Tiny details imperceptible to us decide everything!

― W.G. Sebald

The standards of practice in verification of computer codes and applied calculations are generally appalling. Most of the time when I encounter work, I’m just happy to see anything at all done to verify a code. Put differently, most of the published literature accepts a slip shod practice in terms of verification. In some areas like shock physics, the viewgraph norm still reigns supreme. It actually rules supreme in a far broader swath of science, but you talk about what you know. The missing element in most of the literature is the lack of quantitative analysis of results. Even when the work is better and includes detailed quantitative analysis, the work usually lacks a deep connection with numerical analysis results. The typical best practice in verification only includes the comparison of the observed rate of convergence with the theoretical rate of convergence. Worse yet, the result is asymptotic and codes are rarely practically used with asymptotic meshes. Thus, standard practice is largely superficial, and only scratches the surface of the connections with numerical analysis.

The Devil is in the details, but so is salvation.

― Hyman G. Rickover

The generic problem is that it rarely occurs at all much less being practiced well, then we might want to do it with genuine excellence. Thus, the first step to take is regular pedestrian application of standard analysis. Thus, what masquerades as excellence today is quite threadbare. We verify order of convergence in code verification under circumstances that usually don’t meet the conditions where they formally apply. The theoretical order of convergence only applies in the limit where the mesh is asymptotically fine. Today, the finite size of the discretization is not taken directly into account. This can be done, I’ll show you how below. Beyond this rather great leap of faith, verification does not usually focus on the magnitude of error, numerical stability, or the nature of the problem being solved. All of these are available results through competent numerical analysis, in many cases via utterly classical techniques.

A maxim of verification that is important to emphasize is that the results are a combination of theoretical expectations, the finite resolution and the nature of the problem being solved. All of these factors should be considered in interpreting results.

Before I highlight all of the ways we might make verification a deeper and more valuable investigation, a few other points are worth making about the standards of practice. The first thing to note is the texture within verification, and its two flavors. Code verification is used to investigate the correctness of a code’s implementation. This is accomplished by solving problems with an analytical (exact or nearly-exact) solution. The key is to connect the properties of the method defined by analysis with the observed behavior in the code. The “gold standard” is verifying that the order of convergence observed matches that expected from analysis.

Truth is only relative to those that ignore hard evidence.

― A.E. Samaan

The second flavor of verification is solution (calculation) verification. In solution verification, the objective is to estimate the error in the numerical solution of an applied problem. The error estimate is for the numerical component in the overall error separated from modeling errors. It is an important component in the overall uncertainty estimate for a calculation. The numerical uncertainty is usually derived from the numerical error estimate. The rate or order of convergence is usually available as an auxiliary output of the process. Properly practiced the rate of convergence provides context for the overall exercise.

One of things to understand is that code verification also contains a complete accounting of the numerical error. This error can be used to compare methods with “identical” orders of accuracy for levels of numerical error, which can be useful in making decisions about code options. By the same token solution verification provides information about the observed order of accuracy. Because the applied problems are not analytical or smooth enough, they generally can’t be expected to provide the theoretical order of convergence. The rate of convergence is then an auxiliary result of the solution verification exercise just as the error is an auxiliary result for code verification. It contains useful information on the solution, but it is subservient to the error estimate. Conversely, the error provided in code verification is subservient to the order of accuracy. Nonetheless, the current practice simply scratches the surface of what could be done via verification and its unambiguous ties to numerical analysis.

Little details have special talents in creating big problems!

― Mehmet Murat ildan

If one looks at the fundamental (or equivalence( theorem of numerical analysis, the two aspects of theorem are stability and consistency implying convergence (https://williamjrider.wordpress.com/2016/05/20/the-lax-equivalence-theorem-its-importance-and-limitations/ ). Verification usually uses a combination of error estimation and convergence testing to imply consistency. Stability is merely assumed. This all highlights the relatively superficial nature of the current practice. The result being tested is completely asymptotic, and the stability is merely assumed and never really strictly tested. Some methods are unconditionally stable, which might also be tested. In all cases the lack of stress testing the results of numerical analysis is short-sighted.

One of the most important results in numerical analysis is the stability of the approximation. Failures of stability are one of the most horrific things to encounter in practice. Stability results should be easy and revealing to explore via verification. It also offers the ability to explore what failure of a method looks like, and the sharpness of the estimates of stability. Tests could be devised to examine the stability of a method and confirm this rather fundamental aspect of a numerical method. In addition to confirming this rather fundamental behavior, the character of instability will be made clear if it should arise. Generally, one would expect calculations to diverge under mesh refinement and the instability to manifest itself earlier and earlier as the mesh is refined.  I might suggest that stability could be examined via mesh refinement, and observing the conditions where the convergence character changes.

One of the most unpleasant issues with verification is the deviations of the observed rate of convergence from what is expected theoretically. No one seems to have a good answer to how close, is close enough? Sometimes we can observe that we systematically get closer and closer as the mesh is refined. This is quite typical, but systematic

deviations are common. As I will show, the deviations are expected and may be predicted by detailed numerical analysis. The key is to realize that the effects of finite resolution can be included in the analysis. As such for simple problems we can predict the rate of convergence observed and its deviations for the asymptotic rate. Beyond the ability to predict the rate of convergence, this analysis provides a systematic explanation for this oft-seen results.

This can be done very easily using classical methods for numerical analysis (see previous blog post https://williamjrider.wordpress.com/2014/07/15/conducting-von-neumann-stability-analysis/). We can start with the knowledge that detailed numerical analysis uses an analytical solution to the equations as its basis. We can then analyze the deviations from the analytical and their precise character including the finite resolution. As noted in that previous post, the order of accuracy is examined via a series expansion in the limit where the step size or mesh is vanishingly small. We also know that this limit is only approached and never actually reached in any practical calculation.

For the simple problems amenable to these classical analyses, we can derive the exact rate of convergence for a given step size (this result is limited to the ideal problem central to the analysis). The key part of this approach is using the exact solution to the model equation and the numerical symbol providing an error estimate. Consider the forward Euler method for ODE’s, , the error is . We can now estimate the error for any step size and analytically estimate the convergence rate we would observe in practice. If we employ the relatively standard practice of mesh halving for verification, we get the estimate of the rate of convergence, . A key point to remember is that the solution with the halved time step takes twice the number of steps. Using this methodology, we can easily see the impact of finite resolution. For the forward Euler method, we can see that steps larger than zero raise the rate of convergence above the theoretical value of one. This is exactly what we see in practice.

When one starts to examine what we expect through analysis, a number of interesting things can be observed. If the coarsest step size is slightly unstable, the method will exhibit very large rates of convergence. Remarkably, we see this all the time. Sometimes results of verification produces seemingly absurdly high rates of convergence. Rather than being indicative of everything being great, it is an indication that the calculation is highly suspect. The naïve practitioner will often celebrate the absurd result as being triumphant when it is actually a symptom of problems requiring greater attention. With the addition of a refined analysis, this sort of result can be seen as pathological.

Immediately recognize that we have yielded a significant result with the analysis of perhaps the simplest numerical method in existence. Think of the untapped capacity for explaining the behavior observed in computational practice. Moreover, this significant result explains a serious and far pernicious problem in verification, the misreading of results. Even where the verification practice is quite good, the issue of deviation of convergence rates from the theoretical rates is pervasive. We can easily see that this is a completely expected behavior that falls utterly in line with expectations. This ought to “bait the hook” to conducting more analysis, and connecting it to the verification results.

There is a lot more that could be done here, I’ve merely scratched the surface.

The truth of the story lies in the details.

― Paul Auster

 

When truth is replaced by silence, the silence is a lie.

― Yevgeny Yevtushenko

We all live in a golden age for the abuse of power. Examples abound and touch everyone’s life. We see profound abuses in politics daily. Recently the decades of abuse committed by Harvey Weinstein have come to light. Apparently, it barely existed in the shadows for years and years as one of Hollywood’s worst kept secrets. Weinstein preyed on women with virtual impunity with his power and prestige acting to keep his actions in the dark. The promise and threat of his power in that industry gave him virtual license to act. The silence of the myriad of insiders who knew about the pattern of abuse allowed the crimes to continue unabated. Only after the abuse came to light broadly and outside the movie industry did the unacceptability arise. When the abuse stayed in the shadows, and its knowledge limited to industry insiders, it continued.

Cowardice rightly understood begins with selfishness and ends with shame.

― José Rizal

The power of the online world and our free press should be a boon to exposing and bringing these monsters down. People can be alerted to unacceptable behavior and demand action to remove these abominations from power. It is not working out this way. Instead the power of information has been turned on its head, and the modern world of information has empowered them to new heights. We only need to look at the occupant of the Oval Office for proof. People in power have access to resources and talent not available to others. This power can be turned to marketing and shaping the message to allow them to retain power. Power has its privileges and among these are access to wealth, information and (sexual) favors, most of us can’t even dare to dream of. The abusers turn all of this into a machine that retains and even grows their power. The modern world of interconnection is the latest tool in their arsenal of power. The powerful have largely controlled the media forever, but this control has taken on a new character with Facebook and Twitter.

People follow leaders by choice. Without trust, at best you get compliance.

― Jesse Lyn Stoner

I see it at work in small ways. Sometimes it’s the desire of those in power to keep their poor leadership or stewardship from being open to criticism, i.e., an honest peer review. More recently we have seen ourselves subjected to training on information security that was merely an excuse to be threatened by the United States Attorney General, Jeff Sessions. It was a gross waste of resources to provide a platform for abuse of power (at the cost many millions of dollars to threaten people, and help crush morale). Ostensibly the training was to highlight the importance of protecting sensitive and classified information. This is a topic that we are already trained heavily on, and we are acutely aware of in our daily work. Given our ongoing knowledge of the topic, the whole training was provided to silence the critics of the administration, who will now misuse information control to hide their crimes.

Compliance” is just a subset of “governance” and not the other way around.

― Pearl Zhu

The United States has gone through a virtual orgy of classification since 9/11. This is an extremely serious issue and its tendrils permeate this entire topic. I’ve written in the past about our problems in this regard. Our government, related organizations, and contractors are running wild classifying everything in sight. Increasingly the classification is used to hide and bury information. Quite often things are labeled with the “official use only” category because it is basically unregulated. There is no standard and the tendency is to simply hide as much as possible. It is primed for wide scale abuse. I’m fairly certain that the abuse is already happening on an enormous scale. It is quite often used to keep embarrassing, criminal or simply slip-shod work out of sight and away from scrutiny. It is exactly the sort of thing the current (and past) administrations would use to hide stuff from view. Of course, higher levels of classification have been used similarly. The prime example is the cover-up of the slaughter of innocents by the military in Iraq central to the whole Chelsea Manning case. It wasn’t classified, it was criminal and embarrassing, yet classification was used to attempt to bury the evidence.

Those who lack the courage will always find a philosophy to justify it.

― Albert Camus

Our current President is serial abuser of power whether it be the legal system, women, business associates or the American people, his entire life is constructed around abuse of power and the privileges of wealth. Many people are his enablers, and nothing enables it more than silence. Like Weinstein, his sexual misconducts are many and well known, yet routinely go unpunished. Others either remain silence or ignore and excuse the abuse a being completely normal.

I better use some Tic Tacs just in case I start kissing her. You know, I’m automatically attracted to beautiful — I just start kissing them. It’s like a magnet. Just kiss. I don’t even wait. And when you’re a star, they let you do it. You can do anything…. Grab ’em by the pussy. You can do anything.

– President Donald Trump

We are all enablers. We knew what kind of monster we were electing to the highest office, and people stood by silent. Worse yet they gave him their tacit approval. Weinstein is no different. Sessions is no different either, no sexual misconduct known there, but ignorance and racism is a clear part of his repertoire. In addition, he has direct conflicts of interests with vast prison stocks in his portfolio, and the power to improve his stock holdings through his office and his direct actins. Since his boss is a walking monument to conflict of interest, thus nothing will happen. He will abuse the power of his office with impunity. At this point the entire Nation has forgiven their crimes through willful ignorance. The mandated threat masquerading as training is simply the latest, and by no means the biggest violation of standards of conduct. The threat is designed to silence any documentation of violations and assist in their continued violation through our continued silence and tacit acceptance of their power.

The standards of conduct under the Trump Administration is headed straight to hell. The only thing that they are opposed to is threats to their power and ability to abuse it. They are an entire collection of champion power abusers. Like all abusers, they maintain their power through the cowering masses below them. When we are silent their power is maintained. They are moving the squash all resistance. My training was pointed at the inside of the institutions and instruments of government where they can use “legal” threats to shut us up. They have waged an all-out assault against the news media. Anything they don’t like is labeled as “fake news” and attacked. The legitimacy of facts has been destroyed, providing the foundation for their power. We are now being threatened to cut off the supply of facts to base resistance upon. This training was the act of people wanting to rule like dictators in an authoritarian manner.

I am personally quite concerned about how easily we accept this authoritarian approach to leadership. We seem all too willing and able to simply salute and accept the commands of corrupt overlords. We are threatened with extreme consequences, and those in power can do as they please with virtual impunity. For those abusing power, the set-up is perfect. They are the wolves and we, the sheep, are primed for slaughter. Recent years have witnessed an explosion in the amount of information deemed classified or sensitive. Much of this information is controlled because it is embarrassing or uncomfortable for those in power. Increasingly, information is simply hidden based on non-existent standards. This is a situation that is primed for abuse of power. People is positions of power can hide anything they don’t like. For example, something bad or embarrassing can be deemed to be proprietary or business-sensitive, and buried from view. Here the threats come in handy to make sure that everyone keeps their mouths shut. Various abuses of power can now run free within the system without risk of exposure. Add a weakened free press and you’ve created the perfect storm.

The mantle of National security and patriotism works to compliment the systematic abuse of power. One of the primary forms of abuse is financial gain. The decision making behind the flow of money is typically hidden. No one benefiting from the flow of money is too keen on the details of who got the money and why getting out. All one has to do is look at the AG’s finances. He benefits greatly through other people’s misery. More and longer prison sentences raise the stock prices he holds and enriches him. No one even asks the question, and the abuse of power goes unchecked. Worse yet, it becomes the “way things are done”. This takes us full circle to the whole Harvey Weinstein scandal. It is a textbook example of unchecked power, and the “way we do things”.

The real division is not between conservatives and revolutionaries but between authoritarians and libertarians.

― George Orwell

The only way to rid a system of these abuses is the exposure to clear view. When people see the way that those in power abuse the system, the abusers need to change their way or lose their power. This is the exact reason why the abusers are so keen to squash any disclosures (i.e., leaks). They like being able to run their shadow empires with impunity. Without the risk of exposure, the abusers can simply take their abuses to new levels. For systems needs genuine information control and security, the threat of abuse of power is extreme. If the system is not run with the highest level of ethics and integrity, the abuse of power, and the genuine need for secrecy are on a collision course. In taking my training, the threats from someone completely lacking any visible ethics or integrity is chilling. Simply put, the Attorney General is proven racist, white supremacist apologist, and serial abuser of power. He has no ethical basis to issue his threats, only unbridled power. He has been given free reign purely on ideological grounds.

Democracy can exist only in the countries where people are brave! Coward nations always live under the authoritarian regimes!

― Mehmet Murat ildan

This value system is in complete and total collision with the values that the United States is supposedly based on. This value system is unfortunately consistent with the actual values in the United States today. We are struggling between who we should be and who we are. At work this runs headlong into the fundamental character of the institutions and the people employed. We have a generally docile workforce who are easily cowered by power. We have installed authoritarian monsters in positions of power who are more than willing to abuse this power. So the sheep bow their heads to the wolves and simply welcome the slaughter. Our institutions are similarly cowered by money. The federal government pays the bills, and with that buys what is moral and ethical. A good and logical question is where does this end. What is the point where we rise up and say “NO!”? What is the point where the actions by those in power are too much to tolerate? We aren’t there yet, and I shudder thinking of where that line is. Worse yet, I’m not entirely sure there is a line that our employees, management or the institutions themselves would enforce. This is truly a terrifying prospect.

The strategic adversary is fascism… the fascism in us all, in our heads and in our everyday behavior, the fascism that causes us to love power, to desire the very thing that dominates and exploits us.

― Michel Foucault

No matter the reasons for the training I took, the tone and approach set by those governing my workplace should be chilling to contemplate. We are creating an environment where unethical conduct and rampant abuse of power go unchecked. Abusers can thrive and run free. We can be sure that their power will grow and extend to other areas. Without the checks and balances that exposure brings, the abuse of power is free to run wild and grow. We have installed an utterly repugnant serial abuser as President. He has a history of financial, personal and sexual ethics violations. He now has more power than ever, and is actively destroying the vehicles for oversight. He has surrounded himself with people of similar morality and ethics, or parasitic enablers who trade their own morality and ethics for power (like VP Pence, who like the religious right make their bed with this devil freely).

A fascist is one whose lust for money or power is combined with such an intensity of intolerance toward those of other races, parties, classes, religions, cultures, regions or nations as to make him ruthless in his use of deceit or violence to attain his ends.

― Henry A. Wallace

This sort of deal making isn’t simply for the rampant misogyny of the right wing to answer for. Those on the left have their own crimes to answer for as the Weinstein scandal makes clear. All those people standing up for Women’s rights in the World, but tolerating his private abuse of women are no better. All of their public stands for liberal causes are made shallow through the acts of private cowardice. They are equally guilty and no better than the so-called Christians embracing Trump. Some things are not acceptable, no matter who does them or their belief system. If you are a liberal Hollywood elite and you stood by while Weinstein abused and assaulted women, your morality is for sale. If you’re an evangelical Christian who voted for Trump, you are no better. Both men are monstrous abusers and morally unacceptable. Too often we make the case that their misdeeds are acceptable because of the power they grant to your causes through their position. This is exactly the bargain Trump makes with the right wing, and Weinstein made with the left.

Of course, I ask myself, am I really any better? I take my paycheck with the assumption that the terms of my employment mean they own me. What is the price of my silence? I have duty to my loved ones and my support for them. This keeps me as compliant as I am. I need to ask myself what too far looks like? I’ve asked what the limits for my employers are, and I fear there is no limit; I fear they will comply to almost anything. We are rapidly approaching a moral chasm if we haven’t already gone over the edge. Will we simply fall in, and let our nation become a kleptocracy with a wink and a nod toward our standards, ethics and morality while standing by and letting the abusers run wild. For the greater part, I think that we are already there. It is terrifying to think about how much worse it’s going to get.

Terror is a powerful means of policy and one would have to be a hypocrite not to understand this.

― Leon Trotsky

 

 

 

 

 

 

 

 

Those who deny freedom to others, deserve it not for themselves

― Abraham Lincoln

I’d like to be independent empowered and passionate about work, and I definitely used to be. Instead I find that I’m generally disempowered compliant and despondent these days. The actions that manage us have this effect; sending the clear message that we are not in control; we are to be controlled, and our destiny is determined by our subservience. With the National environment headed in this direction, institutions like our National Labs cannot serve their important purpose. The situation is getting steadily worse, but as I’ve seen there is always somewhere worse. By the standards of most people I still have a good job with lots of perks and benefits. Most might tell me that I’ve got it good, and I do, but I’ve never been satisfied with such mediocrity. The standard of “it could be worse” is simply an appalling way to live. The truth is that I’m in a velvet cage. This is said with the stark realization that the same forces are dragging all of us down. Just because I’m relatively fortunate doesn’t mean that the situation is tolerable. The quip that things could be worse is simply a way of accepting the intolerable.

What is going on, and how did we get here?

When you read management theory, and I do, you would think that good management would do the opposite. I certainly like the feeling of being empowered and valued, it makes me feel like coming to work and doing my best. It is good to feel a real sense of purpose and value in work. I have had this experience and it is incredible. When I am creative, my voice is heard and my actions lead to positive outcomes, work is a real joy. I have definitely experienced this, but not recently. If working well, a management system would strive to engage people in this manner. The current management approach acts are pretty much completely opposite to this end. The entire system seems to be geared to putting us into a confined and controlled role. Our actions are limited because of fear, and the risk that something “bad” might happen. We are simply members of a collective and our individuality is more of a threat than a resource.

Today as always, men fall into two groups: slaves and free men. Whoever does not have two-thirds of his day for himself, is a slave, whatever he may be: a statesman, a businessman, an official, or a scholar.

― Friedrich Nietzsche

Why are we managed in such an inherently destructive short-sighted and inhumane manner?

Star Trek introduced us to the Borg, a race of semi-robotic aliens who simply absorb beings (people) into a hive where their basic humanity and individuality is lost. Everything is controlled and managed for the good of the collective. Science Fiction is an allegory for society, and the forces of depersonalized control embodied by the Borg have only intensified in our world. Even people working in my chosen profession are under the thrall of a mindless collective. Most of the time it is my maturity and experience as an adult that is called upon. My expertise and knowledge should be my most valuable commodity as a professional, yet they go unused and languishing. They come to play in an almost haphazard catch-what-catch-can manner. Most of the time it happens when I engage with someone external. It is never planned or systematic. My management is much more concerned about me being up on my compliance training than productively employing my talents. The end result is the loss of identity and sense of purpose, so that now I am simply the ninth member of the bottom unit of the collective, 9 of 13.

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

― Henry David Thoreau

Increasingly, the forces that have demoralized the blue collar working class world and propelled White Nationalism to the forefront of American politics have worked their way to me. The forces doing this are relentless and bipartisan both the right and left are doing this, but in different ways. Conservatives prize control and order with a horrid authoritarian streak naturally leading to Trump. Fear is a political tool wielded like a scalpel, enslaving the population to the security state. Generally speaking, the rise of the management class at work comes from this side of the spectrum along with their devaluation of the rank and file people. We see a tendency toward command-driven management, and being told what to do. Workers are simply meaningless details interchangeable and disposable. The management class is the heart of importance, and value. The rest of us aren’t really worth much effort simply being necessary cogs to get the work done.

The left has their own crimes to answer for. Much of the right-ward movement is a reaction to the systematic over-reach of the bureaucratic state. Political correctness and the thought police also serve to undermine societal confidence and tolerance in the “elite”. Management is part of this elite so derided to today and each subgroup within the elite has their own axe to grind. The big crime of the left is that they seem to think that every ill and danger can be regulated out of existence. Little or no thought is put into the cost of the regulation or the opportunity lost in the process. This is similar to the behavior of the right with respect to the National Security state. In the process, the individual is lost; the individual is not valued; the individual is not trusted. The value of work and the dignity of labors toward support of the family and the good of society is not honored. Work becomes a complete waste of time. Productivity and meaning in work ceases to be prioritized. Life is too precocious to waste doing this.

A big part of the overall problem is the value of my time. For every single thing I do, I trade it against doing something else. Increasingly, my time is spend doing unproductive and useless things. Every useless thing I have to do displaces something else. Time is a valuable resource, and today my management, my institutions treat my time with flagrant disregard and outright disrespect. This is the rotten core of the problem, the disregard for the cost of making me do stupid useless things. We engage it pointless, wasteful box checking exercises rather than reject pointless activities. It is not the stupid things as much as the valuable things they displace. Almost all the time at work I could spend doing something more valuable, or more gratifying, or more satisfying, or simply something that brings me happiness and joy. I could create, solve problems, learn and find meaning and value. Instead I am enslaved to someone’s idea of what I should do. I am saddled with numerous “terms of employment” based tasks that have no value or meaning. Those saddling me always have the excuse of “it is a good idea to do this”. This sentiment is valid except it completely and utterly discounts what that time could be spent doing that is better.

The difference between technology and slavery is that slaves are fully aware that they are not free

― Nassim Nicholas Taleb

I spend complete days doing nothing, but bullshit. It is mostly what other people think is a good idea, or worse yet some sort of ass covering exercise. I can spend an entire day doing nothing productive at all, and yet I’m doing exactly what I supposed to do. This is a huge problem! Managers do this almost every day. They rarely do what they need to do, actually manage the work going on and the people doing the work. They are managing our compliance and control, not the work; the work we do is mere afterthought that increasingly does not need me any competent person would do. At one time work felt good and important with a deep sense of personal value and accomplishment. Slowly and surely this sense is being under-mined. We have gone on a long slow march away from being empowered and valued as contributing individuals. Today we are simply ever-replicable cogs in a machine that cannot tolerate a hint of individuality or personality.

All over the place, from the popular culture to the propaganda system, there is constant pressure to make people feel that they are helpless, that the only role they can have is to ratify decisions and to consume.

― Noam Chomsky

Work place education exemplifies all of these trends. My training is almost 100% compliance oriented. Nothing I am ever trained on is job related, it is all telling me what I shouldn’t do. This training is a good avatar for priorities, and my actual work is not a priority at all. All the training that develops a person is management related. For the rank and file personal development is completely optional and hardly prioritized. We are there to get our stuff done, and the stuff we do is increasingly shit. They have lots and lots of training, and from what I see use almost none of it. It is full of the best theory that cannot be applied to the workplace. Their training would tell them to harness my passion and empower me, yet none of that ever happens. The entire system is completely oriented toward the opposite. The training system signals very clear values to all of us, the rank and file need to comply and submit, managers are the only employees’ worth developing even if the development is all delusional.

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

― Noam Chomsky

The management literature is full of the gospel of the value of human talent. It sounds great, and I believe in it. Management should be the art of enabling and working to get the most out of employees. If the system was working properly this would happen. For some reason society has removed its trust for people. Our systems are driven and motivated by fear. The systems are strongly motivated to make sure that people don’t fuck up. A large part of the overhead and lack of empowerment is designed to keep people from making mistakes. A big part of the issue is the punishment meted out for any fuck ups. Our institutions are mercilessly punished for any mistakes. Honest mistakes and failures are met with negative outcomes and a lack of tolerance. The result is a system that tries to defend itself through caution, training and control of people. Our innate potential is insufficient justification for risking the reaction a fuck up might generate. The result is an increasingly meek and subdued workforce unwilling to take risks because failure is such a grim prospect.

People get used to anything. The less you think about your oppression, the more your tolerance for it grows. After a while, people just think oppression is the normal state of things. But to become free, you have to be acutely aware of being a slave.

― Assata Shakur

One of the key things that drives the system is a complete lack of cost-benefit analysis

. The cost of a given measure to control a risk is rarely accounted for. The TSA is a prime example. One asshole tries to blow up a shoe, and forever we can’t take toothpaste on a plane. It is patently absurd on the face of it. We give up freedom, we give up time and we expend enormous effort to control minuscule risks. In the process, we have made a wonder of technology and the modern world, something to be hated. So much of the wonder of the modern world is being sacrificed to fear designed to control risks that are so small to be ridiculous. In the process, the vast benefits of modernity are lost. The vast benefits of easy and convenient air travel are overrun by a system designed to control irrational fears. Our fears are things that are completely out of control, and the vast opportunity cost is never considered. The result is a system that destroys our time and productivity in a disproportionate manner.

If one is forever cautious, can one remain a human being?

― Aleksandr Solzhenitsyn

The same thing is happening to our work. Fear and risk is dominating our decision-making. Human potential, talent, productivity, and lives of value are sacrificed at the altar of fear. Caution has replaced boldness. Compliance has replaced value. Control has replaced empowerment. In the process work has lost meaning and the ability for an individual to make a difference has disappeared. Resistance is futile, you will be assimilated.

To be given dominion over another is a hard thing; to wrest dominion over another is a wrong thing; to give dominion of yourself to another is a wicked thing.

― Toni Morrison

 

The law that entropy always increases holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell’s equations — then so much the worse for Maxwell’s equations. If it is found to be contradicted by observation — well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.

— Sir Arthur Stanley Eddington

AS USUAL WORD PRESS’ LaTeX is annoying.  I need to post and go to bed.

Rather than continue to talk about global issues I’ll get back in the weeds this week and get into a technical conversation. I don’t really know where this is going, so this is a bit of stream of consciousness in thinking about a topic and developing a new idea. The inspiration for this came from my talk at the Multimat 2017 meeting, and considering how to fix problems I’ve seen with rarefactions. As a reminder, I had seen that some solvers produce solutions with small, but consistent violations of the second law of thermodynamics in their solution of expansions (e.g. rarefactions). Nothing catastrophic is observed, but it is a troubling failure from nominally robust solvers. This study was itself motivated by the observation of a systematic failure of these solvers to produce convergent solutions to very strong rarefactions, and examine what sort of character the solutions have under more ideal circumstances.

A few points are worth making about the solvers used and how they have been tested in the past. Mostly I’ve worked with finite volume codes, sort of the gold standard of production codes for fluid dynamics. These codes are very reliable, and well understood.  For the most part, the focus of test problems has been shock waves where bad methods can result in catastrophic instability for the codes. Rarefactions are far less studied and tested because they are generally benign, and don’t threaten the stability of the code. As a result, the rarefaction focused test problems are mostly missing. We do know that expansions can produce unphysical solutions for Eulerian codes at critical points (where the characteristic speeds go to zero, and numerical dissipation may vanish). Bad solutions can arise with strong rarefactions, but no one has pointed out that these solutions actually violate the second law of thermodynamics before. The end result is a relative ignorance about shortcomings of the code, and a potentially important outlet for improvement of the methods.

Von Neumann told Shannon to call his measure entropy, since “no one knows what entropy is, so in a debate you will always have the advantage.

― Jeremy Campbell

One of my suggestions about examining this problem is examining the solution to these problems with a wider variety of codes. This would include codes that do not use pure finite volume methods.  One example are methods based on flux differencing where the code can go to formally high-order accuracy for nonlinear problems. Control volume codes are limited to second-order accuracy and the leading nonlinear truncation error can produce the entropy-condition violating energy transfer in expansions, . For almost every control volume code these terms are dissipative in shock waves, thus providing additional stability to the codes in this dangerous configuration. The opposing reaction in expansions can go unnoticed because any imperfections in the solution are modulated by the physics of the problem. For this reason, the failing has gone completely unnoticed for decades.  A reasonable question to explore is whether codes based on different design principles exhibit the same problems, or produce solutions that satisfy the second law of thermodynamics more uniformly.

An important technique in defining flux difference schemes of high-order accuracy is flux splitting (more than second-order accuracy). The core idea is that approximating the fluxes to high order can produce higher formal accuracy then the variables. The question is does this produce solutions of a fundamentally different character with respect to entropy. Simply put, a flux splitting is a decomposition of the fluxes being differenced into negative and positive moving contributions. These fluxes are then differenced and then recomposed into the total flux. The splitting techniques add a directionality to the approximation needed for numerical stability associated with upwinding the approximation. The flux splitting techniques are closely related to Riemann solvers, but here to fore only include a small number of simple linearized Riemann solutions. I’d like to explore some greater generalization of this concept including flux splittings based on exact Riemann solvers.

The Riemann problem is the exact solution to the interaction of two discontinuous states described by hyperbolic equations. This analytic information can be used to develop numerical methods that encode this physically relevant information into the solution. In terms of numerical methods, the Riemann solution is a generalization of the principle of upwinding, where the physical direction of propagation is taken into account. The first person to describe this approach to numerical methods was SK Godunov in 1959. Godunov’s method was first-order accurate and used the exact solution to the Riemann problem. It was soon realized that one only needed to approximate the Riemann solution. This became a key development in the methods over time and allowed great progress. Over time it was realized that it also allowed great flexibility too.

In science if you know what you are doing you should not be doing it. In engineering if you do not know what you are doing you should not be doing it. Of course, you seldom, if ever, see the pure state.

– Richard Hamming

A simple Riemann solver can be defined by linearizing the problem, $ ( u_r – u_l ) $. The quantity is the nonlinear flux, are the states to the left and right of the interface. The dissipation is defined by the eigen-decomposition of the flux Jacobian, . This decomposition is contained of the right and left eigenvectors and the eigenvalues, . These eigenvalues are the characteristic velocities, which for gas dynamics are being the velocities and the sound speeds, . This basic decomposition is the basis of flux splitting techniques.

The basic flux splitting takes the flux and decomposes it into right and left moving pieces, . One was to do this is choose a velocity, , and create contributions where and . A simple choice of creates the Lax-Friedrichs flux, the simplest (and most dissipative) Riemann solver. For the general linearized Riemann solver the flux splitting is and . The choice of the left and right states to evaluate the flux Jacobian defines the flux splitting. For example, if the states are evaluated using Roe’s recipe, we get the Roe flux splitting. If we evaluate the eigenvalues in a bounding fashion we can get the local Lax-Friedrichs method.

Another approach to generating a flux splitting does not use the variables in the expression of splitting, and only uses the fluxes. In this case the expressions are developed in terms of the sign of the eigenvalues/characteristic velocity. The splitting then works only as a scaling by eigenvector decomposition of the flux Jacobian. The expressions are somewhat simplified, as and $ f^- = \frac{1}{2} ( f – R \mbox{sign(\lambda) L ) f$. We not in passing that the smooth or soft version of the sign function might be extremely useful in this type of splitting and introducing a continuously differentiable function (https://williamjrider.wordpress.com/2017/03/24/smoothed-operators/ https://williamjrider.wordpress.com/2017/03/24/smoothed-operators/). By the same token, the absolute value used in the usual flux splitting approach could also be smoothed to similar effect. We need to take care in our choices to assure that the accuracy of the resulting numerical method is not negatively impacted. We get into some very big problems when we want to generalize to other Riemann solvers. Examples of these solvers are the HLL family of solvers, and the most classical Riemann solver, the exact solver or close approximations to that approach (e.g., a single iteration of the nonlinear Newton’s method used in the exact solver). How can these important methods be utilized in flux splitting methods? For very strong wave interactions these classes of methods are extremely valuable and not presently possible to be used effectively in flux splitting.

 Nature not only suggests to us problems, she suggests their solution.

—Henri Poincare´

We can start with the simpler case of the HLL type of flux, which has an algebraic description. The HLL flux is defined using the space-time diagram by integrating the equations to derive a flux. The simplest form of the flux uses bounds for the wave speeds and neglecting all of the structure inside the Riemann fan resulting in a simple closed form expression for the flux, $f_{lr} = \left[a_r f_l – a_l f_r + a_l a_r \left( u_r – u_l \right)\right]/(a_r – a_l)$. The flux is quite simple, but dependent on the estimates for the smallest and largest wave speeds in the system. The left wave speed, is the smallest wave speed and needs to be bounded at zero (i.e., it is negative). The right most wave speed is and is bounded below by zero. The HLL flux has the benefit of reducing to simple upwind flux for the system if all the wave speeds are either negative or positive. For a flux splitting we need to take this apart into negative and positive moving pieces for the purposes of splitting nearby fluxes as we did with the Roe, or flavors of Lax-Friedrichs.

The flux splitting can be defined almost by inspection. The positive flux is $ f^+ = (a_r f – a_l a_r u) / (a_r – a_l) $. The negative flux is $ f^- = (- a_l f + a_l a_r u) / (a_r – a_l) $. This is a wonderfully simple result, and meets all the basic requirements for a flux splitting. Unfortunately, the HLL flux is extremely dissipative, thus lacking some degree of practical utility. Still we expect this flux splitting to be quite robust especially for strong waves with the proviso that the wave speed estimates bound the physical wave speeds. This is a much more delicate estimate than usually recognized. The case of a reflected wave can produce wave speeds nonlinearly that exceed the wave speeds in the initial data.

The harder case is the class of exact Riemann solvers that are defined algorithmically and do not have a closed form. After using an exact Riemann solver we do have a set of initial left and right states, and the resolved state at the centering point . If we desire a flux splitting, it needs to be defined in terms of these variables. The trick in this endeavor is choosing an algebraic structure to help produce a workable flux splitting technique. We build upon the experience of the HLL flux partially because we can incorporate the knowledge arising from the exact solution into the algebraic structure to good effect. In particular, the nature of the one-sided differencing can be reproduced effectively. This requires the wave speed bounds to use the interior states of the solution.

The exact flux is different than the HLL flux, and this will be defined by changing the dissipation vector in the flux. Our chosen structure is a flux defined by $ f_{lr} = \(a_r f_l – a_l f_r – D  \left( u_r – u_l \right)) / (a_r – a_l) $. If we can derive the form for our work will be done. The positive flux is $ f^+ = (a_r f + D u) / (a_r – a_l})$. The negative flux is $ f^- = (- a_l f – D u) / (a_r – a_l) $. Now we just have a little bit of algebra to arrive at our final expression. The math is nice and straightforward, $ D = (a_r f_l – a_l f_r – (a_r – a_l) f_{lr} ) / (u_r – u_l) $. A couple comments are needed at this point. When the states become equal, the solver becomes ill defined, . Fortunately, this is exactly where the linearized flux splitting approaches or HLL would be ideal.

The secret to being wrong isn’t to avoid being wrong! The secret is being willing to be wrong. The secret is realizing that wrong isn’t fatal.

― Seth Godin

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

Van Leer, Bram. “Flux-vector splitting for the Euler equations.” In Eighth international conference on numerical methods in fluid dynamics, pp. 507-512. Springer Berlin/Heidelberg, 1982.

Harten, Amiram, Peter D. Lax, and Bram Van Leer. “On upstream differencing and Godunov-type schemes for hyperbolic conservation laws.” In Upwind and High-Resolution Schemes, pp. 53-79. Springer Berlin Heidelberg, 1997

Shu, Chi-Wang, and Stanley Osher. “Efficient implementation of essentially non-oscillatory shock-capturing schemes, II.” Journal of Computational Physics 83, no. 1 (1989): 32-78..

Jiang, Guang-Shan, and Chi-Wang Shu. “Efficient implementation of weighted ENO schemes.” Journal of computational physics126, no. 1 (1996): 202-228.

Quirk, James J. “A contribution to the great Riemann solver debate.” International Journal for Numerical Methods in Fluids 18, no. 6 (1994): 555-574.