I'm trying to solve a problem where I have a initial and final distribution of tumor, and my goal is to find the best map of parameters (diffusion and reaction terms) for a reaction-diffusion equation, which explain at best the final distribution, knowing the initial one.

By doing some literature search, I figured out that the concept I'm dealing with is the so called "pde-constrained optimization problem" and I saw some works using things like "adjoint variables", Lagrangian, etc... that I'm not familiar with.

My problem is that I can't find any good tutorial, online course, or article explaining these concepts and how to use them in practice. Maybe some of you can give me insights about such lesson available ?

I have a fairly background level in mathematics and programmation, so I'm not looking for only understanding "roughly" the field, but indeed being able to apply solutions to my real-life work...

Many thank for any help you can give me...

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    $\begingroup$ That's a bit of a broad question, but I can recommend the books by Fredi Tröltzsch, Optimal control of Partial Differential Equations (AMS); Juan Carlos De los Reyes, Numerical PDE-Constrained Optimization (Springer); Hinze et al., Optimization with PDE Constraints (Springer). $\endgroup$ Dec 6, 2017 at 9:06
  • $\begingroup$ Thank you... Actually I'm really looking for any help and therefore the question is very broad indeed... But also for this reason your answer was really helpful also;-) thx $\endgroup$
    – david guez
    Dec 6, 2017 at 11:23
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    $\begingroup$ Actually, the problem you describe is a typical inverse problem (or parameter estimation problem). These are conceptually different from optimal control problems, although they are solved by similar techniques. $\endgroup$ Dec 6, 2017 at 14:08

1 Answer 1


I found the following books pretty helpful:

While it's not about PDE inverse problems per se, you will at some point have to either use or implement numerical optimization procedures, for which Nocedal and Wright is an invaluable reference.

That said, I haven't found any one perfect book on the topic. If you post a detailed explanation of your problem in another question here, someone will probably be able to offer you more explanation of the underlying math or, at the very least, offer some more specific references.

  • $\begingroup$ Yes, my question is indeed very broad and vague... All your help however helps me a lot and I start to understand better why I want to, I'll post a more precise question in a few minutes since now I can determine more precisely the point where I'm stuck... Thx! $\endgroup$
    – david guez
    Dec 7, 2017 at 19:48
  • $\begingroup$ I just edited a more precise question here . If someone have time to unlock me it would be great! Thx $\endgroup$
    – david guez
    Dec 8, 2017 at 0:07

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