F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig Fast and Accurate Tensor Approximation of Multivariate Convolution with Linear Scaling in Dimension Fast and Accurate Tensor Approximation of 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/@[email protected]?, e^-^@l^@?^x^@? and e^-^@l^@?^x^@?/@[email protected]? with [email protected]?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(dR"1R"2nlogn) 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 nxnxn grid in the range [email protected]?16384.