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Bachelor/Master Seminar:

The Calderon Problem

WS 2025/26

Lecturer:

Dr. Chun-Kai Kevin Chien
Prof. Dr. Gustav Holzegel
Prof. Dr. Benedikt Wirth

Information on the seminar
HIS/LFS or HIS/LFS

Time, location:	Tu 12:00, highrise building (Einsteinstr. 62), 5th floor
Content:	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.
Topics:	You find the topics here, they will be complemented with more applied presentations from the following list: The D-bar regularized reconstruction method Jennifer Mueller, Samuli Siltanen: Linear and Nonlinear Inverse Problems with Practical Applications Chapter 12 (list of applications, derivation from Maxwell equations, discussion of nonlinearity and ill-posedness, practical aspects) Chapter 15 (regularized reconstruction by the D-bar method based on Nachman's constructive uniqueness theorem; from Dirichlet-to-Neumann operator to complex geometric optics solutions to scattering transform to conductivity) Chapter 16 (variants of the D-bar method and reconstruction in restricted spaces) Andy Adler, David Holder: Electrical Impedance Tomography - Methods, History and Applications Chapter 6 (general regularization methods for inverse problems and relation to EIT) Chapter 9 (D-bar methods) Chapter 10 (image processing tricks) Abschnitt III-IV (applications) Jennifer Mueller, Samuli Siltanen: The D-bar method for electrical impedance tomography - demystified Übersicht über D-bar-Methoden Sarah Jane Hamilton, Andreas Hauptmann: Deep D-bar: Real time electrical impedance tomography imaging with deep neural networks B. von Harrach: The Calderon problem with finitely many unknowns is equivalent to convex semidefinite optimization, SIAM J. Math. Anal. 55 (5), 5666-5684, 2023. reconstruction of shape and position of conductivity anomalies B. von Harrach: Recent progress on the factorization method for electrical impedance tomography quantitative examination of the degree of ill-posedness G. Alessandrini: Stable determination of conductivity by boundary measurements (Under source conditions the inverse map has at least a negative power of the logarithm as continuity modulus.) N. Mandache: Exponential instability in an inverse problem for the Schrödinger equation (... at most ...) Existence and uniqueness results M. Cheney, D. Isaacson, and J.C. Newell (1999), Electrical impedance tomography, SIAM Rev. 41, 85-101 Älterer, guter Übersichtsartikel. 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), 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