Abstract

In this talk we analyze the inverse problem of recovering the space-dependent potential coefficient in an elliptic/parabolic problem from distributed observation. We establish a 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 H1-seminorm penalty, which is then discretized by the continuous piecewise linear finite elements in space, and also backward Euler method in time for parabolic problems. We present a detailed analysis of the discrete scheme, and provide convergence rates in a weighted L^2 norm for discrete approximations with respect to the exact potential. The error bounds are explicitly dependent on the noise level, regularization parameter and discretization parameter. Under suitable conditions, we also derive error estimates in the standard L^2 and interior L^2 norms. Several numerical experiments are given to complement the theoretical analysis