Bayesian inversion of eikonal equations

Abstract

We consider Bayesian inversion for eikonal equations subjected to the point source and Soner Boundary conditions. For every setting, we show Hadamard well-posedness rigorously by exploiting Lagrangian duality.

In the isotropic setting, discretisation errors from finitely truncating the slowness expansion and approximating forward solutions via the Fast Marching Method (FMM) are derived in Hellinger distance for the uniform, loguniform, and lognormal prior cases. For star shaped priors, the induced error from taking approximations via the FMM is quantified. In the Riemannian anisotropy generalisation, the error induced by finite truncation of the metric expansion is derived for analogous uniform and lognormal priors. A modified FMM is employed to numerically approximate the posterior.

We develop complexity optimal Multilevel Markov Chain Monte Carlo (MLMCMC) to approximate posterior expectations. Numerical examples corroborate the optimality of MLMCMC and show that the algorithm is well capable of recovering the slowness from observation data.