Abstract
Abstract
In this work we analyze the inverse problem of recovering a space-dependent potential coefficient in an elliptic / parabolic problem from distributed observation. We establish novel (weighted) conditional stability estimates under very mild conditions on the problem data. Then we provide an error analysis of a standard reconstruction scheme based on the standard output least-squares formulation with Tikhonov regularization (by an H
1-seminorm penalty), which is then discretized by the Galerkin finite element method with continuous piecewise linear finite elements in space (and also backward Euler method in time for parabolic problems). We present a detailed error analysis of the discrete scheme, and provide convergence rates in a weighted
L
2
(
Ω
)
for discrete approximations with respect to the exact potential. The error bounds explicitly depend on the noise level, regularization parameter and discretization parameter(s). Under suitable conditions, we also derive error estimates in the standard
L
2
(
Ω
)
and interior L
2 norms. The analysis employs sharp a priori error estimates and nonstandard test functions. Several numerical experiments are given to complement the theoretical analysis.
Funder
Engineering and Physical Sciences Research Council
National Natural Science Foundation of China
Hong Kong Research Grants Council
Subject
Applied Mathematics,Computer Science Applications,Mathematical Physics,Signal Processing,Theoretical Computer Science
Cited by
2 articles.
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