Low-rank Kronecker-product Approximation to Multi-dimensional Nonlocal Operators. Part II. HKT Representation of Certain Operators

Abstract

This article is the second part continuing Part I [16]. We apply the **-matrix techniques combined with the Kronecker tensor-product approximation to represent integral operators as well as certain functions F(A) of a discrete elliptic operator A in a hypercube (0,1) d ∈ ℝ d in the case of a high spatial dimension d. We focus on the approximation of the operator-valued functions A−σ, σ>0, and sign (A) for a class of finite difference discretisations A ∈ ℝN×N. The asymptotic complexity of our data-sparse representations can be estimated by ** (np log qn), p = 1, 2, with q independent of d, where n=N1/d is the dimension of the discrete problem in one space direction.