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Sparse Reconstructions for Inverse PDE Problems

Authors: Thorsten Raasch

Published in: Dagstuhl Seminar Proceedings, Volume 8492, Structured Decompositions and Efficient Algorithms (2009)

Abstract

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

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Thorsten Raasch. Sparse Reconstructions for Inverse PDE Problems. In Structured Decompositions and Efficient Algorithms. Dagstuhl Seminar Proceedings, Volume 8492, pp. 1-8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{raasch:DagSemProc.08492.8,
  author =	{Raasch, Thorsten},
  title =	{{Sparse Reconstructions for Inverse PDE Problems}},
  booktitle =	{Structured Decompositions and Efficient Algorithms},
  pages =	{1--8},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2009},
  volume =	{8492},
  editor =	{Stephan Dahlke and Ingrid Daubechies and Michal Elad and Gitta Kutyniok and Gerd Teschke},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.08492.8},
  URN =		{urn:nbn:de:0030-drops-18784},
  doi =		{10.4230/DagSemProc.08492.8},
  annote =	{Keywords: Adaptivity, sparse reconstructions, l1 minimization, parameter identification}
}

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Sparse Reconstructions for Inverse PDE Problems

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