Authors:
Ionut-Gabriel Farcas | TU München | Germany
Jonas Latz | TU München | Germany
Prof. Elisabeth Ullmann | TU München | Germany
Dr. Tobias Neckel | TU München | Germany
In the Bayesian approach to inverse problems, one of the major challenges is the post-processing of the posterior distribution; the posterior incorporates the underlying model, usually given in terms of partial differential equations, therefore the numerical evaluation of a quantity of interest defined in terms of the posterior translates into the evaluation of the mathematical model, which is computationally expensive.
In this contribution, we propose an approach based on multilevel decompositions and sparse grid collocation for the Bayesian inversion of computationally expensive problems. Our main goal is to construct a multilevel surrogate of the underlying forward model such that we sequentially update the prior with previously obtained knowledge to obtain a more accurate approximation. Although sparse grids or multilevel methodologies have been previously employed in Bayesian inversion, our approach represents one of the first attempts to create multilevel sparse collocation surrogates in the context of Bayesian inversion.
We summarize our approach as follows. On the one hand, to discretize the problem's domain, we use a hierarchy of finite element grids with a resolution increasing by a constant factor. On the other hand, to discretize the stochastic domain, we use sparse approximations based on weighted Leja sequences. On the first level, we construct weighted Leja points using the initial prior distribution as the weight, we create a sparse approximation of the underlying model using these points, and we compute the posterior distribution on the initial level. Starting with the second level, we repeat the aforementioned process, but using the previously obtained posterior as the prior, and therefore as weight function in the Leja point construction. Hence, as the level increases, we construct surrogates based on priors closer and closer to the true posterior solution. Finally, to obtain a comprehensive overview of our approach, we employ it in several test problems.