Abstract: The Bayesian approach to inverse problems provides a rigorous framework for the incor-poration and quantification of uncertainties in measurements, parameters and models. However, sampling from or integrating w.r.t. the resultung posterior measure can become computationally challenging. In recent years, a lot of effort has been spent on deriving dimension-independent methods and to combine efficient sampling strategies with multilevel or surrogate methods in order to reduce the computational burden of Bayesian inverse problems.
In this talk, we are interested in designing numerical methods which are robust w.r.t. the size of the observational noise, i.e., methods which behave well in case of concentrated posterior measures. The concentration of the posterior is a highly desirable situation in practice, since it relates to informative or large data. However, it can pose as well a significant computational challenge for numerical methods based on the prior or reference measure. We propose to employ the Laplace approximation of the posterior as the base measure for numerical integration in this context. The Laplace approximation is a Gaussian measure centered at the maximum a-posteriori estimate (MAPE) and with covariance matrix depending on the Hessian of the log posterior density at the MAPE. We discuss convergence results of the Laplace approximation in terms of the Hellinger distance and analyze the efficiency of Monte Carlo methods based on it. In particular, we show that Laplace-based importance sampling and quasi-Monte-Carlo as well as Laplace-based Metropolis-Hastings algorithms are robust w.r.t. the concentration of the posterior for large classes of posterior distributions and integrands whereas prior-based Monte Carlo sampling methods are not.