Convergence rate analysis of Galerkin approximation of inverse potential problem

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(?) 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(?) 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.

Automated summary

In this work, we address the inverse problem of recovering a space-dependent potential coefficient in elliptic/parabolic problems from distributed observation. We establish novel stability estimates and analyze the error of a reconstruction scheme based on the output least-squares formulation with Tikhonov regularization. The analysis includes convergence rates and error estimates in different norms, and is supported by numerical experiments.