Sequential Monte Carlo samplers for semi-linear inverse problems and application to magnetoencephalography

Abstract

We discuss the use of a recent class of sequential Monte Carlo methods for\nsolving inverse problems characterized by a semi-linear structure, i.e. where\nthe data depend linearly on a subset of variables and nonlinearly on the\nremaining ones. In this type of problems, under proper Gaussian assumptions one\ncan marginalize the linear variables. This means that the Monte Carlo procedure\nneeds only to be applied to the nonlinear variables, while the linear ones can\nbe treated analytically; as a result, the Monte Carlo variance and/or the\ncomputational cost decrease. We use this approach to solve the inverse problem\nof magnetoencephalography, with a multi-dipole model for the sources. Here,\ndata depend nonlinearly on the number of sources and their locations, and\ndepend linearly on their current vectors. The semi-analytic approach enables us\nto estimate the number of dipoles and their location from a whole time-series,\nrather than a single time point, while keeping a low computational cost.\n