1,304 research outputs found
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 .
The target object is assumed to have a sparse expansion with respect to a wavelet system in space-time, being equivalent to a priori information on the regularity of in a certain scale
of Besov spaces . For the recovery of the unknown coefficient array , 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, being a noisy version of .
Since any application of the forward operator 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 and 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 . Moreover, the sequence of iterates linearly converges to the minimizer of the functional, a result which can also be proved
for the special case , 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, 200
A penalty method for PDE-constrained optimization in inverse problems
Many inverse and parameter estimation problems can be written as
PDE-constrained optimization problems. The goal, then, is to infer the
parameters, typically coefficients of the PDE, from partial measurements of the
solutions of the PDE for several right-hand-sides. Such PDE-constrained
problems can be solved by finding a stationary point of the Lagrangian, which
entails simultaneously updating the paramaters and the (adjoint) state
variables. For large-scale problems, such an all-at-once approach is not
feasible as it requires storing all the state variables. In this case one
usually resorts to a reduced approach where the constraints are explicitly
eliminated (at each iteration) by solving the PDEs. These two approaches, and
variations thereof, are the main workhorses for solving PDE-constrained
optimization problems arising from inverse problems. In this paper, we present
an alternative method that aims to combine the advantages of both approaches.
Our method is based on a quadratic penalty formulation of the constrained
optimization problem. By eliminating the state variable, we develop an
efficient algorithm that has roughly the same computational complexity as the
conventional reduced approach while exploiting a larger search space. Numerical
results show that this method indeed reduces some of the non-linearity of the
problem and is less sensitive the initial iterate
Semivariogram methods for modeling Whittle-Mat\'ern priors in Bayesian inverse problems
We present a new technique, based on semivariogram methodology, for obtaining
point estimates for use in prior modeling for solving Bayesian inverse
problems. This method requires a connection between Gaussian processes with
covariance operators defined by the Mat\'ern covariance function and Gaussian
processes with precision (inverse-covariance) operators defined by the Green's
functions of a class of elliptic stochastic partial differential equations
(SPDEs). We present a detailed mathematical description of this connection. We
will show that there is an equivalence between these two Gaussian processes
when the domain is infinite -- for us, -- which breaks down when
the domain is finite due to the effect of boundary conditions on Green's
functions of PDEs. We show how this connection can be re-established using
extended domains. We then introduce the semivariogram method for estimating the
Mat\'ern covariance parameters, which specify the Gaussian prior needed for
stabilizing the inverse problem. Results are extended from the isotropic case
to the anisotropic case where the correlation length in one direction is larger
than another. Finally, we consider the situation where the correlation length
is spatially dependent rather than constant. We implement each method in
two-dimensional image inpainting test cases to show that it works on practical
examples
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