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Krylov subspace methods for linear systems with tensor product structure

by D. Kressner and Ch. Tobler

(Report number 2009-16)

Abstract
The numerical solution of linear systems with certain tensor product structures is considered. Such structures arise, for example, from the finite element discretization of a linear PDE on a d-dimensional hypercube. Linear systems with tensor product structure can be regarded as linear matrix equations for d = 2 and appear to be their most natural extension for d>2. A standard Krylov subspace method applied to such a linear system suffers from the curse of dimensionality and has a computational cost that grows exponentially with d. The key to breaking the curse is to note that the solution can often be very well approximated by a vector of low tensor rank. We propose and analyse a new class of methods, so called tensor Krylov subspace methods, which exploit this fact and attain a computational cost that grows linearly with d.

Keywords:

@Techreport{KT09_403,
  author = {D. Kressner and Ch. Tobler},
  title = {Krylov subspace methods for linear systems with tensor product structure},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2009-16},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2009/2009-16.pdf },
  year = {2009}
}

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