This thesis is concerned with the expressibility of high dimensional densities arising in Bayesian inverse problems by employing low-rank tensor techniques and controlling the approximation process a priori and a posteriori. The Bayesian paradigm offers well-posed solutions to inverse problems by statistical regularization but the produced posterior density is usually expensive to evaluate.
If the inversion process is additionally constrained to a partial differential equation (PDE), the accessibility of marginals, moments and other integral quantities is usually constrained to slow-converging sampling methods or expensive high-dimensional quadrature schemes. In contrast, the modern class of low-rank tensor formats such as the hierarchical Tucker decomposition offer near immediate access to the desired quantities, once it can be employed as a surrogate. In general, we distinguish two major challenges which are treated in the following, namely the forward and backward problem.
As the theoretical formulation of the posterior density relies on random variables as input for the underlying forward problem, the solvability, convergence and stability in the desired format has to be examined. Based on the underlying random PDE, a unified Galerkin scheme offers adaptive discretisation techniques and the definition of a posteriori error estimators. Employing a tensor space formulation and proximate low-rank formats is canonical when dealing with independent random variables representing an affine coefficient field. More involved settings, such as the lognormal case or the transformation based random domain approach are treated in this thesis. The analysis of the forward solution is essential when considering the Bayesian inverse or backward formulation, as for instance regularity is preserved.
To exploit this fact, interpolation and regression techniques exhibit the potential to efficiently create substitute models, replacing the posterior density and its involved forward problem. Thus, a priori error estimates in terms of Hellinger distance and adaptive refinement strategies are discussed in this thesis. On the one hand for the case of an affine parametric PDE coefficient involving interpolation estimates and on the other hand for more general, non-uniform prior densities while relying on regression and transport/ transformation techniques.
Zitierung
Marschall, M. (2020). Explicit and adaptive Bayesian inversion in hierarchical tensor format [Dissertation, Technische Universität Berlin, 2020]. https://doi.org/10.14279/depositonce-10670
