  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.
