Keywords:
Forschungsbericht
Description / Table of Contents:
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.
Type of Medium:
Online Resource
Pages:
1 Online-Ressource (27 Seiten, 364,96 KB)
Series Statement:
Preprint / Weierstraß-Institut für Angewandte Analysis und Stochastik no. 2844
URL:
https://doi.org/10.20347/WIAS.PREPRINT.2844
DOI:
10.20347/WIAS.PREPRINT.2844
Language:
English
Note:
Literaturverzeichnis: Seite 22-25
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