Numerical Solution of a Coefficient Identification Problem in the Poisson equation Submission

In this Bachelor thesis, the problem of determining the coefficient q ∈ Q := {q ∈ L∞(Ω) : 0 < q ≤ q(x) ≤ q a.e. in Ω} such that it satisfies the Poisson equation −∇ · (q∇u) = f in Ω u = 0 on ∂Ω for given inexact data uδ (instead of exact data u) with ‖u−u‖H1(Ω) ≤ δ is analyzed and solved numerically. It is implemented in c++ using the Distributed and Unified Numerics Environment (DUNE). The origin of the ill-posed nature of this inverse problem is illustrated in 1D. In order to remedy this ill-posedness, Tikhonov regularization is applied together with Morozov’s Discrepancy Principle and, instead of trying to minimize the non-convex least-squares Tikhonov functional, the new convex energy functional found in [1] is used. Experiments performed with this implementation agree with the theory found in [5]: For smooth coefficients a convergence rate of O( √ δ) in the L2(Ω)-Norm is observed and for non-smooth coefficients the reconstruction and the convergence rates are found to be much worse.