Inverse problems and uncertainty quantification

TL;DR

This paper introduces a sampling-free, nonlinear Bayesian update method for inverse problems and uncertainty quantification, leveraging spectral approximations to improve computational efficiency and demonstrate effectiveness on complex models.

Contribution

It develops a novel variational approach for Bayesian updates that avoids Monte Carlo sampling, using polynomial spectral approximations for efficient computation.

Findings

The method effectively handles complex inverse problems.

Spectral approximations improve computational speed.

Application to Lorenz 84 model shows robustness.

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. This is especially the case as 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 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 quadratic Bayesian update on the small but taxing example of the chaotic Lorenz 84 model, where we experiment with the influence of different observation or measurement operators on the update.