IPI
Analysis of the Hessian for inverse scattering problems. Part III: Inverse medium scattering of electromagnetic waves in three dimensions
Tan Bui-Thanh Omar Ghattas
Inverse Problems & Imaging 2013, 7(4): 1139-1155 doi: 10.3934/ipi.2013.7.1139
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.
keywords: electromagnetic wave propagation adjoint Gauss-Newton. compact operators Hessian Newton potential Inverse medium scattering ill-posedness compact embeddings Riesz-Fredholm theory
IPI
A scalable algorithm for MAP estimators in Bayesian inverse problems with Besov priors
Tan Bui-Thanh Omar Ghattas
Inverse Problems & Imaging 2015, 9(1): 27-53 doi: 10.3934/ipi.2015.9.27
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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