Seminar: Bayesian Inversion of Eikonal Equation

Time:

Venue/Location: Phòng C102, VIASM

Báo cáo viên: Hoàng Việt Hà (Nanyang Technological University, Singapore) 

Tóm tắt: We study the Bayesian inverse problem of a forward eikonal equation given noisy observation on its solution. In the isotropic case, we infer the slowness function; and in the anisotropic case of an eikonal equation on a manifold, we infer the manifold metric. We consider the Gaussian prior probability for the log-slowness (or log-metric), which is expressed as a countable linear expansion of mutually independent normal random variables. Using the variational formulation of the eikonal equation, we establish the well- posedness of the inverse problem. We approximate the posterior by finitely truncating the expansion of the log-slowness (log-metric), with an explicit error estimate in the Hellinger metric with respect to the truncation level. Solving the truncated eikonal equation by the Fast Marching Method, we obtain an approximation for the posterior in terms of the truncation level and the discrete grid size in the Fast Marching Method resolution. Using this result, we develop and justify the convergence of a Multilevel Markov Chain Monte Carlo (MLMCMC) method. Using the heap sort procedure for the Fast Marching Method, our MLMCMC method achieves a prescribed level of accuracy for approximating the posterior expectation of quantities of interest, requiring only an essentially optimal level of complexity, which is equivalent to that of the forward solver. This reduces the computation complexity drastically, in comparison to the plain Markov Chain Monte Carlo method where a large number of realizations of the forward equation are solved with equal high accuracy. Numerical examples confirm the theoretical results on the convergence rate of the method and the optimal complexity.