Solving an inverse parabolic problem by optimization from final measurement data

We consider an inverse problem of reconstructing the coefficient q in the parabolic equation u t - Δ u + q ( x ) u = 0 from the final measurement u ( x , T ) , where q is in some subset of L 1 ( Ω ) . The optimization method, combined with the finite element method, is applied to get the numerical solution under some assumption on q. The existence of minimizer, as well as the convergence of approximate solution in finite-dimensional space, is proven. The new ingredient in this paper is that we do not need uniformly a priori bounds of H 1 -norm on q. Numerical implementations are also presented.