Stochastic solutions to mixed linear and nonlinear inverse problems

We derive an efficient stochastic algorithm for 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. The derivation is based on the maximum likelihood regularization parameter selection which we generalize to the case where the underlying linear operator is rectangular and depends on a nonlinear parameter. Unlike in previous studies, 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 propose and accept or reject 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. Finally, our new algorithm is illustrated by solving an inverse problem in seismology. We show that the results obtained by our algorithm are more accurate than those found using Generalized Cross Validation or using the discrepancy principle, and that our algorithm has the capability to quantify uncertainty.