Low-rank tensor reconstruction of concentrated densities with application to Bayesian inversion
- Eigel, Martin
- Gruhlke, Robert
- Marschall, Manuel
2010 Mathematics Subject Classification
- 62F15 62G07 35R60 60H35 65C20 65N12 65N22 65J10
- Tensor train, uncertainty quantification, VMC, low-rank, reduced order model, Bayesian inversion, partial differential equations with random coefficients
A novel method for the accurate functional approximation of possibly highly concentrated probability densities is developed. It is based on the combination of several modern techniques such as transport maps and nonintrusive reconstructions of low-rank tensor representations. The central idea is to carry out computations for statistical quantities of interest such as moments with a convenient reference measure which is approximated by an numerical transport, leading to a perturbed prior. Subsequently, a coordinate transformation leads to a beneficial setting for the further function approximation. An efficient layer based transport construction is realized by using the Variational Monte Carlo (VMC) method. The convergence analysis covers all terms introduced by the different (deterministic and statistical) approximations in the Hellinger distance and the Kullback-Leibler divergence. Important applications are presented and in particular the context of Bayesian inverse problems is illuminated which is a central motivation for the developed approach. Several numerical examples illustrate the efficacy with densities of different complexity.