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

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 ∈ ℝ ^{N × N} . The asymptotic complexity of our data-sparse representations can be estimated by (n ^p log ^q n), p = 1, 2, with q independent of d, where n=N ^{1/ d} is the dimension of the discrete problem in one space direction.