iMechanica - Uncertainty quantification in mechanics - Comments

Re: Uncertainty and Monte Carlo

Biswajit Banerjee — Tue, 05 Aug 2008 18:18:30 -0400

Cosma Shalizi has a nice intro to Monte Carlo methods and pointers to other sources in his Notebooks. See http://cscs.umich.edu/~crshalizi/notebooks/monte-carlo.html.

-- Biswajit

Uncertainity qualification of aircraft structures

Rezwanur Rahman — Mon, 14 Jul 2008 17:10:33 -0400

Please look at the paper:

A probabilistic approach to fatigue risk assessment in aerospace components : G. Cavallini , R. Lazzeri

It is a nice source of idea on typical uncertainity qualification method in structural mechanics. The schematic diagram (figure-1 in the paper) can give a nice instant idea about formal uncertainity qualification method.

I remain very "uncertain" about this "uncertain mechanics" ! ;)

Mike Ciavarella — Mon, 14 Jul 2008 09:42:38 -0400

michele ciavarella
www.micheleciavarella.it

Re: Concentration of measure

Rezwanur Rahman — Sun, 13 Jul 2008 20:51:56 -0400

Hi Dr Biswajit

Thanks for the reply. I am sorry if my comments sounded rude. It is nice to share and discuss about some topic. That is why the blog imechanica for. We can share knowledge and experiences. Definitely it is my pleasure to discuss about a topic. However, lets discuss a more about the topic. Actually the relation of concentration of mesure with Banach space is given nicely in the book: The Concentration of Measure Phenomenon: Michel Ledoux

In the first chapter, the term "Concentration function" is used. The relation with Banach space is discussed in the third chapter. But I did not talk about this book because it is totally abstract. That is why I just left a comment. The relation comes between these two terms by eigenvalues.

I understood the question regarding unknown unknown. Actually in terms of mathematics you can call them "blind terms". As I told you, if you deal with a physical problem there should be some parameters that you can define from the physical phenomena. But, the blind terms are determind in a different way. I will not say that these techniques are sufficient. Because in the researh on "experimental data analysis" people are still working.In order to find out blind terms, firsly we need to see an important thing: I have some known parameters, how they affect the output function? Say we know some "n" parameters from the physical phenomena. But these are may not be enough. Some extra parameters are there which we do not know. For these cases you can use several (usual and well known) methods such as:

ANOVA, ANCOVA, mulifactor experimental design, Bayesian estimation, Bayesian Network, Principle component analysis, Clustering, Decision tree, Hidden Markov model etc.

Unfortunately, these methods sometimes do not work for some specific data. Then some other techniques in data classification literature are used. (such as Independent component analysis or blind decomposition, and lots more).

Using ANOVA, ANCOVA,MANOVA or experimental design you may find answers of your questions:

How many parameters are affecting the output?, Which of them are highly correlated?, and What is the pattern of correlation? etc. From this analysis you may have an estimation that "n" parameters are known but some extra parameters can be there. (Just for an example: may be you know "n" parameters,but among them "m" parameters are providing equvalent bahavior, hence you can replace these "m" parameters by one parameter, so you may have n-m+1 parameters indeed , so you may need actually some more parameters in the experiments). Way may not know what parameter it is but we may suspect that some extra parameters are weakly active. This is just an example. In some cases the blind parameters can be easily discarded. Geometrically, the region that we are dealing is a circular disk (number of dimensions:2) cut from a soild sphere (number of dimensions:3).

I would suggest you to go through a paper "Comparative study of deconvolution algorithms with applications in
non-destructive testing" Nandi, A.K.; Mampel, D.; Roscher, B. it is kind of old work but some idea can be generated. Now these techniques are quite popular in digital image processing, pattern recognition etc. They detect actual signals from blurred signal using these techniques. Also please have a look on the website: http://www.statsoft.com/textbook/stexdes.html. You may consult a book also: Design of Experiments for Engineers and Scientists: Jiju Antony. Hidden markov model is also very active one in predicting unknown unknowns (Answering kind of : Is there any unknown paramter here?). You may also see the book: Inference in Hidden Markov Models: Olivier Cappe, Eric Moulines, Tobias Ryden.

However, these methods need data which comes from experiments. To get meaningful output higher number of experiment is prefered when the parameters are random. The use of concentration of measure can be a way to get the successful experimental dataset without doing a large number of experiment. Sometimes you may know the mean, sometimes you may not know the mean of the output. They showed the applicability of the theory of "concentration of measure" in these cases.

I agree that the paper on concentration of measure did not give very well visible and practical solution of a real life problem. But I would appreciate that they have used the idea for using the concept of concentration of measure. According to my understanding and knowlege on this term, it has a great potential of clustering the probability. Ususlally, all the above methods that I mentioned, they work on classiying or grouping the parameters but concentration of measure can classify the probability of a a parameter arround a physical value. I would say the better analogy (weak analogy though) is with: estimating range for mean or standard deviation done in statistical analysis.

Let me know about further questions or comments.

Rezwan

Re: Concentration of Measure

Biswajit Banerjee — Sun, 13 Jul 2008 18:14:00 -0400

Hi Rezwan,

I think you've misunderstood what I meant to say in my previous comment. When I say "losing the thread" I mean that I am not being able to follow the thread your argument.

Let me give you an example. You have said in one of your comments that "However, concentration of
measure has a nice relation with Banach space. Theory of probability
is one kind of application of functional analysis; in general."

That comment is a general statement of things that most of us in iMechanica probably know. However, the implication of the first statement is not clear. What is the relation to Banach space and why is that interesting to a non mathematician? That general statement does not help to clarify in my mind what the shortcomings of the concentration of measure approach are.

I seek a physical and clear understanding of the underlying ideas. Even though I have a first year math graduate level understanding of functional analysis and measure theory - I don't understand them well enough to readily connect new aspects of to physical problems.

Also, I think you misunderstood what I mean by "unknown unknowns". Because of the fact that they are unknown, you can't identify what these unknown variables are!

What you have talked about in your comment is about "known unknowns" (in Rumsfeld's terms) rather than "uknown uknowns". Bayesian priors can be used to take care of "unknown unknowns" but it's not clear at all how such priors should be calculated even for relatively simple engineering problems. If I have not understood you correctly, please explain again.

You also say " I am not reviewing the paper, so I should not make any comments about its realistic or non-relistic outcomes." But comments about those outcomes are precisely what I'm looking for! In my current position I have to rapidly evaluate papers for their usefulness or otherwise to my work. My personal evaluation has been that the concentration of measure work is essentially useless for practical purposes (as of now). I would like to make sure that I have understood the material correctly and am not making a wrong judgement call.

Also, thanks for taking the time to discuss the matter and providing pointers.

-- Biswajit

Re: Concentration of Measure

Rezwanur Rahman — Sat, 12 Jul 2008 17:15:49 -0400

Hi Dr. Biswajit

Firstly, thanks for uploading the paper. I know abstract mathematics is not very clearly visible. I am kind of loosing the interest in this discussion too. Beacuse, the
question that was asked requires some specific approach and the comments regarding that are little offensive. Anyway, let us clarlify the questions that you have asked:

a) As I have mentioned, the significant issue is to define random variables has some "Practical Engineering", background. If we know a function has some certain number of variables and how they are related, life will become quite easier. But realistic modeling is not that straight forward. However, this issue is more significant when we deal with random variables. I will certainly not go for the abstract mathematical explaination. When we define a problem, firstly we define the variables. Similar thing happens in probabilistic modeling. The issues those arise are:

- classify the variables: which are known, which are unknown. you can not simply discard the unknown variables !!!! It contributes to the output. Once you discard such variables, you will get some garbage results and the blame will go to abstract mathematical techniques!!! You have to classify the variables which are significant, which are not. Mathematics is a tool but the success depends on how you utilize this tool.

- Now the unknown variables are to be predicted. If you really do not know which one is the unknown, you need to do the output analysis from some experiments. The paper on "concentration of measure" gives a concept about making a trade off between your experiments and objective. The pattern or output will give some estimation that how many variables that system can contain. If you suspect a parmater to be a variable but you do not have any information about it, you need to go for estimating prior information from different probabilistic inference techniques (such as Bayesian..). As we are dealing with a physical problem definitely the relevent parameters are to be suspected as random variables. If we are dealing with a fracture in polymer composite, definitely, the physical parameters such as crack growth, velocity, energy release rate, temperature etc are to be accounted. We can not bring "turbulance" by any chance !!!!

- Then you have to classify the independent and dependentrandom variables. If you assume that you have two variables which are not totally independent and you assume the probability distribution (not the conditional probability) is normal keeping in your mind that they are independent, I think the output will give some unrealistic results. So, it is also an important issue.

- If you classify the independent or dependent variables; now they relation among them is to be determined. And so on.

So, it is not very easy that we think. However, besides defining the random variables, we need to define the behavir of these random variables. Suppose, if we design an aircraft wing and keep the loads as a well defined sinusoidal functions; I think this concept is rather unrealistic than defining the loads as random functions. Moreover, if the random variables follow a non-gaussian distribution, we should not deal the stochastic process containing those random variables with gaussian distribution. There are several things to be considered to make your solution "Realistic".

Random processes itself gets involved in a differential equation describing a problem. That why it becomes stochastic or random differential equation. I discussed this in earlier post by giving an simplest example of Euler-Bernoulli Beam. But, the random process gets involved in the ODE or PDE shuld be well understood. Random process can be anything. If my random process is not Brownian motion, I can not include it in my governing ODE or PDE.

However, the question that you posted in this discussion demands abstract mathematical explainations. Because, (as I mentioned earlier) "concentration of measure" is not a well visible term, it originates from the concept of measure theory and functional analysis. So in order to keep the relevency of your question I followed the abstract mathematical explaination. Without this type of explaination I think you can not deal with it. It is not like, I draw a picture and that reveals the whole idea. You can not say any technique in the world as a perfect one. We have limitations. Classical mechanics can not work for electrons. But we can not call it useless. People realized this and quantum mechanics came out.

The paper on "Concentration of measure" that you posted, the discussion started based on that paper. I am not reviewing the paper, so I should not make any comments about its realistic or non-relistic outcomes. As you mentioned earlier, most of the papers with abstract mathematics are non-realistic, I expect the example or explaination that leads you give such comments.

Re: Concentration of Measure

Biswajit Banerjee — Sat, 12 Jul 2008 00:32:10 -0400

Rezwan,

I've added Lugosi's PDF book to the main post.

At this stage I'm, like Mike, losing the thread because the discussion has become too general. Maybe I should make my questions a bit more specific.

In your comment (c) above you say that " The most significant issue is defining the random variable and then defining the random process".

Why is defining the random variable an issue? Don't we already know the variables which have known variability? The unknown uknowns can be ignored since we can't say anything useful about them. The problem that I see is that the epistemic (known) uncertainty cannot be quantified well enough. We cannot assign a probability distribution function based on the three-five tests that experimentalists perform for a particular configuration. And doing more tests is expensive and hence never done in real life. What should we do given these constraints?

Also, I don't understand what you mean by the random process in this context. Are you suggesting that the physics cannot be described by differential equations (with coefficients which are random variables)? Could you clarify?

-- Biswajit

Re: Highly abstract mathematics vs. more engineering-oriented

Biswajit Banerjee — Sat, 12 Jul 2008 00:21:57 -0400

Hi Mike,

The problem of higly abstract math. vs engineering oriented approaches is one that I face on a regular basis. It takes a lot of effort to learn the math. But once I've learnt the math well enough, I realize that the papers that use the math could well have done without much of the general formalism. Often, I find that the results presented in mathematically couched papers may be trivial or irrelevant.

So I've decide to take the easy way out and ask experts on iMechanica to clear out the forest of ideas from measure theory and show me whether or not the paper in question presents a realistic aproach or not. I have my doubts after spending a lot of time on such buzzwords as spectral stochastic finite elements and polynomial chaos expansions - which are all very nice but essentially useless for practical problems.

On the other hand, give the fact that these problems are hard, the easiest way forward seems to be to learn the math and then use it properly.

-- Biswajit

I don't follow here.But anyway let's move on.You pose a problem?

Mike Ciavarella — Fri, 11 Jul 2008 15:50:42 -0400

michele ciavarella
www.micheleciavarella.it

Thanks for the papers.

Rezwanur Rahman — Fri, 11 Jul 2008 13:28:32 -0400

Thanks for the papers. Weibull distribution is not only a statistical tool. The most important issue in theoretical modeling is the correct model for correct case. It is often seen that the physical phenomena is thought to be a deterministic one and suddenly people use statistcs to overcome some defeciencies. As I am working on composites, actually I saw the use of Weibull distribution in several wroks. People tried to define the strength of the fiber-matrix interface in terms of Weibull distribution based on some test. The important thing is to be noted: this is not exactly a statistical modeling. Can you send me a paper which accounts the "Goodeness of Fit" in approximating the strength with Weibull distribution? However, this is not probabilistic modeling. Stochastic modeling or Probabilistic modeling has a visible difference with fitting a data set to Weibull distrubution; this is rather statistical data analysis.It does not account the actual phenomena happening inside the material. The uncertainity of a material is more significant at meso/micro level rather than at macro level; I mentioned before. As far as thin gaussian distribution (mild) is concern; it is good to approximate the distribution with Laplacian distribution.

Mathematically, a term "almost certain" has a difference with "certain".We can not say a random variable with probability of 0.999999...99, a deterministic variable. The risk is : (1-0.999999...99).

Statistical approach is not the same as probabilistic approach. As long as you deal with probabilistic or statistical approach, you can not expect 100% perfect output. That is why we have "mean;median and standard deviation". Probabilistic modeling or statistical modeling makes you get agree at the beginning when you defined a random variable having mean "μ" and standard deviation "σ".

Well in the future there will be "virtual conferences" - anyway

Mike Ciavarella — Fri, 11 Jul 2008 03:36:58 -0400

You are quite rigth. One cannot go to all conferences! In the energy sector, I think there was recently a conference entirely on the web recently.

However, I pointed you to that NAFEMS organization, there are documents there, you can get in touch with people there, maybe you find something useful.

Keep searching.

I have noticed this post oscillates between highly abstract mathematics and more engineering-oriented one. It is difficult to conciliate.

Re: Concentration of Measure

Rezwanur Rahman — Thu, 10 Jul 2008 20:06:00 -0400

a) I can give another link: http://www.lsp.ups-tlse.fr/Proba_Winter_School/Lugosi.pdf.

b) Once you deal with probability theory in terms of measure theory, you can bring the concept of functional analysis. I agree that the Chernoff inequality, Hoeffding inequality, or McDiarmid's inequality etc are well known in probability theory, but if you want to deal with the inequalities, you can use some theories of functional analysis to deal with the probability because measure of a set is one kind of operator. Mathematically, probability in a sample space can be referded to probability space. Hence there are relations which can be easily seen. However, concentration of measure has a nice relation with Banach space. Theory of probability is one kind of application of functional analysis; in general.

c) Uncertainity qualification is not a straight forward approach. The most significant issue is defining the random variable and then defining the random process.

d) For further reading I can suggest:

Introduction to Probability Models: Sheldon M. Ross

Re: Concentration of Measure

Biswajit Banerjee — Thu, 10 Jul 2008 19:16:09 -0400

1) I've attached a pdf version of a book (by Dubhasi and Panconesi) to the original post. The book discusses concentration of measure inequalities in an accessible manner.

2) Regarding the paper in question, the problem that they have considered is relatively simple. Very few uncertainty quantification exercises on realistic problems can be found in the literature - most papers deal with toy problems. Also, much of the recent literature suggests that we have to use thousands of processors to compute uncertainties accurately. Very few researchers have access to such huge computing resources.

I don't see the paper addressing a realistic problem or providing a realistic solution at this stage.

3) Also I wouldn't call the Chernoff inequality, Hoeffding inequality, or McDiarmid's inequality well known inequalities in functional analysis in general (though they are well known to probability theorists). For our readers, a starting point towards understanding these inequalities is Markov's inequality. See Eric Weisstein's Mathworld for a simple proof .

Re: V&V NAFEMS course

Biswajit Banerjee — Thu, 10 Jul 2008 18:59:56 -0400

Hi Mike,

Thanks for the pointer to the course.

The cost of the course is prohibitive. Also, I have decided not to travel to the US or Europe until the visa procedures become less insulting to my dignity. So any conferences/courses in those places are irrelevant to me. It will be more useful to me if you can point me to any courses in India.

-- Biswajit

Concentration of Measure

Rezwanur Rahman — Thu, 10 Jul 2008 16:29:22 -0400

Hi Dr. Biswajit

I have looked at the paper. At the first phase
I had some observations. I will present the second phase (more detail)
later on. Firstly, the term concentration of measure is a very
important from the perspective of functional analysis as well as probability theory. The idea is to sense the enrichment of any mathematical entity (dimension or variables) in a measure space such as Metric space, Hausdorff space and so on. Typically if we think to roll a dice once, the probability of getting side with "six" is 1/6. If we roll it twice this probability increases because the sample space is increased. Hence, if we roll the dice (say for an example) 10,000 times; definitely this probability of getting a "six" is increased quite highly. It is clear that the sample space gets big and the event "Getting the side with six" becomes almost certain when we roll the dice almost infinitely. Mathematically, according to measure theory the the measure of the probability in the probability space (the triplet, (Ω,F,P)) gets entrapped in a region/interval according to the concept of concentration of measure. On the other way we can say the concentration of the "measurable sets" get higher leads to very small deviation rather giving a densed interval or may be a point. Actually, in real life problems related to uncertainity, we look for the higher concentration of measurable sets in the probability (or measure) space. Now the question comes, should we do millions of experiments to converge to a almost certain output? Practically the answer is negetive. For that reason this paper was publised, I am working on the damage model and other researchers on uncertainity qualification too.

The authors focused on the parameter CF (Confedence factor) is the ratio of difference (may be eucledian distance or metric) "M" between mean,threshold of failure and the Uncertainity "U"; besed on concentration of measure. For several cases they calculated this term. They showed the demand of trials or experiments is expensive to extract a probability of failure, whereas it is welly managed by using concentration of measure inequalities.

The idea behind this paper is to use the consept of "concentration of measure" so that the model gets closer to less uncertainity as the number of trials (measure) is increased. But side by side a large number of trials is tedious and costly. The smaller failure tolerance "ε" leads to higher trials. The concentration of measure inequalities are giving an upper bound of the failure probability instead of doing a huge number of experiments.

They have defined, the probability measure of safe events "A" as:

P(Y to be safe) is not exactly "1" in real due to the fact that the probability space is not compact. Hence the failure probability P(Y fails) is defined under an upper bound or tolerance (ε) (eq 1,3 in the paper). So the concentration of measure inequalities actually gives the interval estimation in terms of inequality of the failure probability because the failure probability is supposed to be unknown in prior. The concentration of measure is nothing but a probability density itself in the form of inequality.And the final step looks for validation or activeness of the model. They showed this application of concentration of measure in some exact as well as non-exact scenarios. in some case the mean or mathematical expected value is known, in some cases it is unknown. They showed that the model can work for all of the cases. However, I will give more explaination after the weekend.

To go through the paper I would suggest to have some introductory reading on:

- Advanced Set theory (specially concept of supremum "sup" and infimum "inf")

- Measure theory (What is measure of a set? What happens when a set is not compact or

What is non-compact measure?)

- Some idea on some famous inequalities used in functional analysis such as:

Chernoff inequality,Hoeffding inequality,McDiarmid's inequality, etc.

I will send you the soft copies of some articles on concentration of measure inequality.

Thanks

Rezwan

Uncertainty quantification in mechanics

Biswajit Banerjee — Tue, 08 Jul 2008 22:22:18 -0400

Modern composite structures have a wide spread in their failure stress. Advanced multiphysics codes can have a wide range of predicted behavior for nominally the same inputs. How do we certify the design of such structures or the accuracy of such codes?

The quantification of uncertainties in engineering design has garnered som interest in recent years. The most accurate method of quantifying the spread of outcomes of an experiment is the Monte Carlo approach. However, the cost of Monte Carlo simulations has caused most researchers to use some form of reliability analysis (no pun intended). Such approaches reduce the number of tests that are needed to quantify the behavior of a structure. However, some probability distribution function has to be assumed for the input parameters. An alternative is to solve a stochastic set of differential equations - often using the Stochastic finite element method.

A recent paper by Lucas, Owhadi, Ortiz takes a slightly different tack. They claim to provide tight upper bounds on the uncertainty through concentration of measure inequalities. Does anyone have a good idea of what these are and could they explain it to a lay audience? I'll also attempt to explain this idea as I learn more about it.

Links will be added as I learn about them.

-- Biswajit