Krylov Subspace Methods for Linear Systems with Tensor Product Structure

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. * Supported by the SNF research module Preconditioned methods for large-scale model reduction within the SNF ProDoc Efficient Numerical Methods for Partial Differential Equations.

Solution of linear systems in high spatial dimensions

“…The second approach has a drawback which will be discussed next. First we mention the following articles about conjugate gradient methods in the context of tensor methods: Tobler [29], Kressner-Tobler [23][24][25] , and Savas-Eldén [28].…”

Fast tensor product solvers for optimization problems with fractional differential equations as constraints

“…In [83] it was shown that the matrix-vector product with the matrix (14) can maintain the low-rank structure, only requiring an additional truncation step that can be performed cheaply compared to the cost of the full method. The construction of such a truncation function T ε for the matrix case is further described in [46,83].…”

“…As we noted, algebraic operations (matrix, scalar products and additions) and the SVD re-compression procedure in the TT format allow to rewrite any classical iterative method, keeping all vectors in the tensor format and performing only structured operations [64,4,21,46]. Let us denote the compression (or truncation) procedure from a vector y to a vector y ≈ y as y = T ε (y),…”

An Error Analysis of Galerkin Projection Methods for Linear Systems with Tensor Product Structure

“…The convergence of the Galerkin method on tensor products of (rational) Krylov subspaces for Lyapunov and Sylvester equations has been analysed in [2,10,21,20,27,28]. For the extensions considered in this paper the framework developed in [2] appears to be most suitable.…”

“…Such linear systems arise, for example, from the discretization of a separable d-dimensional PDE with tensorized finite elements [14,21]. Moreover, methods for (5) can be used as preconditioners in iterative methods for more general linear systems [19,4] and eigenvalue problems [22].…”