Bayesian Inversion of Log-normal Eikonal Equations

by   Zhan Fei Yeo, et al.

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.


page 1

page 2

page 3

page 4


Multilevel Gibbs Sampling for Bayesian Regression

Bayesian regression remains a simple but effective tool based on Bayesia...

Bayesian inversion for Electrical Impedance Tomography by sparse interpolation

We study the Electrical Impedance Tomography Bayesian inverse problem fo...

Multilevel adaptive sparse Leja approximations for Bayesian inverse problems

Deterministic interpolation and quadrature methods are often unsuitable ...

Conditioning by Projection for the Sampling from Prior Gaussian Distributions

In this work we are interested in the (ill-posed) inverse problem for ab...

Low-rank tensor reconstruction of concentrated densities with application to Bayesian inversion

Transport maps have become a popular mechanic to express complicated pro...

Bayesian inversion of convolved hidden Markov models with applications in reservoir prediction

Efficient assessment of convolved hidden Markov models is discussed. The...

Multilevel Markov Chain Monte Carlo for Bayesian Elliptic Inverse Problems with Besov Random Tree Priors

We propose a multilevel Monte Carlo-FEM algorithm to solve elliptic Baye...

Please sign up or login with your details

Forgot password? Click here to reset