Fast and accurate tensor approximation of a multivariate convolution with linear scaling in dimension

Abstract

In the present paper we present the tensor-product approximation of a multidimensional convolution transform discretized via a collocation–projection scheme on uniform or composite refined grids. Examples of convolving kernels are provided by the classical Newton, Slater (exponential) and Yukawa potentials, 1 / ‖ x ‖ , e − λ ‖ x ‖ and e − λ ‖ x ‖ / ‖ x ‖ with x ∈ R d . For piecewise constant elements on the uniform grid of size n d , we prove quadratic convergence O ( h 2 ) in the mesh parameter h = 1 / n , and then justify the Richardson extrapolation method on a sequence of grids that improves the order of approximation up to O ( h 3 ) . A fast algorithm of complexity O ( d R 1 R 2 n log n ) is described for tensor-product convolution on uniform/composite grids of size n d , where R 1 , R 2 are tensor ranks of convolving functions. We also present the tensor-product convolution scheme in the two-level Tucker canonical format and discuss the consequent rank reduction strategy. Finally, we give numerical illustrations confirming: (a) the approximation theory for convolution schemes of order O ( h 2 ) and O ( h 3 ) ; (b) linear-logarithmic scaling of 1D discrete convolution on composite grids; (c) linear-logarithmic scaling in n of our tensor-product convolution method on an n × n × n grid in the range n ≤ 16384 .