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Use of tensor formats in elliptic eigenvalue problems
Author(s) -
Hackbusch Wolfgang,
Khoromskij Boris N.,
Sauter Stefan,
Tyrtyshnikov Eugene E.
Publication year - 2012
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.793
Subject(s) - mathematics , eigenvalues and eigenvectors , tensor (intrinsic definition) , rank (graph theory) , operator (biology) , univariate , computation , logarithm , algebra over a field , pure mathematics , mathematical analysis , combinatorics , quantum mechanics , algorithm , multivariate statistics , biochemistry , chemistry , physics , statistics , repressor , transcription factor , gene
SUMMARY We investigate approximations to eigenfunctions of a certain class of elliptic operators in R d by finite sums of products of functions with separated variables and especially conditions providing an exponential decrease of the error with respect to the number of terms. The consistent use of tensor formats can be regarded as a base for a new class of rank‐truncated iterative eigensolvers. The computational cost is almost linear in the univariate problem size n , while traditional method scale like n d . Tensor methods can be applied to solving large‐scale spectral problems in computational quantum chemistry, for example, the Schrödinger, Hartree–Fock and Kohn–Sham equations in electronic structure calculations. The results of numerical experiments clearly indicate the linear‐logarithmic scaling of the low‐rank tensor method in n . The algorithms work equally well for the computation of both minimal and maximal eigenvalues of the discrete elliptic operators. Copyright © 2011 John Wiley & Sons, Ltd.