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 $Psi={psi_lambda}$ in space-time, being equivalent to a priori information on the regularity of $u=mathbf u^ opPsi$ in a certain scale of Besov spaces $B^s_{p,p}$. For the recovery of the unknown coefficient array $mathbf u$, we miminize a Tikhonov-type functional begin{equation*} min_{mathbf u}|Kmathbf u^ opPsi-f^delta|^2+alphasum_{lambda}omega_lambda|u_lambda|^p end{equation*} by an associated thresholded Landweber algorithm, $f^delta$ being a noisy version of $f$. Since any application of the forward operator $K$ and its adjoint involves the numerical solution of a PDE, perturbed versions of the iteration have to be studied. In particular, for reasons of efficiency, adaptive applications of $K$ and $K^*$ are indispensable cite{Ra07}. By a suitable choice of the respective tolerances and stopping criteria, also the adaptive iteration could recently be shown to have regularizing properties cite{BoMa08a} for $p>1$. Moreover, the sequence of iterates linearly converges to the minimizer of the functional, a result which can also be proved for the special case $p=1$, see [DaFoRa08]. We illustrate the performance of the resulting method by numerical computations for one- and two-dimensional inverse heat conduction problems. References: [BoMa08a] T. Bonesky and P. Maass, Iterated soft shrinkage with adaptive operator evaluations, Preprint, 2008 [DaFoRa08] S. Dahlke, M. Fornasier, and T. Raasch, Multiscale Preconditioning for Adaptive Sparse Optimization, in preparation, 2008 [Ra07] T.~Raasch, Adaptive wavelet and frame schemes for elliptic and parabolic equations, Dissertation, Philipps-Universit"at Marburg, 2007