Accurate computation of conditional expectation for highly non-linear problems

TL;DR

This paper introduces a novel conditioned expectation method that accurately computes conditional expectations in highly non-linear inverse problems, improving upon traditional filters like Kalman by localizing the operator at measurement points.

Contribution

The paper presents a reformulation of conditional expectation that localizes the operator, enabling accurate predictions in highly non-linear problems and ensuring positive definite covariance matrices.

Findings

The conditioned expectation (CdE) accurately predicts conditioned mean and covariance in non-linear problems.

The method guarantees positive definite covariance matrices with straightforward numerical integration.

Numerical examples confirm the theoretical advantages of the proposed approach.

Abstract

This paper focuses on inverse problems to identify parameters by incorporating information from measurements. These generally ill-posed problems are formulated here in a probabilistic setting based on Bayes's theorem because it leads to a unique solution of the updated distribution of parameters. Many approaches build on Bayesian updating in terms of probability measures or their densities. However, the uncertainty propagation problems and their discretisation within the stochastic Galerkin or collocation method are naturally formulated for random vectors which calls for updating of random variables, i.e. a filter. Such filters typically build on some approximation to conditional expectation (CE). Specifically, the approximation of the CE with affine functions leads to the familiar Kalman filter which works best on linear or close to linear problems only. Our approach builds on a reformulation, which allows to localise the operator of the CE to the point of measured value. The resulting conditioned expectation (CdE) predicts correctly the quantities of interest, e.g. conditioned mean and covariance, even for general highly non-linear problems. The novel CdE allows straight-forward numerical integration; particularly, the approximated covariance matrix is always positive definite for integration rules with positive weights. The theoretical results are confirmed by numerical examples.