Adaptive Multilevel Sparse Leja Approximations for Bayesian inverse problems

Published:

Ionut-Gabriel Farcas (Oden Institute, UT Austin), Elisabeth Ullmann (TUM), Tobias Neckel (TUM), Hans-Joachim Bungartz (TUM) and I have published an article in the SIAM Journal of Scientific Computing (SISC), part A. We propose an efficient, deterministic strategy allowing us to find sparse grid approximations to posterior measures in Bayesian inverse problems. Deterministic strategies tend to be inefficient, if the posterior measure is highly concentrated: finding the area of concentration requires many model evaluations which are infeasible. Our algorithm uses computationally cheap model discretisations (‘multilevel approach’) to find this area of concentration computationally cheaply. We combine this idea with adaptive sparse grids that are constructed with weighted Leja points and address problems such as the separation of posterior measures. Finally, we validate the approach in numerical experiments, considering, e.g., an elliptic inverse problem in a 3D domain. Click here for the journal publication or click here for the preprint version.