Quantized tensor FEM for multiscale problems: diffusion problems in two and three dimensions
METADATA ONLY
Loading...
Author / Producer
Date
2020-06
Publication Type
Report
ETH Bibliography
yes
Citations
Altmetric
METADATA ONLY
Data
Rights / License
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.
Permanent link
Publication status
published
Editor
Book title
Journal / series
SAM Research Report
Volume
2020-33
Pages / Article No.
Publisher
Seminar for Applied Mathematics, ETH Zurich
Event
Edition / version
Methods
Software
Geographic location
Date collected
Date created
Subject
multiscale problems; homogenization; exponential convergence; tensor decompositions; quantized tensor trains
Organisational unit
03435 - Schwab, Christoph / Schwab, Christoph