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 spacetime, 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 Tikhonovtype functional
begin{equation*}
min_{mathbf u}Kmathbf u^ opPsif^delta^2+alphasum_{lambda}omega_lambdau_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 twodimensional 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, PhilippsUniversit"at Marburg, 2007
BibTeX  Entry
@InProceedings{raasch:DSP:2009:1878,
author = {Thorsten Raasch},
title = {Sparse Reconstructions for Inverse PDE Problems},
booktitle = {Structured Decompositions and Efficient Algorithms},
year = {2009},
editor = {Stephan Dahlke and Ingrid Daubechies and Michal Elad and Gitta Kutyniok and Gerd Teschke},
number = {08492},
series = {Dagstuhl Seminar Proceedings},
ISSN = {18624405},
publisher = {Schloss Dagstuhl  LeibnizZentrum fuer Informatik, Germany},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2009/1878},
annote = {Keywords: Adaptivity, sparse reconstructions, l1 minimization, parameter identification}
}
2009
Keywords: 

Adaptivity, sparse reconstructions, l1 minimization, parameter identification 
Seminar: 

08492  Structured Decompositions and Efficient Algorithms

Issue date: 

2009 
Date of publication: 

2009 