Krylov Subspace Methods for Linear Systems with Tensor Product Structure

Krylov Subspace Methods for Linear Systems with Tensor Product Structure

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...

Low-Rank Tensor Krylov Subspace Methods for Parametrized Linear Systems

We consider linear systems A(α)x(α) = b(α) depending on possibly many parameters α = (α1, . . . ,αp). Solving these systems simultaneously for a standard discretization of the parameter space would require a computational effort growing exponentially in the number of parameters. We show that this curse of dimensionality can be avoided for sufficiently smooth parameter dependencies. For this pur...

Krylov subspace methods for solving linear systems

Inexact Krylov Subspace Methods for Linear Systems

There is a class of linear problems for which the computation of the matrix-vector product is very expensive since a time consuming approximation method is necessary to compute it with some prescribed relative precision. In this paper we investigate the effect of an approximately computed matrix-vector product on the convergence and accuracy of several Krylov subspace solvers. The obtained insi...

Krylov Subspace Methods for Tensor Computations

A couple of generalizations of matrix Krylov subspace methods to tensors are presented. It is shown that a particular variant can be interpreted as a Krylov factorization of the tensor. A generalization to tensors of the Krylov-Schur method for computing matrix eigenvalues is proposed. The methods are intended for the computation of lowrank approximations of large and sparse tensors. A few nume...