Inverse Problems in a Bayesian Setting

TL;DR

This paper presents a computationally efficient, sampling-free Bayesian filtering method for inverse problems and uncertainty quantification, utilizing conditional expectations and polynomial approximations to avoid slow Monte Carlo sampling.

Contribution

It introduces a novel nonlinear Bayesian update filter based on variational principles and polynomial discretization, improving computational efficiency in inverse problems.

Findings

The filter effectively handles complex inverse problems.

It outperforms linear updates in nonlinear scenarios.

Demonstrated on multiple examples with increasing complexity.

Abstract

In a Bayesian setting, inverse problems and uncertainty quantification (UQ) --- the propagation of uncertainty through a computational (forward) model --- are strongly connected. In the form of conditional expectation the Bayesian update becomes computationally attractive. We give a detailed account of this approach via conditional approximation, various approximations, and the construction of filters. Together with a functional or spectral approach for the forward UQ there is no need for time-consuming and slowly convergent Monte Carlo sampling. The developed sampling-free non-linear Bayesian update in form of a filter is derived from the variational problem associated with conditional expectation. This formulation in general calls for further discretisation to make the computation possible, and we choose a polynomial approximation. After giving details on the actual computation in the framework of functional or spectral approximations, we demonstrate the workings of the algorithm on a number of examples of increasing complexity. At last, we compare the linear and nonlinear Bayesian update in form of a filter on some examples.