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 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‖1/‖x‖, e−λ‖x‖ and e−λ‖x‖/‖x‖ with x∈Rdx∈Rd. For piecewise constant elements on the uniform grid of size ndnd, we prove quadratic convergence O(h2)O(h2) in the mesh parameter h=1/nh=1/n, and then justify the Richardson extrapolation method on a sequence of grids that improves the order of approximation up to O(h3)O(h3). A fast algorithm of complexity O(dR1R2nlogn)O(dR1R2nlogn) is described for tensor-product convolution on uniform/composite grids of size ndnd, where R1,R2R1,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)O(h2) and O(h3)O(h3); (b) linear-logarithmic scaling of 1D discrete convolution on composite grids; (c) linear-logarithmic scaling in nn of our tensor-product convolution method on an n×n×nn×n×n grid in the range n≤16384n≤16384.