Milton Friedman and took hold like a leech. It arguable that it is the most moronic idea ever in business (“the dumbest idea ever”). Nonetheless it has become the lifeblood of business thought, and by virtue of being a business mantra, lifeblood of government thinking. It has been poisoning the proverbial well ever since. It has become the reason for the vampiric obsession with short-term profits, and a variety of self-destructive business practices. The only “positive” side has been its role in driving the accumulation of wealth within chief executives, and financial services. Stock is no longer held for any significant length of time, and business careers hinge upon the quarterly balance sheet. Whole industries have been ground under the wheels of the quarterly report. Government research in a lemming like fashion has followed suit and driven research to be slaved to the quarterly report too.

The consequences for the American economy have been frightening. Aside from the accumulation of wealth by upper management, we have had various industries completely savaged by the practice, rank and file workers devalued and fired, and no investment in future value. The stock trading frenzy created by this short-term thinking has driven the creation of financial services that produce nothing of value for the economy, and have succeeded in destabilizing the system. As we have seen in 2008 the results can be nearly catastrophic. In addition, the entire business-government system has become unremittingly corrupt and driven by greed and influence peddling. Corporate R&D used to be a vibrant source of science funding and form a pipeline for future value. Now it is nearly barren with the great corporate research labs fading memories. The research that is funded is extremely short-term focused and rarely daring or speculative. The sorts of breakthroughs that have become the backbone of the modern economy no longer get any attention.

The government has been similarly infested as anything that is “good” business practice is “good” for government management. Science is no exception. We now have to apply similar logic to our research and submit quarterly reports. Similar to business we have had to strip mine the future and inflate our quarterly bottom line. The result has been a systematic devaluing of the future. The national leadership has adopted the short-term perspective whole cloth.

At least in some quarters there is recognition of this trend and a push to reverse this trend. It is going to be a hard path to reversing the problem as the short-term focus has been the “goose that laid the golden egg” for many. These ideas have also distorted the scientific enterprise in many ways. The government’s and business’ investment in R&D has become inherently shortsighted. This has caused the whole approach to science to become radically imbalanced. Computational modeling and simulation is but one example that I’m intimately familiar with. It is time to turn things around.

“Fear is the mind-killer.” — Frank Herbert

The United States likes to think of itself as a courageous country (a country full of heros). This picture is increasingly distant from the reality of a society of cowards who are almost scared of their own shadows. Why? What is going on in our society to drive trend to be scared of everything?Calling the United States a bunch of cowards seems rather hyperbolic, and it is. The issue is that the leadership of the nation is constantly stoking the fires of irrational fear as a tool to drive political goals. By failing to aspire toward a spirit of shared sacrifice and duty, we are creating a society that looks to avoid anything remotely dangerous or risky. The consequences of this cynical form of gamesmanship are slowly ravaging the United States’ ability to be a dynamic force for anything good. In the process we are sapping the vitality that once brought the nation to the head of the international order. In some ways this trend is symptomatic of our largess as the sole military and economic superpower of the last half of the 20th Century. The fear is drawn from the societal memory of our fading roll in the World, and the evolution away from the mono-polar power we once represented.

Where is the national leadership that calls on citizens to reach for the stars? Where are the voices asking for courage and sacrifice? Once upon a time we had leaders who asked much of us.

“For of those to whom much is given, much is required. “
and
“And so, my fellow Americans: ask not what your country can do for you — ask what you can do for your country.” – John F. Kennedy.

The consequences of this fear go well beyond mere name-calling or implications associated with the psychological aspects of fear, but undermine the ability of the Country to achieve anything of substance, or spend precious resources rationally. The use of fear to motivate people’s choices by politicians is rampant as is the use of fear in managing work. Fear moves people to make irrational choices, and our Nation’s leader whether in government or business want people to choose irrationally in favor of outcomes that benefit those in power. Fear is a powerful way to achieve this. All of this is a serious negative drain on the nation. In almost any endeavor trying to do things you are afraid of leads to diminished performance. One works harder to avoid the negative outcome than achieve the positive one. Fear is an enormous tax on all our efforts, and usually leads to the outcomes that we feared in the first place. We live in a world where broad swaths of public policy are fear-driven. It is a plague on our culture.

Like many of you, my attention has been drawn to the event in Iraq (and Syria) with the onslaught of ISIS. A chorus of fear mongering by politicians bent of scaring the public to support military action to stem the tide of anti-Western factions in the region has coupled this. Supposedly ISIS is worse than Al Qaeda, and we should be afraid. You are so afraid that you will demand action. In fact that hamburger you are stuffing into your face is a much larger danger to your well being than ISIS will ever be. Worse yet, we put up with the fear-mongers whose fear baiting is aided and abetted by the new media because they see ratings. When we add up the costs, this chorus of fear is savaging us and it is hurting our Country deeply.

“Stop letting your fear condemn you to mediocrity.” ― Steve Maraboli,

We have collectively lost the ability to judge the difference between a real threat and an unfortunate occurrence. Even if we include the loss of life on 9-11 the threat to you due to terrorism is minimal. Despite this reality we expend fast sums of money, time, effort and human lives trying to stop it. It is an abysmal investment of all of these things. We could do so much more with those resources. To make matters worse, the “War on Terror” has distorted our public policy in numerous ways. Stating with the so-called Patriot act we have sacrifice freedom and privacy at the altar of public safety and national security. We create the Department of Homeland Security (a remarkably Soviet sounding name at that), which is a monument to wasting taxpayer money. Perhaps the most remarkable aspect of the DHS is that entering the United States is now more arduous than entering the former Soviet Union (Russia). This fact ought to absolutely be appalling to the American psyche. Meanwhile, numerous bigger threats go completely untouched by action or effort to mitigate their impact.

For starters as the news media became more interested in ratings than news, they began to amplify the influence of the exotic events. Large, unusual, violent events are ratings gold, and their presence in the news is grossly inflated. The mundane everyday things that are large risks are also boring or depressing, and people would just as soon ignore them. In many cases the mundane everyday risks are huge moneymakers for the owners and advertisers in the media, and they have no interest in killing their cash cow even at the expense of human life (think the medical-industrial complex, and agri-business). Given that people are already horrific at judging statistical risks, these trends have only tended to increase the distance between perceived and actual danger. Politicians know all these things and use them to their advantage. The same things that get ratings for the news grab voter’s attention, and the cynics “leading” the country know it.

When did all this start? I tend to think that the tipping point was the mid-1970’s. This era was extremely important for the United States with a number of psychically jarring events taking center stage. The upheaval of the 1960’s had turned society on its head with deep changes in racial and sexual politics. The Vietnam War had undermined the Nation’s innate sense of supremacy while scandal ripped through the government. Faith and trust in the United States took a major hit. At the same time it marked the apex of economic equality with the beginnings of the trends that have undermined it ever since. This underlying lack of faith and trust in institutions has played a key roll in powering our decline. The anti-tax movement that set in motion public policy that drives the growing inequality in income and wealth began then arising from these very forces. These coupled to the insecurities of national defense, gender and race to form the foundation of the modern conservative movement. These fears have been used over and over to drive money and power into the military-intelligence-industrial-complex at a completely irrational rate.

“They who can give up essential liberty to obtain a little temporary safety deserve neither liberty nor safety.” ― Benjamin Franklin

The so-called Patriot Act is an exemplar of the current thinking. There seems to be no limit to the amount of freedom American will sacrifice to gain a marginal and inconsequential amount of safety. The threat of terrorism in no way justifies the cost. Dozens of other issues are a greater threat to the safety of the public, yet receive no attention. We can blame the unholy alliance of the news media and politicians for fueling this completely irrational investment in National security coupled to a diminishment of personal, and societal liberty. We have created a nation of cowards who will be declared to be heros by the same forces that have fueled the unrelenting cowardice. The fear that 9-11 engendered in the culture unleashed a number of demons on our culture that we continue to hold onto. In addition to the reduction in the freedoms we supposedly cherish, we have allowed our nation to conduct themselves in manner opposed to our deepest principles for more than a decade.

“If failure is not an option, then neither is success.” ― Seth Godin

We are left with a society that commits major resources and effort into managing inconsequential risks. Our public policy is driven by fear instead of hope. Our investments are based on fear, and lack of trust. Very little we end up doing now is actually bold or far-sighted. Instead we are over-managed and choose investments with a guarantee of payoff however small it might be.

Fear of failure is killing progress. Research is about doing new things, things that have never been done before. This entails a large amount of risk of failure. Most of the time there is a good reason why things haven’t been done before. Sometimes it is difficult, or even seemingly impossible. At other times technology is opening doors and possibilities that didn’t exist. Nonetheless the essence of good research is discovery and discovery involves risk. The better the research is, the higher the chance for failure, but the potential for higher rewards also exists. What happens when research can’t ever fail? It ceases being research. More and more our public funding of research is falling prey to the fear-mongering, risk avoiding attitudes, and suffering as a direct result.

At a deep level research is a refined form of learning. Learning is powered by failure. If you are not failing, you are not learning or more deeply stretching yourself. One looks to put themselves into the optimal mode for learning by stretching themselves beyond their competence just enough. Under these conditions people should fail a lot, not so much as to be disastrous, but enough to provide feedback. Research is the same. If research isn’t failing it is not pushing boundaries and the efforts are suboptimal. This nature of suboptimality defines the current research environment. The very real conclusion is that our research is not failing nearly as much as it needs to. Too much success is actually a sign that the management of the research is itself failing.

A huge amount of the problem is directly related to the combination of short-term thinking where any profit made now is celebrated regardless of how the future works out. This is part of the whole “maximize shareholder value” mindset that has created a pathological business climate. Future value and long-term planning has become meaningless in business because any money invested isn’t available for short-term shareholder value. More than this, the shareholder is free to divest themselves of their shares once the value has been sucked away. Over the long-term this has created a lot of wealth, but slowly and steadily hollowed out the long-term future prospects for broad swaths of the economy.

To make matters worse government has become addicted to these very same business practices. Research funding is no exception. The results must be immediate and any failure to give an immediate return is greeted as a failure. The quality and depth of long-term research is being destroyed by the application of these ideas. These business ideas aren’t good for business either, but for science they are deadly. We are slowly and persistently destroying the vitality of the future for fleeting gains in the present.

“Anyone who says failure is not an option has also ruled out innovation.” ― Seth Godin

Maybe if the United States continues to proudly proclaim itself as the “home of the brave and the land of the free” we might make an effort to actually act like it. Instead we just proclaim it like another empty slogan. Right now this slogan is increasingly false advertising.

“Theories might inspire you, but experiments will advance you.” ― Amit Kalantri

This week I have a couple of opportunities to speak directly with my upper management. At one level this is nothing more than an enormous pain in the ass, but that is my short sighted monkey-self speaking. I have to prepare two talks and spend time vetting them with others. It is enormously disruptive to getting “work” done.

On the other hand, a lot of my “work” is actually a complete waste of time. Really. Most of what I get paid for is literally a complete waste of a very precious resource, time. So it might be worthwhile making good use of these opportunities. Maybe something can be done to provide work more meaning, or perhaps I need to quit feeling the level of duty to waste my precious time on stupid meaningless stuff some idiot calls work. Most of the time wasting crap is feeding the limitless maw of the bureaucracy that infests our society.

Now we can return to the task at hand. The venues for both engagements are somewhat artificial and neither is ideal, but its what I have to work with. At the same time, it is the chance to say things that might influence change for the better. Making this happen to the extent possible has occupied my thoughts. If I do it well, the whole thing will be worth the hassle. So with hope firmly in my grasp, I’ll charge ahead.

I always believe that things can get better, which could be interpreted as whining, but I prefer to think of this as a combination of the optimism of continuous improvement and the quest for excellence. I firmly believe that actual excellence is something we have a starkly short supply of. Part of the reason is the endless stream of crap that gets in the way of doing things of value. I’m reminded of the phenomenon of “bullshit jobs” that has been recently been observed (http://www.salon.com/2014/06/01/help_us_thomas_piketty_the_1s_sick_and_twisted_new_scheme/). The problem with bullshit jobs is that they have to create more work to keep them in business, and their bullshit creeps into everyone’s life as a result. Thus, we have created a system that works steadfastly to keep excellence at bay. Nonetheless in keeping with this firmly progressive approach, I need to craft a clear narrative arc that points the way to a brighter, productive future.

High performance computing is one clear over-arching aspect of what I work on. Every single project I work on connects to this. The problem is that to a large extent HPC is becoming increasingly disconnected from reality. Originally computing was an important element in various applied programs starting with the Manhattan project. Computing had grown in prominence and capability through the (first) nuclear age in supporting weapons and reactors alike. NASA also relied heavily on contributions from computing, and the impact of computation modeling improved the efficiency of delivery of science and engineering. Throughout this period computing was never the prime focus, but rather a tool for effective delivery of a physical product. In other words there was always something real at stake that was grounded in the physical “real” world. Today, more and more there seems to have been a transition to a World where the computers became the reality.

More and more the lack of support for the next supercomputer is taking on the tone and language of the past, as if we have “supercomputer gap” with other countries. The tone and approach is reminiscent of the “missile gap” of a generation ago, or the “bomber gap” two generations ago. Both of those gaps were BS to a very large degree, and I firmly believe the supercomputer gap is too. These gaps are effective marketing ploys to garner support for building more of our high performance computers. Instead we should focus on the good high performance computing can do for real problem solving capability, and let the computing chips fall where they may.

There is a gap, but it isn’t measured in terms of FLOPS, CPUs, memory, it is measured in terms of our practice. Our supercomputers have lost touch with reality. Supercomputing needs to be connected to a real tangible activity where the modeling assists experiments, observations and design in producing something that services a societal need. These societal needs could be anything from national defense, cyber-security, space exploration, to designing better more fuel-efficient aircraft, or safer more efficient energy production. The reality we are seeing is that each of these has become secondary to the need for the fastest supercomputer.

A problem is that the supercomputing efforts are horribly imbalanced having become primarily a quest for hardware capable of running the LINPAC benchmark the fastest. LINPAC does not reflect the true computational character of the real applications supercomputers use. In many ways it is almost ideally suited towards demonstrating high operation count. Ironically it is nearly optimal in its lack of correspondence to applications. As a result of the dynamic that has emerged is that real application power has become a secondary, optional element in our thinking about supercomputing.

These developments highlight our disconnect from reality. In the past, the reality of the objective was the guiding element in computing. If the computing program got out of balance, reality would intercede to slay any hubris that developed. This formed a virtuous cycle where experimental data would push theory, or computed predictions would drive theorists to explain, or design experiments to provide evidence.

In fact, we have maimed this virtuous cycle by taking reality out of the picture.

The Stockpile Stewardship program was founded as the alternative to the underground testing of nuclear weapons, and supercomputing was its flagship. We even had a certain official say that a computer could be “Nevada* in box” and pushing the return key would be akin to pressing the button on a nuclear test. It was a foolish and offensive thing to say, almost everyone else in the room knew it was; yet this point of view has taken root, and continues to wreck havoc. Then and now, the computer hardware has become nearly to sole motivation with a loss of the purpose for the entire activity far too common. Everything else needed to be successful has been short-changed in the process. With the removal of the fully integrated experiments of the nuclear test from the process, the balance in everything else needed to be carefully guarded. Instead, this balance was undermined almost from the start. We have not put together a computing program with sufficient balance, support and connections to theory and experiment to succeed, as the Country should demand.

“The real world is where the monsters are.” ― Rick Riordan

I have come to understand that there is something essential in building something new. In the nuclear reactor business, the United States continues to operate old reactors, and fails to build new ones. Given the maturity of the technology, the tendency in high performance computing is to allow highly calibrated models to be used. These models are highly focused on working within a parameter space that is well trodden and containing to be the focus. If the United States were building new reactors with new designs the modeling would be taxed by changes in the parameter space. The same is true for nuclear weapons. In the past there were new designs and tests that either confirmed existing models, or yielded a swift kick to the head with an unexplained result. It is the continued existence of the inexplicable that would jar models and modeling out of an intellectual slumber. Without this we push ourselves into realms of unreasonable confidence in our ability to model things. Worse yet we allow ourselves to pile all our uncertainty into calibration, and then declare confidently that we understand the technology.

At the core of the problem is the simple, easy and incorrect view that bigger, faster supercomputers are the key. The key is deep thought and problem solving approach devised by brilliant scientists exercising the full breadth of scientific tools available. The computer in many ways is the least important element in successful stewardship; it is necessary, but woefully insufficient to provide success.

“Never confuse movement with action.” ― Ernest Hemingway

Supercomputing was originally defined as the use of powerful computers to solve problems. Problem solving was the essence of the activity. Today this is only true by fiat. Supercomputing has become almost completely about the machines, and the successful demonstration of the machines power on stunt applications or largely irrelevant benchmarks. Instead of defining the power of computing by problems being solved, the raw power of the computer has become the focus. This has led to a diminishment in the focus on algorithms and methods, which has actually a better track record than Moore’s law for improving computational problem solving capability. The consequence of this misguided focus is a real diminishment in our actual capability to solve problems with supercomputers. In other words, our quest for the fastest computer is ironically undermining our ability to use computers effectively as possible.

The figure below shows how improvements in numerical linear algebra have competed with Moore’s law over a period of nearly forty years. This figure was created in 2004 as part of a DOE study (the Scales workshop URL?). The figure has several distinct problems: the dates are not included, and the algorithm curve is smooth. Adding texture to this is very illuminating because the last big algorithmic breakthrough occurred in the mid 1980’s (twenty years prior to the report). Previous breakthroughs occurred on an even more frequent time scale, 7-10 years. Therefore in 2004 we were already overdue for a new breakthrough, which has not come yet. On the other hand one might conclude that multigrid is the ultimate linear algebra algorithm for computing (I for one don’t believe this). Another meaningful theory might be that our attention was drawn away from improving the fundamental algorithms towards a focus on making these algorithms work on massively parallel supercomputers. Perhaps improving on multigrid is a difficult problem, and the problem might be that we have already snatched all the low hanging fruit. I’d even grudgingly admit that multigrid might be the ultimate linear algebra methods, but my faith is that something better is out there waiting to be discovered. New ideas and differing perspectives are needed to advance. Today, we are a full decade further along without a breakthrough, and even more due for a breakthrough. The problem is that we aren’t thinking along the lines of driving for algorithmic advances.

I believe in progress; I think there are discoveries to be made. The problem is we are putting all of our effort into moving our old algorithms to the new massively parallel computers of the past decade. Part of the reason for this is the increasingly perilous nature of Moore’s law. We have had to increase the level of parallelism in our codes by immense degrees to continue following Moore’s law. Around 2005 the clock speeds in microprocessors stopped their steady climb. For Moore’s law this is the harbinger of doom. The end is near, the combination of microprocessor limits and parallelism limits are conspiring to make computers amazingly power intensive, and the continued rise as in the past cannot continue. At the same time, we are suffering from the failure to continue supporting the improvements in problem solving capability from algorithmic and method investments that had provided more than Moore’s law-worth in increased capability.

A second piece of this figure that is problematic is the smooth curve of advances in algorithm power. This is not how it happens. Algorithms have breakthroughs and in the case of numerical linear algebra it is how the solution time scales with the number of unknowns. This results is quantum leaps in performance when a method allows us to access a new scaling. In between these leaps we have small improvements as the new method is made more efficient or procedural improvements are made. This is characteristically different than Moore’s law in a key way. Moore’s law is akin to a safe bond investment that provides steady returns in a predictable safe manner. Program managers and politicians love this because it is safe whereas algorithmic breakthroughs are like tech stocks; sometimes it pays off hugely, most of the time the return is small. This dynamic is beginning to fall apart; Moore’s law will soon fail (or maybe it won’t).

I might even forecast that the demise of Moore’s law even for a short while might be good for us. Instead of relying on power to grow endlessly, we might have to think a bit harder about how we solve problems. We won’t have an enormously powerful computer that will simply crush problems into submission. This doesn’t happen in reality, but listening to supercomputing proponents you’d think it is common. Did I mention bullshit jobs earlier?

The truth of the matter is that computing might benefit from a discovery that will allow the continuation of the massive progress of the past 70 years. There is no reason to believe that some new technology will bail us out. The deeper issue regards the overall balance of the efforts. The hardware and software technologies have always worked together in a sort of tug-of-war that bares similarity to what we see in tension between theoretical and experimental science. One field drives the other depending on the question and the availability of emergent ideas or technologies that opens new vistas. Insofar as computing is concerned my concern is plain: hardware concerns have had preeminence for twenty or thirty years while algorithmic and method focus has waned. The balance has been severely compromised. Enormous value has been lost to this lack of balance.

This gets to the core of what computing is about. Computing is a tool. It is a different way to solve problems, manage or discover information and communicate. For some computing has become an end unto itself rather than a tool for modern society. We have allowed this perspective to infect scientific computing as a discipline because of the utility of acquiring new supercomputers outweighs using them effectively. This is the root of the problem and the cause of the lack of balance we see at present. This is coupled to a host of other issues in society, not the least of which is a boundless superficiality that drives a short-term focus and disallows real achievement because of the risk of failure has been deemed unacceptable.

We should work steadfastly to restore the necessary balance and perspective for success. We need to allow risk to enter into our research agenda and set more aggressive goals. Requisite with this risk we should provide greater freedom and autonomy to those striving for the goals. Supercomputing should recognize that the core of its utility is computing as a problem solving approach that relies upon computing hardware for success. There is an unfortunate tendency to simply state supercomputing as a national security resource regardless of the actual utility of the computer for problem solving. These claims border on being unethical. We need computers that are primarily designed to solve important problems. Problems don’t become important because a computer can solve them.

* Nevada is the location of the site the United States used for underground nuclear testing.

“To avoid criticism say nothing, do nothing, be nothing.” ― Aristotle

Peer review is undergoing a bit of a crisis these days. As with peer review itself a hard sharp look at a topic is a good thing. It is a key professional responsibility that is done without pay and little explicit appreciation. With the cost of academic journals skyrocketing people are rightly asking tough questions about the system where institutions pay for journals twice: once for the access to the journals, and a second time through the labor of their employees in conducting the peer review. Over the past twenty years I have also seen the standard of peer review for organizations up close. It is backsliding for the simple reason that the powers that be do not understand or appreciate peer review’s basic role. Because the consequences of a negative review have become so severe real hard-hitting peer review is rarely done, and never on the record. The powers that be have come to be completely intolerant of anything that looks like a mistake or failure.

“If failure is not an option, then neither is success.” ― Seth Godin

Critical hard-hitting peer review is necessary for the successful conduct of science. This is almost never questioned until one starts to peel away the most superficial layers of the scientific enterprise. In its purest form, peer review works, but reality soon intrudes as the issue is examined. It is broadly acknowledged that a positive, but critical review is enormously challenging. It is also utterly essential to self-improvement. Too often the delivery of reviews is off the mark and either comes off as parochial and mean-spirited, or perhaps worse yet, completely without depth and integrity.

In the cold light of day we should greet a well-done critical review as a gift. A well done review won’t just tell you how great you are, or how good you ideas are, but point out the weaknesses in your work, and suggest how to improve. If you’re just hearing how great you are in all likelihood the review isn’t being done well and the praise you are getting is bullshit. Moreover this bullshit is doing you a great disservice.

For me, I open email containing the review of a paper with dread. It is always personal, at the same time it isn’t usually nearly as bad as I fear. In the long run, my own work is greatly improved by peer review, and to be honest it would suffer greatly without it. Ultimately it is a great force for the good, it keeps me sharp, honest and exacting in the quality of my work. Nonetheless it is a difficult thing to deal with and the weaker part of me would gladly avoid it at times. My thinking self intervenes and takes it as an important part of self-improvement. Occasionally there is a deep disagreement, and the reviewer is wrong, but these points are usually contentious and not fully decided by the community. A good review is always an opportunity to learn and grow.

There are other issues with peer review worth taking note of. For example, certain people become quite renowned and unfortunately get immunized from peer review. A friend of mine had two fairly famous advisors and submitted a paper with them. The paper received no real review, the reviews just said “great work” and “publish”. It was a good paper, but in this case the process was broken. These non-reviews were a severe disservice to the community, my friend and even his famous advisors. Because the reviewers were anonymous, we don’t know who they were, but they did no one a favor. Even this paper could stand to be improved. I can say that my reviews are never that glib. On the other hand there are times I could do better as a reviewer, but I’m under the view that I could always do better.

The places I’ve worked are themselves reviewed in keeping with the basic scientific attitude that peer review is a necessity. The idea is good, and could be just as valuable for my employers as the peer review is for me as a scientist. Like many good ideas, the execution of the idea is flawed. Over time the flaws have grown until it is fair to say that the system is broken. The technical work is highly scripted, shows the best the organization has and never receives any more than a smattering of critique. The review only hits around the edges, and the marks given to the organization are always “World Class”. What criticism is received doesn’t really need to be addressed anyway. Rarely, if ever, does the review lead to anything except a declaration “we are great again this year”.

How did we get this way? This whole attitude has two sources: the lack of understanding of how science works, and the unwillingness to accept failure of any sort. There is a lack of understanding that mistakes happen when people or organizations stretch and challenge themselves. If you aren’t making mistakes you probably aren’t applying yourself, or trying very hard. We also systematically lowball all our expectations for achievement to avoid the possibility of mistakes or failure. This is a chronic condition that is slowly sucking the vitality from our research institutions. It is literally a crisis.

As most educators note, mistakes are necessary for success as they are the foundation of learning. If one isn’t making mistakes, or outright failing, they aren’t pushing their own limits. We are increasingly defining a system that takes mistakes and the possibility of failure off the table. The consequences of this are grave in that these mistakes and failures are the engine of success. This is not to say that we should allow the problems of malpractice or lack of seriousness of effort to creep into our work. I am saying that we need to encourage the possibility of failure and mistakes arising from honest, earnest efforts without the current threats of repercussions. By driving mistakes and failures from the system we guarantee mediocrity. If I look around me at the system we have created I see boundless mediocrity. We are becoming a milquetoast Nation. Gone are the bold initiatives that made our country proud. Now everyone is afraid of screwing up, which is precipitating the biggest screw-up of them all.

What needs to be done? How do we get out of this horrible bind?

We need to start by meaningfully differentiating between mistakes and failures due to incompetence from those coming from ambition. If we continue to punish ambitious efforts that fail in the manner we do, we will kill ambition. In many cases one glorious success is worth 100 noble failures. Our attitude toward any of these things in making sure that we have 100 mediocre successes that can be spun into seeming competence. We need to demand high standards, ask hard questions and focus on doing our best. We need to recognize that the lack of mistakes and failures is actually a bad thing and sign that things are not working.

We need to encourage our critics to find our flaws, demand we fix them and tell us where we can do better. We are not good enough, not by a long shot.

What comes to mind when you think of a scientist at a conference? What about when the conference is being held somewhere nice like Hawaii, a ski resort, Italy, or France? Does this make it a boondoogle? Should the government severely regulate or stand in the way of scientists attending these meetings?

Well they do, and it is harming the quality of science in the United States. In addition to harming science it isn’t saving any money, but rather costing more money. The scandal isn’t scientists attending conferences, but rather the government’s mismanagement of the scientific enterprise to such a massive degree.

I attend a number of conferences each year (probably on the order of five to eight a year). As a working scientist this is an absolute necessity for success. It is a responsibility as a researcher to actively participate in the presentation of new research findings and as part of the peer audience. Additionally, it is an essential form of continuing professional education. As I’ve matured, the organization of conferences and associated sub-meeting or mini-symposia has become a staple of my professional work. It is important work, and the challenges have become excessive lately.

The attendance at conferences of those who either work for, or are funded by the federal government is being heavily scrutinized. The reason was a General Services Administration (GSA) conference in 2010 that was quite a boondoggle. The GSA management in organizing and structuring the meeting showed exceedingly poor judgment.* They probably should have lost their jobs, which would have been a rational response. As most people know, the governmental response is far from rational.  Like most scandals, the over-reaction has been worse and more expensive than the original scandal itself.  The costs incurred by the administration of conference attendance and extra costs through delays, and unnecessary management attention on the topic makes it clear that money is not the issue. Avoiding the appearance of impropriety is the goal. The system is succeeding in producing an environment that is increasingly hostile to scientific research, and undermines the advance and practice of science. Another more poetic way of stating the approach would be dubbing it as a “circular firing squad.”

So why do scientists need to attend conferences?

We can start by talking about what a conference is and what purpose it serves. Typically a conference is associated with a defined technical field such as “Compressed Sensing” or a professional organization such as the “American Physical Society” or a combination of the two. Conferences come in all shapes and sizes. Some meetings are enormous (think meetings of societies such as the American Geophysical Union) to small topical workshops on emerging fields with 20 or 30 scientists. Each has immense importance to science’s progress. The key aspect of the conference is the exchange of information, with people taking a number of distinct roles: presenter, audience, critic, connector, teacher, and student… A conference is enormously important to the conduct of science. The exchange of ideas and subsequent debate, sharing of common experience, friendships all play a key role in successful research.

Judging by how conference attendance is managed the main goal of attending a conference is giving a talk. Everything else is secondary. This is where the damage crosses the line over into outright malpractice. When a young scientist joins the new community sometimes the best thing to do is have them attend a conference and absorb the breadth and depth of the field. It also provides an avenue to meet their new colleagues, and learn the culture by immersing themselves in it. This is almost impossible today.

The benefits of attending conferences go well beyond the purely technical aspects of the profession. Conferences are where new ideas are presented or different ideas are debated in open forum. Sometimes different points-of-view can be engaged directly leading to breakthroughs that wouldn’t be possible otherwise. There is something special about human beings sharing a meal together that cannot be replicated in other ways. Conferences are key in developing vibrant technical communities that empower the advance of science and technology. My government’s response to a stupid GSA scandal is putting all of these benefits at risk.*

I’ve quipped that we should have a special conference center is some awful place where no one would want to go. That way the Congress and public would know that we go to the conferences to engage in technical work. On the other hand, part of going to conferences involves getting inspired to do better work. Why not go to some place that is inspiring? Why not go to some place that has great restaurants so that the sharing of the meal can be memorable on multiple levels? Why not make the entire event memorable and worthwhile and enriching at a personal level? At the core of the attitude of many in government is a sense that life should be suffered with work being the most unpleasant aspect of them all. It is a rather pathetic point of view that leads to nothing positive. We shouldn’t be punished for working in the public sphere, yet punishment seems to be the objective.

Let me get to the point of attending conferences in foreign countries. Science is international, now more than ever. Thanks to lousy funding, lousy education and lousy management (with the topic here being the latest example) a lot of the best science happens in other countries. It always has, but the balance has been tipped ever more toward Europe, China, India… The mismanagement of conference attendance is some ways is completely consistent with the mindset that is overturning the United States’ supremacy in science. One can argue that like the health of the American middle class, we are already second rate in many regards. The mismanagement of science is simply driving this outcome ever more strongly. Politicians, the citizens who put them in office, and the vested interests funding campaigns care little about the state of science in the United States. We are working to undo the sort of advantage the United States had during most of the 20^th Century. Corporations seem to care little especially considering that they don’t really abide by borders thus science in Europe or China can benefit them as well. It is the rank and file citizens of the United States who will suffer the economic price for the lack of scientific discovery and technological innovation precipitated by the systematic mismanagement we see today.

Scientists are people and we respond to the same things as everyone else. The attendance of conferences is an essential aspect of doing science, the current approach and attitude toward conferences is undermining the quality and effectiveness of science. This should deeply concern every citizen because the quality of science has a direct impact on society as a whole. Whether your concerns are grounded on the health of the economy, or the National security, or our role as World leaders, science plays a key role in success. In the process of our systematic mismanagement of the scientific enterprise we are failing each of these.

* In an earlier version of the post I incorrectly identified the Internal Revenue Service (IRS) as the government agency responsible for the scandalous conference in 2010.

“In other words, it’s a huge shit sandwich, and we’re all gonna have to take a bite.”–Lt. Lockhart, Full Metal Jacket

When I starting to write this the thought occurred to me, “do I really want to do this? This topic is a lightning rod and it’s sure to piss everyone off!” Of course, this is exactly the reason it needs to be discussed. Climate is an enormous scientific and societal problem that has become a horrific “shit sandwich” that we all get to share. It is starting to infect the entirety of the societal engagement with science in a profoundly negative way. A thoughtful and open discussion about the quality and reliability of the underlying science cannot be had. This serves two terrible purposes: it energizes the climate “deniers” who have defacto won the argument by making it completely toxic, and it savages the public image of science, damaging not just the perception of science, but its practice. We are facing the prospect that the only outcome for humanity is bad. So I’m going to grab the proverbial lightning rod with both hands.

Let’s get to one of the elephants in the room right away; the issue of deniers, skeptics and critics needs immediate attention. My contention is that these labels are important and the distinctions are important. First, the positive side of the coin, the critics are essential to progress and an honest dialog on the subject. The current circumstances are drowning out the ability to be critical of climate science. This is dangerous. The science is good, but not good enough; it is never good enough. Because criticism is so muted by the polarized political atmosphere-surrounding climate, the skeptics are rightly energized. Some degree of skepticism is warranted especially considering how the scientific community is characterized. The problem is that as one moves along the spectrum of skepticism, one approaches the third category, the denier. The denier cannot be defended, it is simply the outright denial of facts, of science, but we are creating a situation where the facts are muddied by both sides of the argument. A modern society should rightly, repudiate the deniers; instead cynical and greedy forces are empowering them. To the extent that the scientific community misbehaves, they also empower denial as reasonable.

Science is being horribly politicized these days. No subject is more so than climate science on the topic of climate change or the more pointed and accurate term, global warming. Scientists are not the core of the problem, but they do add to the toxic mix by responding to the environment in a manner that undermines the credibility of science. We are confronted with a situation where the scientific results threaten the ability of greedy people to make lots of money. The greedy people who stand to have their earning power diminished are fighting back. Some of their weapons are scientists who side with them for largely ideological reasons (or outright financial gain). There are always scientists who are willing to sow doubt as hired guns of the greedy, just as both sides of a court cases can get their own experts if the price is right. The truly damaging part of this dialog is the damage the credible scientific community is doing to itself in joining the battle.

Climate science is the archetype of this dynamic much like tobacco was a generation ago. The amazing thing is that some of the same hired guns that are attacking climate scientists today attacked the idea that tobacco was causing cancer then. The majority of these scientists have absolutely zero credibility. Or to put it another way, they have the same credibility I have as a climate expert. My views here are associated with how science is being conducted, and the atmosphere for improvement, or how criticism of the quality of the science can proceed without playing into the hands of the denial industry. As a scientist this is completely within the frame of valid expertise on my part.

I’m not engaging in any sort of false equivalence although one might see this as the conclusion of this piece. The difference in level of damage to science and society by the misbehavior on each side is vastly different. The side of denial has almost the complete absence of merit. Whatever merit it does have is associated with the critical side of skepticism, and nothing from the self-interested parties providing most of the resources for the “movement”. The sins of the climate science community are basically at the margins, but lead to a loss of effectiveness. First and foremost, the World needs to realize that an enormous problem exists, and needs to be addressed quickly and forthrightly. The problem is when the climate community starts to address the actions to be taken to deal with global warming. There are many potential ways of addressing the problem and all of them are expensive and controversial. Some of the ways that would be most effective do not suit the “green” agenda. The problem is that too many who are trying to sound the alarm are also pushing specific solutions, and in particular solutions from the left. This undermines the ability to convince the World that there is a problem that must be solved. Moreover it makes science itself look like a partisan activity with a given political point-of-view.

Before I go further, I will just state up front that my judgment, for what its worth, is that man is the prime element causing the observed global warming through our collective industrial and agricultural activities. The evidence of the warming is rock solid, and the hypothesis that this warming is dominantly anthropogenic is very likely true. This is still a hypothesis and most of my quibbling rests upon discomfort with the level of credibility of climate models. We must be careful with regard to the level of uncertainty of these models, which is likely to be larger than commonly characterized because of the methodology used. This care must include the proviso that the amount of warming could well be much larger than predicted, which would prove catastrophic for humanity.

I have to admit that this issue is personal at some level. My parents are deniers born and bred through watching the propaganda machine known as Fox News. This has led to them being exposed to above-mentioned hired guns as appropriate experts in climate science. I’ve read one of their books (by Fred Singer, which my dad had been reading), and found it to be complete crap, but well enough written to fool an educated person without an appropriate technical background. My advice was to pay attention to the author’s conflicts of interest and past associations. For example previous funding by the tobacco lobby should be a clear “red flag”. My key point is there are more scientifically credible skeptics who make valid critiques. Those critiques are more scientific and not meant to be digested like another source of propaganda. As such they are much more difficult to make serve the biased purposes of the denier’s funding sources.

The climate skeptics are primarily driven and funded by interests that have massive financial interests in continued (or accelerated) use of carbon-based fuel. Others have a conservative world-view associated with Manifest Density (or put more bluntly, God put the Earth here for man to rape to his heart’s content). Fortunately these attitudes are countered by evangelicals who believe in stewardship of the Earth is their divine responsibility. Here in the USA, the old school rape and pillage the Earth types still have the edge. These people aren’t skeptics they are simply greedy or delusional self-centered people who have absolutely no credible argument against the science. Their only goal is to seed doubt in the mind of the untrained masses that form the majority of our society.

This is not to say that every skeptic is simply the willing tool of greedy corporate interests, or ideological zealots devoted to clear cutting as a God-given right. It is a spectrum where one end has honest and meaningful critical assessments, and the other with near denial of all evidence contrary to their opinion. Climate science has a lot of problems today that need to be solved. The honest skeptic is an important voice of criticism that ultimately drives the science to be better. There are some skeptics who have reasonable scientific arguments, but the charlatans drown them out. Moreover too many of the skeptics fail to call out the charlatans for what they are. Others are worse and figuratively get in bed with them. Unfortunately, the climate community isn’t behaving themselves either and has succeeded in helping to produce an appallingly poisonous environment for improving the science.  Worse yet, the climate community’s defense of their conclusions is assisting the poisoning the entire societal dialog on science and its proper role.

Let me be clear, the vast majority of the blame for the current state of affairs lies with greedy corporate interests that want to preserve the status quo that enriches them, the future of humanity be damned. They fund scientific skeptics to undermine science and ally themselves with conservative ideological interests who agree with the outcome they want. Their goals are fundamentally unethical and immoral. They are the worst kind of societal scum. If science doesn’t help line their pockets, they will oppose it. In opposing these forces, the scientific community has lowered their standards by embracing concepts that are unscientific. Key among these is the concept of consensus as defacto proof.

Consensus is not proof, but it plays a thorny role in the scientific process (i.e., peer review). Sometimes the consensus view is not correct. Most of time it is correct, but on occasion it is wrong. Consensus is a reflection of agreement within the community and a statement about what the best technical judgment is (at the moment), but not proof. The problem is that the public is being sold on the idea that consensus is proof, or at least when it comes off that way, no one corrects them (aside from the skeptics). The awful thing about this is that the scientific community is then handing the moral high ground over to the skeptics, even the scummy hired guns. Consensus is simply the critical judgment of the community that a given line of reasoning is favored given all of the evidence. All in all I would add my name to that consensus while remaining critical of the science. It really doesn’t matter that the consensus is 97% or if it were 92% or 99%. These numbers are meaningless insofar as proof is concerned. It is agreement, and nothing more.

Let’s talk about the scientific method and where the climate science sits with respect to it. Proof comes through agreement with observation or experiment. This is the rub. Anthropogenic (man-made) climate change (global warming) is a hypothesis. Observations seem to be unambiguous regarding warming, it is occurring and its magnitude and rate of increase are unprecedented. Arguments about the recent pause in warming are largely irrelevant to this aspect of the discussion. The observation is also highly correlated with man’s industrial activity primarily seen through increased CO2 concentrations in the atmosphere. It is worth noting that correlation does not imply causation; however, we do know unequivocally that CO2 does cause warming via the greenhouse effect. Thus the warming is correlated with a known causal effect.

Nonetheless, the hypothesis testing for the man-made basis for warming comes via modeling the climate on supercomputer. These models consistently show that warming is anthropogenic, but the proof via modeling is far from definitive despite being rather convincing. The relatively larger amount of warming in the Polar Regions is important because the models predict this. The loss of polar ice and melting of permafrost is another important observation that backs up the credibility of the models. It is the texture of this discussion that we are missing. The modeling needs to be better especially with regard to studying the sensitivity and uncertainty of the models. To put it bluntly, this work is not up to scratch, and the effort, dialog and discussion of these matters is not presently productive. Part of the problem is a general unwilliness to admit the flaws in the work publically because of how it would empower skeptics. The scumbag side of the skeptics has no interest in improving the science, and would use this honesty against climate science. This is the start of the toxic spiral because an important aspect of science has been short-circuited by the nature of the dialog. Self-criticism in the climate community is not as sharp, nor as open as it needs to be. The quality of the science will suffer, or has already suffered as a direct consequence.

The modeling community needs to be acutely focused on doing better. It is true that there is vigorous debate inside the climate science community on many aspects of the modeling, but some topics are less open. The big issue is the nature of the projections into the future, are they predictive? And if so, how predictive? How does on grapple with the question? Key to answering these big questions how does calibration enter into the models? What is this calibration doing from a modeling point-of-view? Generally, the modeling does not produce a useful result without the gross calibration, but at what cost?

Again, we get to the heart of the problem. Critics and skeptics are essential to progress, and the climate community seems to be interested in silencing skeptics, at least publically. They are systematically over emphasizing the surety of their work. They are not willing to admit the imperfections in their work openly. For example, the uncertainty in the projections in the global climate is derived by the trajectories of a host of climate models. This is not uncertainty; it is simply a model-based voting scheme, none of which has any assurance of correctness. Each of these models has an innate uncertainty associated with the model, its numerical solution, and other modeling imperfections. The issue not explored is the nature of these models’ intrinsic bias, and their impact on the projections. This entire topic needs a substantially better scientific treatment. This is an old-fashioned way of exploring the issue rather than a reflection of modern computational science. By not providing a path to better methods we are not doing our best. This is basically ceding the debate to the deniers, and empowering the status quo for decades.

Let’s get to the heart of where climate scientists really start to potentially damage their work’s significance. Whenever climate science aligns itself with the left wing of the environmental movement, the general public acceptance of the issue is harmed. Given the science it is reasonable to suggest that carbon neutral policy be pursued; however when nuclear power is rejected the community goes too far. The scientific evidence would clearly point towards reducing the production of energy via carbon-based sources as greatly as possible. Nuclear power is probably the single greatest hope to reduce carbon emissions greatly without wrecking the economy. Solar, wind and other energy sources have their place, but they cannot replace base capacity (today). Nuclear power can do this right now. Solving the energy production issue without carbon is an immense political and technical problem, but it is out of scope for the climate community. Advocating an energy path directly hurts their ability to provide the impact their science needs.

When the advocacy begins to tread into the area of economics and equality (inequality) it has definitively cross over into the political realm. People who use global warming to push these issues are a direct threat to the legitimacy of the whole field. This isn’t to say that inequality isn’t a legitimate issue; it just isn’t at the core of climate change. By coupling the two issues so closely, they simply equate themselves with the skeptics who ally themselves with carbon-spewing oligarchs. Neither extreme has a place in the debate over whether global warming is occurring, occurring due to man-made effects or whether it is a threat. The issues are important, but decoupled. Coupling them creates the toxic blend we have today.

The consensus issue is a problem for science in general. Science is about truth and we don’t vote for what is right. If 97 percent of scientists believe something it doesn’t mean a damn thing. They could be completely wrong, science does not work through consensus, and it works through evidence.

Let’s talk about what the evidence says. The Earth is warming, and warming at a rate that is unprecedented in the natural climate record. Something very dangerous to every inhabitant of the planet is happening. Why it is happening is the issue. The working hypothesis is greenhouse gases, and that is where modeling comes in. Just because all the models seem to agree with they hypothesis does not make it the truth. This focus on consensus as proof is hurting the scientific community in every field because it poisons the public perception of science.

To be clear I’m not saying that I don’t believe in anthropogenic global warming, I do. I believe that the combined effect of man’s activity in burning carbon-based fuel, agriculture and deforestation is driving climate change. This is a hypothesis. It is a compelling scientific argument that makes logical sense and fits the observations. I just cannot prove it. I don’t think climate science has proven it either. The greenhouse effect along with other human activity is the leading contender for the observed warming. While not proven, the evidence is large enough that National and International policy should be reflected in mitigating the activities that are most likely causing it.

What is proven is that the Earth is warming up a lot, and it is almost certainty a very bad thing. By very bad I mean that millions if not billions of people will lose their lives as a result. This ought to spur action. The people paying the majority of the skeptics don’t care; there is too much money to be made.

Something I am an expert on is modeling. I’m also an expert in modeling credibility. The approach that the climate community (IPCC) has taken to demonstrating credibility is very problematic. Basically the models are voting for outcomes, and in this perverse way it shadows the consensus argument in a perverse way. They are not actually providing any credible view of their uncertainty or accuracy. In the end this fails to provide the sort of clear guidance needed to improve the modeling. The whole of their approach does not reflect the best in computational science. Despite this criticism, I’d take their models as being a reasonable reflection to the Earth’s response. I’m questioning the overall quality of their approach and evidence.

The end result is a climate science community that has played to the lowest denominator. We need great science to study this issue. Instead we have devolved into a mindless shouting match that basically hands victory to those who would have us do nothing. This is a tragedy because climate change is an existential threat to our species.

Ultimately science and technology must be healthy if we can hope to deal with the consequences of global warming. Transportation will need dramatic overhaul. We need to produce new options for producing energy in an economically viable manner. Geoengineering may be needed to mitigate the dumping of carbon into the atmosphere. Biological and agricultural sciences are needed to provide relief from the impacts of science. If the ability to progress via science is damaged by our collective dysfunction, the ability to respond in a healthy way to the warming will be harmed.

What would make the whole situation better? If you believe global warming is happening because of the evidence, and that the hypothesis of human causation is the best explanation then work to make the science going into these better by being critical of every weakness. If someone supports this belief because of their worldview and not the science treat them with suspicion. There are supporters of climate change as an issue because it empowers their political goals (like radical environmentalism, Marxism, etc..), not because the science says it is a problem. People with this view hurt the ability of society to deal forthrightly with the issue. If on the other hand, you don’t believe global warming is happening and/or humans are causing it then work to make the science better by being purposefully critical of what undermines your belief. As a skeptic also be critical of people who don’t believe these things because of their worldview. If you don’t then you’ll be put into the same camp as religious zealots and greedy oligarchs. These people work to undermine the legitimacy of any skepticism of the science. In the end the people who choose their answer to global warming based on worldview aren’t interested in the truth, they are interested in winning, everyone else be damned.

The quote from “Full Metal Jacket” applies to both the issue of climate change, and the harm done through the nature of the public dialog. Science is being damaged deeply by the dialog. Those trying to undermine any response to climate change aren’t simply the only ones doing this damage; those who rightly call themselves scientists are causing harm. As a direct result no one on either side is going to win, we are all going to lose.