Sparse Reconstructions for Inverse PDE Problems

We are concerned with the numerical solution of linear parameter identification problems for parabolic PDE, written as an operator equation Ku = f . The target object u is assumed to have a sparse expansion with respect to a wavelet system Ψ = {ψλ} in space-time. For the recovery of the unknown coefficient array, we use Tikhonov regularization with `p coefficient penalties and the associated iterative shrinkage algorithms. Since any application of K and K∗ involves the numerical solution of a PDE, perturbed versions of the iteration have to be studied. In particular, for reasons of efficiency, adaptive operator applications are indispensable. By a suitable choice of the respective tolerances and stopping criteria, also the adaptive iteration converges and it has regularizing properties. We illustrate the performance of the resulting method by numerical computations for oneand two-dimensional inverse heat conduction problems.