Multigrid Accelerated Tensor Approximation of Function Related Multidimensional Arrays

Abstract

Abstract. In this paper, we describe and analyze a novel tensor approximation method for discretized multidimensional functions and operators in R d , based on the idea of multigrid acceleration. The approach stands on successive reiterations of the orthogonal Tucker tensor approximation on a sequence of nested refined grids. On the one hand, it provides a good initial guess for the nonlinear iterations to find the approximating subspaces on finer grids; on the other hand, it allows us to transfer from the coarse-to-fine grids the important data structure information on the location of the so-called most important fibers in directional unfolding matrices. The method indicates linear complexity with respect to the size of data representing the input tensor. In particular, if the target tensor is given by using the rank-R canonical model, then our approximation method is proved to have linear scaling in the univariate grid size n and in the input rank R.T h em e t h o di s tested by three-dimensional (3D) electronic structure calculations. For the multigrid accelerated low Tucker-rank approximation of the all electron densities having strong nuclear cusps, we obtain high resolution of their 3D convolution product with the Newton potential. The accuracy of order 10 −6 in max-norm is achieved on large n × n × n grids up to n =1 .6 · 10 4 ,w i t ht h et i m es c a l ei ns e v e r a l minutes.

Tensors-structured numerical methods in scientific computing: Survey on recent advances

“…The arising discretizations via "formatted" function related tensors typically inherit the separability properties of the initial solutions on the continuous level, usually providing fast exponential convergence in the separation rank. This favorable feature combined with the modern multilinear algebra methods of nonlinear tensor approximation [21,1,42,79,57,55] lead to the new concept of numerical schemes in higher dimensions which scale linearly in the dimension parameter d.…”

Low-rank quadrature-based tensor approximation of the Galerkin projected Newton/Yukawa kernels

“…Figure 5.1 illustrates the tensor rank vs. relative error for best algebraic recompressions via the multigrid orthogonal Tucker decomposition [10] and for those obtained with NFFD applied to our sample tensor. In fact, it is known that the rank-r orthogonal Tucker model provides the lower bound for the canonical rank R, i.e., r ≤ R. Finally, we conclude that this work presents a methodology to approximate L 2 -Galerkin projections of Green kernels in R 3 onto the set of tensor-product basis functions, through canonical tensor-product sums.…”

“…Algorithm TGN was already successfully applied in numerical computations of various 3D convolution integrals [7] included in the Fock operator of the nonlinear Hartree-Fock equation in 3D, see [9,10,11,12]. In particular, this includes fast multiple computations of the Coulomb and exchange convolution integrals in the tensor-structured numerical methods for solving the ab initio Hartree-Fock equation on large n × n × n Cartesian grids, in the range n ≤ 10 4 , see [12].…”

“…In the recent years, the idea of tensor approximation of operators and functions has lead to powerful numerical algorithms in large-scale problems of computational physics, in particular, in electronic structure calculations based on the Hartree-Fock or DFT models [6,10,8]. In these applications one deals with numerical computations of integral transforms which include Green's kernels in R d [5,7,9].…”

“…Note that both exchange and the Hartree potentials contain the 3D convolution transform with the Newton convolving kernel, that has to be computed at each step of the iterations on nonlinearity. Hence, these terms represent the most complicated part in the numerical treatment of the Hartree-Fock equation, see, e.g., [7,9,10,12]. Another popular modification of the Hartree-Fock and Schrödinger equations is based on the so-called Lippmann-Schwinger integral formulation, which contains also the convolution transform with the Yukawa potential e −λ x x (λ ∈ R + ) [6,8].…”