W. Hackbusch, B.N. Khoromskij, S. Sauter, and E. Tyrtyshnikov. Use of tensor formats in elliptic eigenvalue problems. Numerical Linear Algebra with Applications, 19(1):133–151, 2012.
We investigate approximations to eigenfunctions of a certain class of elliptic operators in Rd 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 nd. Tensor methods can be applied to solving large-scale spectral problems in computational quantum chemistry, for example, the Schrodinger, 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
elliptic operators; spectra; eigenfunctions; separation of variables; separable approximations; matrix approximations; low-rank matrices; Kronecker products; multi-dimensional matrices; tensors; numerical methods; iterative algorithms