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IPI

Continuing our previous work [6, Inverse Problems, 2012, 28, 055002] and [5, Inverse Problems, 2012, 28, 055001],
we address the ill-posedness of the inverse scattering problem of
electromagnetic waves due to an inhomogeneous medium by studying the
Hessian of the data misfit. We derive and analyze the Hessian in both
Hölder and Sobolev spaces. Using an integral equation approach based
on Newton potential theory and compact embeddings in Hölder and
Sobolev spaces, we show that the Hessian can be decomposed into three
components, all of which are shown to be compact operators. The
implication of the compactness of the Hessian is that for small data
noise and model error, the discrete Hessian can be approximated by a
low-rank matrix. This in turn enables fast solution of an
appropriately regularized inverse problem, as well as Gaussian-based
quantification of uncertainty in the estimated inhomogeneity.

IPI

We present a systematic construction of FEM-based
dimension-independent (discretization-invariant) Markov chain Monte
Carlo (MCMC) approaches to explore PDE-constrained Bayesian inverse
problems in infinite dimensional parameter spaces. In particular, we
consider two frameworks to achieve this goal:
Metropolize-then-discretize and discretize-then-Metropolize. The
former refers to the method of discretizing function-space MCMC
methods. The latter, on the other hand, first discretizes the Bayesian
inverse problem and then proposes MCMC methods for the resulting
discretized posterior probability density. In general, these two
frameworks do not commute, that is, the resulting finite dimensional
MCMC algorithms are not identical. The discretization step of the
former may not be trivial since it involves both numerical analysis
and probability theory, while the latter, perhaps ``easier'', may not
be discretization-invariant using traditional approaches. This paper
constructively develops finite element (FEM) discretization schemes
for both frameworks and shows that both commutativity and
discretization-invariance are attained. In particular, it shows how to
construct discretize-then-Metropolize approaches for both
Metropolis-adjusted Langevin algorithm and the hybrid Monte Carlo method
that commute with their Metropolize-then-discretize counterparts. The
key that enables this achievement is a proper FEM discretization of
the prior, the likelihood, and the Bayes' formula, together with a
correct definition of quantities such as the gradient and the covariance
matrix in discretized finite dimensional parameter spaces. The
implication is that practitioners can take advantage of the developments
in this paper to straightforwardly construct discretization-invariant
discretize-then-Metropolize MCMC for large-scale inverse
problems. Numerical results for one- and two-dimensional elliptic
inverse problems with up to $17899$ parameters are presented to
support the proposed developments.

IPI

We present a scalable solver for approximating the

*maximum a posteriori*(MAP) point of Bayesian inverse problems with Besov priors based on wavelet expansions with random coefficients. It is a subspace trust region interior reflective Newton conjugate gradient method for bound constrained optimization problems. The method combines the rapid locally-quadratic convergence rate properties of Newton's method, the effectiveness of trust region globalization for treating ill-conditioned problems, and the Eisenstat--Walker idea of preventing oversolving. We demonstrate the scalability of the proposed method on two inverse problems: a deconvolution problem and a coefficient inverse problem governed by elliptic partial differential equations. The numerical results show that the number of Newton iterations is independent of the number of wavelet coefficients $n$ and the computation time scales linearly in $n$. It will be numerically shown, under our implementations, that the proposed solver is two times faster than the split Bregman approach, and it is an order of magnitude less expensive than the interior path following primal-dual method. Our results also confirm the fact that the Besov $\mathbb{B}_{11}^1$ prior is sparsity promoting, discretization-invariant, and edge-preserving for both imaging and inverse problems governed by partial differential equations.
keywords:
Bayesian inversion
,
bound-constrained optimization
,
partial differential
equations
,
discretization-invariant
,
interior point
method
,
deconvolution
,
split Bregman method
,
Besov space priors
,
sparsity
,
trust region
,
MAP
,
wavelet
,
edge-preserving.
,
inverse problem
,
Newton method

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