Sparse Deterministic Approximation of Bayesian Inverse Problems

TL;DR

This paper develops a deterministic, sparse polynomial chaos approach to efficiently approximate the posterior density in Bayesian inverse problems involving infinite-dimensional parameters, enabling better uncertainty quantification.

Contribution

It introduces a generalized polynomial chaos representation for the posterior density and establishes conditions for algebraic convergence rates based on prior sparsity.

Findings

Posterior density can be approximated with algebraic convergence under certain prior conditions.

Sparsity of the input data's coefficients influences the approximation accuracy.

Efficient uncertainty quantification is achieved through the proposed deterministic formulation.

Abstract

We present a parametric deterministic formulation of Bayesian inverse problems with input parameter from infinite dimensional, separable Banach spaces. In this formulation, the forward problems are parametric, deterministic elliptic partial differential equations, and the inverse problem is to determine the unknown, parametric deterministic coefficients from noisy observations comprising linear functionals of the solution. We prove a generalized polynomial chaos representation of the posterior density with respect to the prior measure, given noisy observational data. We analyze the sparsity of the posterior density in terms of the summability of the input data's coefficient sequence. To this end, we estimate the fluctuations in the prior. We exhibit sufficient conditions on the prior model in order for approximations of the posterior density to converge at a given algebraic rate, in terms of the number $N$ of unknowns appearing in the parameteric representation of the prior measure. Similar sparsity and approximation results are also exhibited for the solution and covariance of the elliptic partial differential equation under the posterior. These results then form the basis for efficient uncertainty quantification, in the presence of data with noise.