Sequential Monte Carlo samplers for semilinear inverse problems and application to magnetoencephalography

TL;DR

This paper introduces a semi-analytic sequential Monte Carlo method tailored for semi-linear inverse problems, exemplified by magnetoencephalography, reducing computational costs and improving source localization over time-series data.

Contribution

It develops a semi-analytic SMC approach that marginalizes linear variables, enhancing efficiency in solving semi-linear inverse problems like magnetoencephalography.

Findings

Efficient estimation of dipole number and locations from time-series data.

Reduced Monte Carlo variance and computational cost.

Successful application to magnetoencephalography source localization.

Abstract

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