Low-rank Wasserstein polynomial chaos expansions in the framework of optimal transport
dc.bibliographicCitation.seriesTitle | WIAS Preprints | eng |
dc.bibliographicCitation.volume | 2927 | |
dc.contributor.author | Gruhlke, Robert | |
dc.contributor.author | Eigel, Martin | |
dc.date.accessioned | 2022-07-08T13:04:39Z | |
dc.date.available | 2022-07-08T13:04:39Z | |
dc.date.issued | 2022 | |
dc.description.abstract | A unsupervised learning approach for the computation of an explicit functional representation of a random vector Y is presented, which only relies on a finite set of samples with unknown distribution. Motivated by recent advances with computational optimal transport for estimating Wasserstein distances, we develop a new Wasserstein multi-element polynomial chaos expansion (WPCE). It relies on the minimization of a regularized empirical Wasserstein metric known as debiased Sinkhorn divergence. As a requirement for an efficient polynomial basis expansion, a suitable (minimal) stochastic coordinate system X has to be determined with the aim to identify ideally independent random variables. This approach generalizes representations through diffeomorphic transport maps to the case of non-continuous and non-injective model classes M with different input and output dimension, yielding the relation Y=M(X) in distribution. Moreover, since the used PCE grows exponentially in the number of random coordinates of X, we introduce an appropriate low-rank format given as stacks of tensor trains, which alleviates the curse of dimensionality, leading to only linear dependence on the input dimension. By the choice of the model class M and the smooth loss function, higher order optimization schemes become possible. It is shown that the relaxation to a discontinuous model class is necessary to explain multimodal distributions. Moreover, the proposed framework is applied to a numerical upscaling task, considering a computationally challenging microscopic random non-periodic composite material. This leads to tractable effective macroscopic random field in adopted stochastic coordinates. | eng |
dc.description.version | publishedVersion | eng |
dc.identifier.uri | https://oa.tib.eu/renate/handle/123456789/9685 | |
dc.identifier.uri | https://doi.org/10.34657/8723 | |
dc.language.iso | eng | |
dc.publisher | Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik | |
dc.relation.doi | https://doi.org/10.20347/WIAS.PREPRINT.2927 | |
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 | Tensor train format | eng |
dc.subject.other | Wasserstein metric | eng |
dc.subject.other | polynomial chaos expansion | eng |
dc.subject.other | alternating least squares | eng |
dc.subject.other | numerical upscaling | eng |
dc.subject.other | tensor chain | eng |
dc.subject.other | Sinkhorn divergence | eng |
dc.subject.other | optimal transport | eng |
dc.subject.other | multimodal distribution | eng |
dc.title | Low-rank Wasserstein polynomial chaos expansions in the framework of optimal transport | eng |
dc.type | Report | eng |
dc.type | Text | eng |
dcterms.extent | 35 S. | |
tib.accessRights | openAccess | |
wgl.contributor | WIAS | |
wgl.subject | Mathematik | |
wgl.type | Report / Forschungsbericht / Arbeitspapier |
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