Deterministic interpolation and quadrature methods are often unsuitable to address Bayesian inverse problems depending on computationally expensive forward mathematical models. While interpolation may give precise posterior approximations, deterministic quadrature is usually unable to efficiently investigate an informative and thus concentrated likelihood. This leads to a large number of required expensive evaluations of the mathematical model. To overcome these challenges, we formulate and test a multilevel adaptive sparse Leja algorithm. At each level, adaptive sparse grid interpolation and quadrature are used to approximate the posterior and perform all quadrature operations, respectively. Specifically, our algorithm uses coarse discretizations of the underlying mathematical model to investigate the parameter space and to identify areas of high posterior probability. Adaptive sparse grid algorithms are then used to place points in these areas, and ignore other areas of small posterior probability. The points are weighted Leja points. As the model discretization is coarse, the construction of the sparse grid is computationally efficient. On this sparse grid, the posterior measure can be approximated accurately with few expensive, fine model discretizations. The efficiency of the algorithm can be enhanced further by exploiting more than two discretization levels. We apply the proposed multilevel adaptive sparse Leja algorithm in numerical experiments involving elliptic inverse problems in 2D and 3D space, in which we compare it with Markov chain Monte Carlo sampling and a standard multilevel approximation.
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