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Author: Eigel, Martin × Author: Heidenreich, Sebastian × Start date: 2020 × End date: 2024 × Start date: 2020 × WGL Institute: WIAS ×

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    Efficient approximation of high-dimensional exponentials by tensor networks
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Eigel, Martin; Farchmin, Nando; Heidenreich, Sebastian; Trunschke, Philipp
    In this work a general approach to compute a compressed representation of the exponential exp(h) of a high-dimensional function h is presented. Such exponential functions play an important role in several problems in Uncertainty Quantification, e.g. the approximation of log-normal random fields or the evaluation of Bayesian posterior measures. Usually, these high-dimensional objects are intractable numerically and can only be accessed pointwise in sampling methods. In contrast, the proposed method constructs a functional representation of the exponential by exploiting its nature as a solution of an ordinary differential equation. The application of a Petrov--Galerkin scheme to this equation provides a tensor train representation of the solution for which we derive an efficient and reliable a posteriori error estimator. Numerical experiments with a log-normal random field and a Bayesian likelihood illustrate the performance of the approach in comparison to other recent low-rank representations for the respective applications. Although the present work considers only a specific differential equation, the presented method can be applied in a more general setting. We show that the composition of a generic holonomic function and a high-dimensional function corresponds to a differential equation that can be used in our method. Moreover, the differential equation can be modified to adapt the norm in the a posteriori error estimates to the problem at hand.
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    Adaptive non-intrusive reconstruction of solutions to high-dimensional parametric PDEs
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Eigel, Martin; Farchmin, Nando; Heidenreich, Sebastian; Trunschke, Philipp
    Numerical methods for random parametric PDEs can greatly benefit from adaptive refinement schemes, in particular when functional approximations are computed as in stochastic Galerkin and stochastic collocations methods. This work is concerned with a non-intrusive generalization of the adaptive Galerkin FEM with residual based error estimation. It combines the non-intrusive character of a randomized least-squares method with the a posteriori error analysis of stochastic Galerkin methods. The proposed approach uses the Variational Monte Carlo method to obtain a quasi-optimal low-rank approximation of the Galerkin projection in a highly efficient hierarchical tensor format. We derive an adaptive refinement algorithm which is steered by a reliable error estimator. Opposite to stochastic Galerkin methods, the approach is easily applicable to a wide range of problems, enabling a fully automated adjustment of all discretization parameters. Benchmark examples with affine and (unbounded) lognormal coefficient fields illustrate the performance of the non-intrusive adaptive algorithm, showing best-in-class performance