Quantized tensor FEM for multiscale problems: diffusion problems in two and three dimensions
dc.contributor.author
Kazeev, Vladimir
dc.contributor.author
Oseledets, Ivan
dc.contributor.author
Rakhuba, Maxim
dc.contributor.author
Schwab, Christoph
dc.date.accessioned
2020-10-22T15:18:49Z
dc.date.available
2020-10-22T07:04:08Z
dc.date.available
2020-10-22T15:18:49Z
dc.date.issued
2020-06
dc.identifier.uri
http://hdl.handle.net/20.500.11850/447112
dc.description.abstract
Homogenization in terms of multiscale limits transforms a multiscale problem with n+1 asymptotically separated microscales posed on a physical domain D⊂Rd into a one-scale problem posed on a product domain of dimension (n+1)d by introducing n so-called “fast variables”. This procedure allows to convert n+1 scales in d physical dimensions into a single-scale structure in (n+1)d dimensions. We prove here that both the original, physical multiscale problem and the corresponding high-dimensional, one-scale limiting problem can be efficiently treated numerically with the recently developed quantized tensor-train finite-element method (QTT-FEM). The method is based on restricting computation to sequences of nested subspaces of low dimensions (which are called tensor ranks) within a vast but generic “virtual” (background) discretization space. In the course of computation, these subspaces are computed iteratively and data-adaptively at runtime, bypassing any “offline precomputation”. For the purpose of theoretical analysis, such low-dimensional subspaces are constructed analytically so as to bound the tensor ranks vs. error tolerance τ>0. We consider a model linear elliptic multiscale problem in several physical dimensions and show, theoretically and experimentally, that both (i) the solution of the associated high-dimensional one-scale problem and (ii) the corresponding approximation to the solution of the multiscale problem admit efficient approximation by the QTT-FEM. These problems can therefore be numerically solved in a scale-robust fashion by standard (low-order) PDE discretizations combined with state-of-the-art general-purpose solvers for tensor-structured linear systems. We prove scale-robust exponential convergence, i.e., that QTT-FEM achieves accuracy τ with the number of effective degrees of freedom scaling polynomially in logτ.
en_US
dc.language.iso
en
en_US
dc.publisher
Seminar for Applied Mathematics, ETH Zurich
en_US
dc.subject
multiscale problems
en_US
dc.subject
homogenization
en_US
dc.subject
exponential convergence
en_US
dc.subject
tensor decompositions
en_US
dc.subject
quantized tensor trains
en_US
dc.title
Quantized tensor FEM for multiscale problems: diffusion problems in two and three dimensions
en_US
dc.type
Report
ethz.journal.title
SAM Research Report
ethz.journal.volume
2020-33
en_US
ethz.size
31 p.
en_US
ethz.publication.place
Zurich
en_US
ethz.publication.status
published
en_US
ethz.leitzahl
ETH Zürich::00002 - ETH Zürich::00012 - Lehre und Forschung::00007 - Departemente::02000 - Dep. Mathematik / Dep. of Mathematics::02501 - Seminar für Angewandte Mathematik / Seminar for Applied Mathematics::03435 - Schwab, Christoph / Schwab, Christoph
en_US
ethz.leitzahl.certified
ETH Zürich::00002 - ETH Zürich::00012 - Lehre und Forschung::00007 - Departemente::02000 - Dep. Mathematik / Dep. of Mathematics::02501 - Seminar für Angewandte Mathematik / Seminar for Applied Mathematics::03435 - Schwab, Christoph / Schwab, Christoph
en_US
ethz.identifier.url
https://math.ethz.ch/sam/research/reports.html?id=906
ethz.date.deposited
2020-10-22T07:04:19Z
ethz.source
FORM
ethz.eth
yes
en_US
ethz.identifier.internal
https://math.ethz.ch/sam/research/reports.html?id=906
en_US
ethz.availability
Metadata only
en_US
ethz.rosetta.installDate
2020-10-22T15:19:01Z
ethz.rosetta.lastUpdated
2020-10-22T15:19:01Z
ethz.rosetta.versionExported
true
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