Bayesian Inversion of Log-normal Eikonal Equations

TL;DR

This paper develops a multilevel MCMC approach to efficiently solve the Bayesian inverse problem for the log-normal eikonal equation, providing theoretical error bounds and numerical validation.

Contribution

It introduces a multilevel MCMC method for the Bayesian inverse problem of the eikonal equation, reducing computational cost while maintaining accuracy.

Findings

Multilevel MCMC achieves prescribed accuracy with optimal complexity.

Error bounds are established for the posterior approximation.

Numerical examples confirm theoretical predictions.

Abstract

We study the Bayesian inverse problem for inferring the log-normal slowness function of the eikonal equation given noisy observation data on its solution at a set of spatial points. We study approximation of the posterior probability measure by solving the truncated eikonal equation, which contains only a finite number of terms in the Karhunen-Loeve expansion of the slowness function, by the Fast Marching Method. The error of this approximation in the Hellinger metric is deduced in terms of the truncation level of the slowness and the grid size in the Fast Marching Method resolution. It is well known that the plain Markov Chain Monte Carlo procedure for sampling the posterior probability is highly expensive. We develop and justify the convergence of a Multilevel Markov Chain Monte Carlo method. Using the heap sort procedure in solving the forward eikonal equation by the Fast Marching Method, our Multilevel Markov Chain Monte Carlo method achieves a prescribed level of accuracy for approximating the posterior expectation of quantities of interest, requiring only an essentially optimal level of complexity. Numerical examples confirm the theoretical results.