Sparse deterministic approximation of Bayesian inverse problems

Abstract

We present a parametric deterministic formulation of Bayesian inverse\nproblems with input parameter from infinite dimensional, separable Banach\nspaces. In this formulation, the forward problems are parametric, deterministic\nelliptic partial differential equations, and the inverse problem is to\ndetermine the unknown, parametric deterministic coefficients from noisy\nobservations comprising linear functionals of the solution.\n We prove a generalized polynomial chaos representation of the posterior\ndensity with respect to the prior measure, given noisy observational data. We\nanalyze the sparsity of the posterior density in terms of the summability of\nthe input data's coefficient sequence. To this end, we estimate the\nfluctuations in the prior. We exhibit sufficient conditions on the prior model\nin order for approximations of the posterior density to converge at a given\nalgebraic rate, in terms of the number $N$ of unknowns appearing in the\nparameteric representation of the prior measure. Similar sparsity and\napproximation results are also exhibited for the solution and covariance of the\nelliptic partial differential equation under the posterior. These results then\nform the basis for efficient uncertainty quantification, in the presence of\ndata with noise.\n