A parallel sampling algorithm for some nonlinear inverse problems

Abstract

Abstract We derive a parallel sampling algorithm for computational inverse problems that present an unknown linear forcing term and a vector of nonlinear parameters to be recovered. It is assumed that the data are noisy and that the linear part of the problem is ill-posed. The vector of nonlinear parameters ${m} $ is modeled as a random variable. A dilation parameter $\alpha $ is used to scale the regularity of the linear unknown and is also modeled as a random variable. A posterior probability distribution for $({m}, \alpha )$ is derived following an approach related to the maximum likelihood (ML) regularization parameter selection (Galatsanos, N. P. & Katsaggelos, A. K. (1992). Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation. IEEE Trans. Image Process., 1, 322–336). A major difference in our approach is that, unlike in Galatsanos, N. P. & Katsaggelos, A. K. (1992, Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation. IEEE Trans. Image Process., 1, 322–336), we do not limit ourselves to the maximum likelihood value of $\alpha $. We then derive a parallel sampling algorithm where we alternate computing proposals in parallel and combining proposals to accept or reject them as in Calderhead, B. (2014, A general construction for parallelizing metropolis- hastings algorithms. Proc. Natl Acad Sci, 111, 17408–17413). This algorithm is well suited to problems where proposals are expensive to compute. We then apply it to an inverse problem in seismology. We show how our results compare favorably to those obtained from the ML, the Generalized Cross Validation and the Constrained Least Squares algorithms.