Approximate iterations for structured matrices

Abstract

Important matrix-valued functions f (A) are, e.g., the inverse A −1 , the square root √ A and the sign function. Their evaluation for large matrices arising from pdes is not an easy task and needs techniques exploiting appropriate structures of the matrices A and f (A) (often f (A) possesses this structure only approximately). However, intermediate matrices arising during the evaluation may lose the structure of the initial matrix. This would make the computations inefficient and even infeasible. However, the main result of this paper is that an iterative fixed-point like process for the evaluation of f (A) can be transformed, under certain general assumptions, into another process which preserves the convergence rate and benefits from the underlying structure. It is shown how this result applies to matrices in a tensor format with a bounded tensor rank and to the structure of the hierarchical matrix technique. We demonstrate our results by verifying all requirements in the case of the iterative computation of A −1 and √ A.

A regularized Newton method for the efficient approximation of tensors represented in the canonical tensor format

“…If one makes sure that z k −App ε k ≤ c z k −x * and if x k −−−→ k→∞ x * at least quadratically, we have that y k −−−→ k→∞ x * with the same rate of convergence as x k −−−→ k→∞ x * . For a complete convergence analysis of inexact iterations we refer to [10,Hackbusch,Khoromskij and Tyrtyshnikov]. The computation of the pointwise inverse of a tensor u ∈ T with u i = 0 for all i ∈ {1, .…”

Tensor-Structured Preconditioners and Approximate Inverse of Elliptic Operators in ℝ d

“…3 The Tucker vs. canonical approximation of the 3D Newton/Yukawa kernels Fig. 4 Convergence history for the Tucker model applied to f 1,κ , f 2,κ , κ ∈ [1,15] which can be satisfied only on very coarse grids (recall that for nonoscillating kernels we have r = O(log n)). However, for the case of oscillating kernels we may have κ = Cn, i.e., the canonical decomposition would be preferable.…”

“…Figure 3 represents the convergence history for the best orthogonal Tucker [23] vs. canonical [2] approximations of the Newton/Yukawa potentials on an n × n × n grid for n = 2048. Figure 4 shows the convergence history for the Tucker model applied to f 1,κ , f 2,κ depending on κ ∈ [1,15] and termination criteria with fixed values ε 1 > 0 Convergence for the Tucker and canonical models applied to f 2,κ , κ ∈ [1,15] is presented in Figures 5 and 6, respectively. Figure 5, right clearly demonstrates exponential convergence in the Tucker rank r in the interval r ≥ r 0 = κ (supports the theory).…”

“…[3,4,13,15,19,24]). In this way the construction of spectrally close tensor-structured preconditioners for the class of elliptic integral-differential operators in R d is one of the building blocks in development of efficient numerical methods in the case d ≥ 3.…”

Variational calculus with sums of elementary tensors of fixed rank

“…To overcome this issue, efficient recompression methods have been developed in [1,2,5,8,9] to approximate a given sum of elementary tensors by low rank tensors. Moreover, the convergence of such approximate iterations is known, see [16] for an analysis. The subject of this article is in contrast to this approach.…”