Efficient approximation of high-dimensional exponentials by tensor networks
dc.bibliographicCitation.seriesTitle | WIAS Preprints | eng |
dc.bibliographicCitation.volume | 2844 | |
dc.contributor.author | Eigel, Martin | |
dc.contributor.author | Farchmin, Nando | |
dc.contributor.author | Heidenreich, Sebastian | |
dc.contributor.author | Trunschke, Philipp | |
dc.date.accessioned | 2022-07-05T14:10:48Z | |
dc.date.available | 2022-07-05T14:10:48Z | |
dc.date.issued | 2021 | |
dc.description.abstract | 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. | eng |
dc.description.version | publishedVersion | eng |
dc.identifier.uri | https://oa.tib.eu/renate/handle/123456789/9562 | |
dc.identifier.uri | https://doi.org/10.34657/8600 | |
dc.language.iso | eng | |
dc.publisher | Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik | |
dc.relation.doi | https://doi.org/10.20347/WIAS.PREPRINT.2844 | |
dc.relation.issn | 2198-5855 | |
dc.rights.license | This document may be downloaded, read, stored and printed for your own use within the limits of § 53 UrhG but it may not be distributed via the internet or passed on to external parties. | eng |
dc.rights.license | Dieses Dokument darf im Rahmen von § 53 UrhG zum eigenen Gebrauch kostenfrei heruntergeladen, gelesen, gespeichert und ausgedruckt, aber nicht im Internet bereitgestellt oder an Außenstehende weitergegeben werden. | ger |
dc.subject.ddc | 510 | |
dc.subject.other | Uncertainty quantification | eng |
dc.subject.other | dynamical system approximation | eng |
dc.subject.other | Petrov--Galerkin | eng |
dc.subject.other | a posteriori error bounds | eng |
dc.subject.other | tensor product methods | eng |
dc.subject.other | tensor train format | eng |
dc.subject.other | holonomic functions | eng |
dc.subject.other | Bayesian likelihoods | eng |
dc.subject.other | log-normal random field | eng |
dc.title | Efficient approximation of high-dimensional exponentials by tensor networks | eng |
dc.type | Report | eng |
dc.type | Text | eng |
dcterms.extent | 25 S. | |
tib.accessRights | openAccess | |
wgl.contributor | WIAS | |
wgl.subject | Mathematik | |
wgl.type | Report / Forschungsbericht / Arbeitspapier |
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