Convergence of Probability Densities using Approximate Models for Forward and Inverse Problems in Uncertainty Quantification

TL;DR

This paper proves that approximate models in uncertainty quantification lead to converging probability densities in both forward and inverse problems, supported by numerical demonstrations of optimal convergence rates.

Contribution

It establishes theoretical convergence results for probability densities in forward and inverse UQ problems using approximate models, with numerical validation.

Findings

Convergence of probability densities is proven as models improve.

Numerical results show optimal convergence with sparse grid and PDE/ODE discretizations.

Approximate models reliably estimate true densities in UQ contexts.

Abstract

We analyze the convergence of probability density functions utilizing approximate models for both forward and inverse problems. We consider the standard forward uncertainty quantification problem where an assumed probability density on parameters is propagated through the approximate model to produce a probability density, often called a push-forward probability density, on a set of quantities of interest (QoI). The inverse problem considered in this paper seeks a posterior probability density on model input parameters such that the subsequent push-forward density through the parameter-to-QoI map matches a given probability density on the QoI. We prove that the probability densities obtained from solving the forward and inverse problems, using approximate models, converge to the true probability densities as the approximate models converges to the true models. Numerical results are presented to demonstrate optimal convergence of probability densities for sparse grid approximations of parameter-to-QoI maps and standard spatial and temporal discretizations of PDEs and ODEs.