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When quoting this document, please refer to the following
DOI: 10.4230/DagSemProc.08492.8
URN: urn:nbn:de:0030-drops-18784
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2009/1878/
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Raasch, Thorsten
Sparse Reconstructions for Inverse PDE Problems
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
BibTeX - Entry
@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.dagstuhl.de/opus/volltexte/2009/1878},
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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Keywords: |
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Adaptivity, sparse reconstructions, l1 minimization, parameter identification |
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Collection: |
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08492 - Structured Decompositions and Efficient Algorithms |
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Issue Date: |
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2009 |
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Date of publication: |
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24.02.2009 |
