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

In the present paper we present the tensor-product approximation of multidimensional convolution transform discretized via collocation-projection scheme on the uniform or composite refined grids. Examples of convolving kernels are given by the classical Newton, Slater (exponential) and Yukawa potentials, 1/‖x‖, e−λ‖x‖ and e−λ‖x‖/‖x‖ with x ∈ Rd. For piecewise constant elements on the uniform grid of size nd, we prove the quadratic convergence O(h2) 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(h3). The fast algorithm of complexity O(dR1R2n log n) is described for tensor-product convolution on the uniform/composite grids of size nd, where R1, R2 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(h2) and O(h3); (b) linear-logarithmic scaling of 1D discrete convolution on composite grids; (c) linear-logarithmic scaling in n of our tensor-product convolution method on n× n× n grid in the range n ≤ 16384. AMS Subject Classification: 65F30, 65F50, 65N35, 65F10