Sparse Spectral Methods for Approximating PDE Solutions in Particle Flow

In sequential Monte Carlo, a method for performing the Bayesian computation prior × likelihood is to derive the law of motion of a particle ensemble: a particle flow. This enables sampling from complex distributions while avoiding issues such as particle degeneracy and the need for resampling. However, some particle-flow implementations require solving a partial differential equation (PDE) whose coefficients depend on the density of particles. The solution to this PDE must typically be approximated as analytical solutions are limited to specific cases. Traditionally, spectral methods for approximating the solution are based on the tensor product formulation in which the solution is represented as a weighted sum of products of univariate basis functions. Using NB bases per coordinate of the NX-dimensional domain leads to (NB)^NX unknowns. Thus, even for problems of moderate dimensionality, current computational resources are insufficient to support the full tensor-product grid necessary for an accurate approximation. In this work, we propose an approximation based on a sparse grid/hyperbolic cross technique to solve the PDE in more general settings. The solution is approximated with multivariate polynomials bases whose span is dense in the space of solutions under mild assumptions on the underlying distribution. We show the accuracy of our technique for sampling from Gaussian and non-Gaussian distributions.