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Quantized tensor FEM for multiscale problems: diffusion problems in two and three dimensions

by V. Kazeev and I. Oseledets and M. Rakhuba and Ch. Schwab

(Report number 2020-33)

Abstract
Homogenization in terms of multiscale limits transforms a multiscale problem with \(n+1\) asymptotically separated microscales posed on a physical domain \(D \subset \mathbb{R}^d\) 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 \(\tau>0\). 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 \(\tau\) with the number of effective degrees of freedom scaling polynomially in \(\log \tau\).

Keywords: multiscale problems, homogenization, exponential convergence, tensor decompositions, quantized tensor trains

@Techreport{KORS20_906,
  author = {V. Kazeev and I. Oseledets and M. Rakhuba and Ch. Schwab},
  title = {Quantized tensor FEM for multiscale problems: diffusion problems in two and three dimensions},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2020-33},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2020/2020-33.pdf },
  year = {2020}
}

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