Institute for Analysis and Numerics Applied Mathematics Münster

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Bachelor/Master Seminar:

Mathematical Optimization/The Calderon Problem

WS 2025/26

Lecturer:

Prof. Dr. Gustav Holzegel
Prof. Dr. Benedikt Wirth

Information on the seminar
HIS/LFS or HIS/LFS

Time, location:	to be determined
Content:	Optimization: Many application problems can be formulated as variational or optimization problem. This often also involves partial differential equations as constraints, e.g. during optimal control of biological/chemical/physical/economic processes, the design of engineering structures, or inverse problems of medicine and biology. We will deal with the analysis and numerics for such problems as well as general optimization methods. Calderon problem: A classical type of inverse problem is parameter identification in PDEs: More specifically, one tries to identify one or more spatially varying coefficients of the PDE from observations of (parts of) its solution. A particular example is to identify the conductivity coefficient of Laplace's equation from measurements of the solution and its normal derivative on the domain boundary. An application of this problem is electrical impedance tomography. The problem was posed in 1980 by Calderon, and its unique solvability was shown by Uhlmann and Sylvester. This has sparked a lot of development in inverse problems, but also related fields, e.g. in geometry.
Prerequisites:	Analysis I-III; a specialization in a module on one of the fields numerics, analysis, geometry will be helpful.
Participation:	If you are interested, please contact us by e-mail.
Requirements:	Presentation of 60-90 minutes and up to 10 pages handout for your peers (it should be shown and discussed at least a week before the presentation)
Topics:	Here you find an exemplary list of topics on the Calderon problem: Jennifer Mueller, Samuli Siltanen: Linear and Nonlinear Inverse Problems with Practical Applications Book is available online within the university. Of interest to us are the sections on nonlinear inverse problems and EIT. M. Cheney, D. Isaacson, and J.C. Newell (1999), Electrical impedance tomography, SIAM Rev. 41, 85-101 Review article. KOHN AND M. VOGELIUS, Determining conductivity by boundary measurements, Comm. Pure Appl. Math., 37 (1984), pp. 113-123. NACHMAN, Reconstruction from boundary measurements, Ann. of Math., 128 (1988),Â pp. 531-576. Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math., 143 (1996), pp. 7. SYLVESTER AND UHLMANN, A uniqueness theorem for an inverse boundary value problem in electrical prospection, Comm. Pure Appl. Math., 39 (1986), pp. 91-112 SYLVESTER AND UHLMANN, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), pp. 153-169; SYLVESTER AND UHLMANN, Inverse boundary value problems at the boundary-continuous dependence, Comm. Pure Appl. Math., 41 (1988), pp. 197-22 B. Harrach: The CalderÃ³n problem with finitely many unknowns is equivalent to convex semidefinite optimization, SIAM J. Math. Anal. 55 (5), 5666-5684, 2023. For optimization we might follow chapters in textbooks or a few selected articles in the context of optimizing artificial neural networks such as: Ulbrich, Ulbrich, Hinze, Pinnau: Optimization with PDE constraints Table of contents available on the website -- to have a look inside let us know. Nocedal, Wright: Numerical Optimization Boyd, Vandenberghe: Convex Optimization Conn, Gould, Toint: Trust Region Methods Suh: Convex Optimization for Machine Learning Ruder: An overview of gradient descent optimization algorithms Kingma, Ba: ADAM: A Method for Stochastic Optimization Bottou: Stochastic Learning Dereich, Jentzen, Riekert: Sharp higher order convergence rates for the Adam optimizer