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 ut-Δu+q(x)u=0ut-Δu+q(x)u=0 from the final measurement u(x,T)u(x,T), where q   is in some subset of L1(Ω)L1(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 H1H1-norm on q. Numerical implementations are also presented.