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I would like to master Inverse PDE Problems particularly with the use of Finite Elements. My problem is I don't know where to start. Should I begin by reading a book on Inverse Problems or on PDE-constrianed Optimization!

Could I get some book recommendations and/or route to follow.

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I will use parameter estimation problem as an example to briefly introduce what inverse problems are and then list some of my recommendations on learning this topic.

One famous example would be the Calderón Problem, basically what it concerns is how to use the boundary measurements data to determine the material related parameters of the interior domain. The finite element methods you mentioned are used to solve the forward problem(!) not the inverse problem, a class of problems can be formulated as follows:

Given data $d$ on boundary, the material parameter $\beta$ you would like to estimate, the forward problem is: $$\mathcal{A}(\beta) u = f $$ where the differential operator $\mathcal{A}$ depends on the parameter, also accompanied with some boundary condition(see Remark 1). Forward problem actually can be ill-posed.

Now your inverse problem would be using this PDE as a constraint, to solve a functional minimization problem like the following: $$ \min_{u,\beta} \mathfrak{F}(u,\beta,d,u_h,\beta_h) $$ for example, $\mathfrak{F}$ is your objective functional, could be Tikhonov functional, and others.(see Remark 2)

The prerequisites to conduct research on this field would be:

  • If you would like to do some theoretical work on inverse problems, Functional Analysis is a must, because you will use Fréchet derivatives as your daily snacks. Also some PDE theory is needed. For this purpose, reading Lars Hörmander's four-volume series The Analysis of Linear Partial Differential Operators would be more than nice, but it costs too much time. A rather concise but difficult to read book is Victor Isakov's Inverse problems for partial differential equations, btw despite the fact that Isakov works in a not-so-famous university, he is always considered to be one of the great mathematician working on inverse problems by Gunther Uhlmann, Stanley Osher, etc.(more book recommendation see Remark 3)

  • Since you mentioned finite element, I am guessing you would like to learn this topic from a more numerical point of view, then Stig Larsson's book Partial Differential Equations with Numerical Methods is not hard to read even you are not a numerical analyst, it would get you started in finite element methods. For the connection between the finite element methods and inverse problems, there isn't a dedicated book about this that I know of, however there are many research articles on this. For example the current development in inverse problems Gunther Uhlmann edited: Inside Out: Invserse Problems and Applications shall be a neat introduction to some advanced research area.

To sum up, my suggestion would be: Know what exactly the PDE constraint optimization you are gonna do first, do not bother with the lengthy mathematical theory behind it, then go learning the corresponding finite element methods for the forward problem, lastly go back to the inverse problem to see what you need there.

SOME UPDATES Remark 1: If we just have a single boundary measurement, we could treat the data $d$ as the Dirichlet boundary condition for the forward problem; If what we know is a mapping(!), aka Dirichlet-Neumann map, ie given input electromagnetic wave of certain frequency, we could get certain feedback on the boundary, then the boundary data $d$ is not the boundary condition of the forward problem, but rather that Dirichlet-Neumann map, working like a black-box.

Remark 2: A potential difficulty here, due to the property of high frequency waves, the objective functional may have lots and lots of local minima, and this problem is still an open research area.

Remark 3: I consulted my officemate today who is working on Dirichlet-Neumann mapping, and he recommended two more books, the first one is the yet to be published lecture notes by Gunter Uhlmann and Joel Feldman, this is a more user-friendly read than Isakov's. Another book is by Colton and Kress, Inverse Acoustic and Electromagnetic Scattering Theory, which is even more user-friendly assuming you know some PDE and/or functional analysis and/or finite element.

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  • $\begingroup$ Thanks for the advice. Lars Hörmander's four-volume series does look enticing but as you said it'll take too long. $\endgroup$ – Jason Apr 26 '12 at 10:48
  • $\begingroup$ @Jason You are more than welcome, I discussed with my officemate about my answer and updated some new infos. $\endgroup$ – Shuhao Cao Apr 26 '12 at 21:56
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As a shameless plug, if you're familiar with the finite element method and want to learn about modern numerical methods to solve inverse problems, I've got a paper on that: http://epubs.siam.org/sisc/resource/1/sjoce3/v30/i6/p2965_s1?isAuthorized=no

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  • $\begingroup$ Thanks Wolfgang. I've just downloaded it and will read through it. Do you also have a tutorial on your deal.II website that shows an implementation? $\endgroup$ – Jason May 3 '12 at 17:17
  • $\begingroup$ No, but I have two student projects that worked on a simple model problem. Send me email off-list if you're interested. $\endgroup$ – Wolfgang Bangerth May 4 '12 at 3:05
  • $\begingroup$ @WolfgangBangerth Hi, Wolfgang, really nice paper you've got there, I am also of great interest about the AFEM applications in inverse problems, could I send you email too? $\endgroup$ – Shuhao Cao May 9 '12 at 2:53
  • $\begingroup$ Sure. It's easy to find me online. $\endgroup$ – Wolfgang Bangerth May 9 '12 at 13:21
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There are basically two approaches to attack inverse problem for PDE systems. Either posed it as an optimization problem or Kalman-filtering (probabilistic approach) techniques. The latter method is useful if your PDE systems contain uncertainties. That may include modelling error and etc.

If your PDE systems are large enough, the optimization approach becomes computationally prohibitive (may equally be expensive as Kalman-filtering technique). Therefore, method like adjoint for computing gradient efficiently is very beneficial. Why is this efficient ? Because adjoint method needs two simulations (one forward simulation and one backward simulation) regardless of the number of decision variable. Contrary to computing gradient using finite-difference methods which require N number of simulations, where N is the number of decision variable. Non-gradient (derivative-free) based optimizations for PDE systems, although they are easy to implement, but the optimization can be costly in term of the runtime.

To model the PDE systems, yes, Finite-element can be used. But it depends on the nature of your PDE systems which perfectly fit to the physics.

And don't forget inverse problem for PDE systems is an ill-posed problem. Because you can't get as many as measurement data for the parameter and/or state variables to be estimated. Hence, the solutions are not unique.

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For a Bayesian perspective, Stuart's paper, "Inverse problems: A Bayesian perspective" is really good. It's pretty advanced though. Perhaps one could start reading that paper, then backtrack and learn the necessary background whenever an unfamiliar concept is used.

For an optimization approach, I found the explanation in Bangerth's PhD thesis was quite good and concise (I actually prefer it to the other paper he posted in this thread). Again, this is the sort of thing where you'd have to backtrack constantly to fill in your knowledge if you're just starting learning.

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As Shuhao Cao said, you definitely need functional analysis. A good one that you might want to try out is "Functional Analysis: Applications to Mechanics and Inverse Problems" by Lebedev, Vorovich and Gladwell. Whatever your background in analysis is, this book should be fine, as it starts with the basics.

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