Gutenberg Open Science: Stability of low-rank tensor representations and structured multilevel preconditioning for elliptic PDEs

Please use this identifier to cite or link to this item: http://doi.org/10.25358/openscience-5853

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DC Field	Value	Language
dc.contributor.author	Bachmayr, Markus	-
dc.contributor.author	Kazeev, Vladimir	-
dc.date.accessioned	2021-05-10T07:13:41Z	-
dc.date.available	2021-05-10T07:13:41Z	-
dc.date.issued	2020	-
dc.identifier.uri	https://openscience.ub.uni-mainz.de/handle/20.500.12030/5862	-
dc.description.abstract	Folding grid value vectors of size 2L into Lth-order tensors of mode size 2×⋯×2, combined with low-rank representation in the tensor train format, has been shown to result in highly efficient approximations for various classes of functions. These include solutions of elliptic PDEs on nonsmooth domains or with oscillatory data. This tensor-structured approach is attractive because it leads to highly compressed, adaptive approximations based on simple discretizations. Standard choices of the underlying bases, such as piecewise multilinear finite elements on uniform tensor product grids, entail the well-known matrix ill-conditioning of discrete operators. We demonstrate that, for low-rank representations, the use of tensor structure itself additionally introduces representation ill-conditioning, a new effect specific to computations in tensor networks. We analyze the tensor structure of a BPX preconditioner for a second-order linear elliptic operator and construct an explicit tensor-structured representation of the preconditioner, with ranks independent of the number L of discretization levels. The straightforward application of the preconditioner yields discrete operators whose matrix conditioning is uniform with respect to the discretization parameter, but in decompositions that suffer from representation ill-conditioning. By additionally eliminating certain redundancies in the representations of the preconditioned discrete operators, we obtain reduced-rank decompositions that are free of both matrix and representation ill-conditioning. For an iterative solver based on soft thresholding of low-rank tensors, we obtain convergence and complexity estimates and demonstrate its reliability and efficiency for discretizations with up to 250 nodes in each dimension.	en_GB
dc.language.iso	eng	de
dc.rights	CC BY	*
dc.rights.uri	https://creativecommons.org/licenses/by/4.0/	*
dc.subject.ddc	510 Mathematik	de_DE
dc.subject.ddc	510 Mathematics	en_GB
dc.title	Stability of low-rank tensor representations and structured multilevel preconditioning for elliptic PDEs	en_GB
dc.type	Zeitschriftenaufsatz	de
dc.identifier.doi	http://doi.org/10.25358/openscience-5853	-
jgu.type.dinitype	article	en_GB
jgu.type.version	Published version	de
jgu.type.resource	Text	de
jgu.organisation.department	FB 08 Physik, Mathematik u. Informatik	de
jgu.organisation.number	7940	-
jgu.organisation.name	Johannes Gutenberg-Universität Mainz	-
jgu.rights.accessrights	openAccess	-
jgu.journal.title	Foundations of computational mathematics	de
jgu.journal.volume	20	de
jgu.pages.start	1175	de
jgu.pages.end	1236	de
jgu.publisher.year	2020	-
jgu.publisher.name	Springer	de
jgu.publisher.place	New York, NY	de
jgu.publisher.uri	https://doi.org/10.1007/s10208-020-09446-z	de
jgu.publisher.issn	1615-3383	de
jgu.organisation.place	Mainz	-
jgu.subject.ddccode	510	de
jgu.publisher.doi	10.1007/s10208-020-09446-z
jgu.organisation.ror	https://ror.org/023b0x485
Appears in collections:	JGU-Publikationen

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