An introduction to finite element methods for inverse coefficient problems in elliptic PDEs

DOI10.1365/S13291-021-00236-2arXiv2103.00888WikidataQ114054859 ScholiaQ114054859MaRDI QIDQ6361800

Publication date: 1 March 2021

Abstract: Several novel imaging and non-destructive testing technologies are based on reconstructing the spatially dependent coefficient in an elliptic partial differential equation from measurements of its solution(s). In practical applications, the unknown coefficient is often assumed to be piecewise constant on a given pixel partition (corresponding to the desired resolution), and only finitely many measurements can be made. This leads to the problem of inverting a finite-dimensional non-linear forward operator

m a t h c a l F : m a t h c a l D (m a t h c a l F) s u b s e t e q m a t h b b R^{n} o m a t h b b R^{m}

, where evaluating

m a t h c a l F

requires one or several PDE solutions. Numerical inversion methods require the implementation of this forward operator and its Jacobian. We show how to efficiently implement both using a standard FEM package and prove convergence of the FEM approximations against their true-solution counterparts. We present simple example codes for Comsol with the Matlab Livelink package, and numerically demonstrate the challenges that arise from non-uniqueness, non-linearity and instability issues. We also discuss monotonicity and convexity properties of the forward operator that arise for symmetric measurement settings.

Mathematics Subject Classification ID

Variational methods applied to PDEs (35A15) Stability and convergence of numerical methods for boundary value problems involving PDEs (65N12) Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs (65N30) Numerical methods for inverse problems for boundary value problems involving PDEs (65N21) Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis (65-01)

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