Well-posedness of Bayesian inverse problems for hyperbolic conservation laws

TL;DR

This paper investigates the well-posedness of Bayesian inverse problems for hyperbolic conservation laws, establishing stability and continuity of the posterior distribution with respect to measurements and approximations, supported by numerical experiments.

Contribution

It provides the first rigorous analysis of the stability and Lipschitz continuity of the Bayesian inverse problem for hyperbolic conservation laws.

Findings

Lipschitz continuity of the measurement to posterior map

Stability of the posterior to approximations in Wasserstein distance

Numerical experiments confirming theoretical estimates

Abstract

We study the well-posedness of the Bayesian inverse problem for scalar hyperbolic conservation laws where the statistical information about inputs such as the initial datum and (possibly discontinuous) flux function are inferred from noisy measurements. In particular, the Lipschitz continuity of the measurement to posterior map as well as the stability of the posterior to approximations, are established with respect to the Wasserstein distance. Numerical experiments are presented to illustrate the derived estimates.