Bayesian inverse problems for hyperbolic equations

Abstract

Numerical analysis of Bayesian inverse problems for hyperbolic partial differential equations is analysed in this report. Inverse problems involve constructing a mathematical model while only given limited information on the solution. Furthermore, these information are usually affected by errors caused by the noisy environment. Using classical methods, the inverse problems are typically ill-posed. To make the problems well-posed, a regularising term has to be chosen. However, by treating the error as a random variable, the Bayesian approach guarantees that the problem is well-posed. Given an observation filled with noise which follows a known probability distribution, we seek to find the posterior measure on the coefficient space. We use Markov Chain Monte Carlo method to sample the posterior expectation of a quantity of interest. The forward hyperbolic equation is solved numerically by the finite element method. We analyse the error estimate for the posterior expectation due to MCMC and finite element approximation. Numerical examples confirm the theoretical result. Most of the results in this report are new.