Dr. Martin Eigel – An adaptive tensor reconstruction for Bayesian inversion

WIAS Berlin, Forschungsgruppe „Nichtlineare Optimierung und Inverse Probleme“

Abstract

We consider a functional low-rank representation for probability densities which can be applied in Bayesian inverse problems. While the used tensor compression makes the evaluation of high-dimensional integrals amenable (under appropriate conditions), typical properties of the measures such as non-linearity and high concentration cannot be represented easily in a classical (polynomial) basis. We hence propose to first construct a transport map, providing an approximate density transfer from a convenient reference to a complicated target measure. The respective pull-back yields a perturbed prior density in a new coordinate system for which a layered decomposition can be constructed in the spirit of (FEM) hp-refinements. A low-rank tensor reconstruction („Variational Monte Carlo“) is used for the layer-based representations, leading to highly efficient evaluations of moments with respect to the complicated posterior. Numerical experiments confirm the superior convergence in comparison to Monte Carlo and MCMC in benchmark problems.