Use of tensor formats in elliptic eigenvalue problems

Abstract

We investigate approximations by finite sums of products of functions with separated variables to eigenfunctions of certain class of elliptic operators in higher dimensions, and especially conditions providing an exponential decrease of the error with respect to the number of terms. The results of the consistent use of tensor formats can be regarded as a base for a new class of rank truncated iterative eigensolvers with almost linear complexity in the univariate problem size that improves dramatically the traditional methods of linear scaling in the volume size. Tensor methods can be applied to solving large scale spectral problems in the computational quantum chemistry, for example to the Schrödinger, Hartree-Fock and Kohn-Sham equations in electronic structure calculations. The results of numerical experiments clearly indicate the linear-logarithmic scaling of low-rank tensor method in the univariate problem size.
The algorithms work equally well for the computation of both, minimal and maximal eigenvalues of the discrete elliptic operators.