A stochastic approach to mixed linear and nonlinear inverse problems with applications to seismology
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We derive an efficient stochastic algorithm for computational inverse problems that present an unknown linear forcing term and a set of nonlinear parameters to be recovered. It is assumed that the data is noisy and that the linear part of the problem is ill-posed. The vector of nonlinear parameters to be recovered is modeled as a random variable. This random vector is augmented by a random regularization parameter for the linear part. A probability distribution function for this augmented random vector knowing the measurements is derived. We explain how this derivation is related to the maximum likelihood regularization parameter selection [Galatsanos and Katsaggelos, 1992], which we generalize to the case where the underlying linear operator is rectangular and depends on a nonlinear parameter. A major difference in our approach is that, unlike in [Galatsanos and Katsaggelos, 1992], we do not limit ourselves to the most likely regularization parameter, instead we show that due to the dependence of the problem on the nonlinear parameter, there is a great advantage in exploring all positive values of the regularization parameter. Based on our new probability distribution function, we construct a choice sampling algorithm to compute the posterior expected value and covariance of the nonlinear parameter. This algorithm is greatly accelerated by using a parallel platform where we alternate computing proposals in parallel and combining proposals to accept or reject them as in [Calderhead, 2014]. Finally, our new algorithm is illustrated by solving an inverse problem in seismology. We show how our algorithm performs in that example and how it is able to compute marginal posterior probability functions even in the presence of strong noise. We discuss why this problem can not be approached by using the Generalized Cross Validation method or the discrepancy principle.
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