Low-rank tensor reconstruction of concentrated densities with application to Bayesian inversion

Abstract

This paper presents a novel method for the accurate functional approximation of possibly highly concentrated probability densities. It is based on the combination of several modern techniques such as transport maps and low-rank approximations via a nonintrusive tensor train reconstruction. The central idea is to carry out computations for statistical quantities of interest such as moments based on a convenient representation of a reference density for which accurate numerical methods can be employed. Since the transport from target to reference can usually not be determined exactly, one has to cope with a perturbed reference density due to a numerically approximated transport map. By the introduction of a layered approximation and appropriate coordinate transformations, the problem is split into a set of independent approximations in seperately chosen orthonormal basis functions, combining the notions h- and p-refinement (i.e. “mesh size” and polynomial degree). An efficient low-rank representation of the perturbed reference density is achieved via the Variational Monte Carlo method. This nonintrusive regression technique reconstructs the map in the tensor train format. An a priori convergence analysis with respect to the error terms introduced by the different (deterministic and statistical) approximations in the Hellinger distance and the Kullback–Leibler divergence is derived. Important applications are presented and in particular the context of Bayesian inverse problems is illuminated which is a main motivation for the developed approach. Several numerical examples illustrate the efficacy with densities of different complexity and degrees of perturbation of the transport to the reference density. The (superior) convergence is demonstrated in comparison to Monte Carlo and Markov Chain Monte Carlo methods.

An hp‐adaptive multi‐element stochastic collocation method for surrogate modeling with information re‐use

“…16 The algorithm stops once the criterion (51) is not met. This happens if the error indicator (40) and/or the elements are sufficiently small, equivalently, 𝜚 k is sufficiently small.…”

“…Another possibility is to use the spectral stochastic embedding (SSE) method 37 . Other approaches are based on transport maps 40,41 . We will show in Section 5, that our multi‐element stochastic collocation method is also capable of handling such highly concentrated likelihood functions.…”

“…The error indicator (40) relates the highest-order partial variances to the total variance in an element d k , see Figure 2A. Thus, if the estimation on the local variance 𝜎 2 k is sufficiently accurate, the coefficients in the nominator will be small, thus resulting in a small error indicator value as well.…”

“…In the case of an anisotropic basis, for example, obtained with a dimension-adaptive algorithm, the only restriction on the local multi-index set Λ (k) is that it is downward closed. Then, the error indicator (40) might not be informative regarding the accuracy of the local polynomial approximation, for example, in the case of multi-index sets with a single dominant dimension. Considering the arbitrary shape of the basis, the employed coefficients may originate from different polynomial levels, unlike in (40).…”

“…Then, the error indicator (40) might not be informative regarding the accuracy of the local polynomial approximation, for example, in the case of multi-index sets with a single dominant dimension. Considering the arbitrary shape of the basis, the employed coefficients may originate from different polynomial levels, unlike in (40). It could therefore be beneficial to employ only admissible coefficients.…”