A Multilinear Singular Value Decomposition
SIAM Journal on Matrix Analysis and Applications
On the Best Rank-1 and Rank-(R1,R2,. . .,RN) Approximation of Higher-Order Tensors
SIAM Journal on Matrix Analysis and Applications
Rank-One Approximation to High Order Tensors
SIAM Journal on Matrix Analysis and Applications
On tensor approximation of Green iterations for Kohn-Sham equations
Computing and Visualization in Science
Tucker Dimensionality Reduction of Three-Dimensional Arrays in Linear Time
SIAM Journal on Matrix Analysis and Applications
Tensor decomposition in electronic structure calculations on 3D Cartesian grids
Journal of Computational Physics
Linear algebra for tensor problems
Computing
Multigrid Accelerated Tensor Approximation of Function Related Multidimensional Arrays
SIAM Journal on Scientific Computing
Tensor Decompositions and Applications
SIAM Review
Numerical Solution of the Hartree-Fock Equation in Multilevel Tensor-Structured Format
SIAM Journal on Scientific Computing
Approximation of the electron density of Aluminium clusters in tensor-product format
Journal of Computational Physics
On the computational benefit of tensor separation for high-dimensional discrete convolutions
Multidimensional Systems and Signal Processing
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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^-^@l^@?^x^@? and e^-^@l^@?^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(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 n@?16384.