Tensor-Structured Preconditioners and Approximate Inverse of Elliptic Operators in ℝ d

Abstract

In the present paper we analyze a class of tensor-structured preconditioners for the multidimensional second-order elliptic operators in R d , d ≥ 2. For equations in a bounded domain, the construction is based on the rank-R tensor-product approximation of the elliptic resolvent B R ≈ (L − λI ) −1 , where L is the sum of univariate elliptic operators. We prove the explicit estimate on the tensor rank R that ensures the spectral equivalence. For equations in an unbounded domain, one can utilize the tensor-structured approximation of Green's kernel for the shifted Laplacian in R d , which is well developed in the case of nonoscillatory potentials. For the oscillating kernels e −iκ x / x , x ∈ R d , κ ∈ R + , we give constructive proof of the rank-O(κ) separable approximation. This leads to the tensor representation for the discretized 3D Helmholtz kernel on an n × n × n grid that requires only O(κ | log ε| 2 n) reals for storage. Such representations can be applied to both the 3D volume and boundary calculations with sublinear cost O(n 2 ), even in the case κ = O(n).Numerical illustrations demonstrate the efficiency of low tensor-rank approximation for Green's kernels e −λ x / x , x ∈ R 3 , in the case of Newton (λ = 0), Yukawa (λ ∈ R + ), and Helmholtz (λ = iκ, κ ∈ R + ) potentials, as well as for the kernel functions 1/ x and 1/ x d−2 , x ∈ R d , in higher dimensions d > 3. We present numerical results on the iterative calculation of the minimal eigenvalue for the d-dimensional finite difference Laplacian by the power method with the rank truncation and based on the approximate inverse B R ≈ (− ) −1 , with 3 ≤ d ≤ 50.

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

“…In general, the best R-term nonlinear approximation shows rather slow polynomial convergence in the rank parameter R, [92], and it can be calculated by the simple (but non-robust) greedy-type incremental algorithms. For the class of (physically relevant) analytic multivariate functions and Green's kernels the exponential convergence in the separation rank R can be proved [25,93,49,53,54], that leads to the asymptotically optimal bound on the canonical rank, R = O(log N ), implying the approximation rate O(N −α ), with α ≥ 1.…”

“…Fully tensorized numerical approach for solving the Hartree-Fock equation, discretized over N × N × N grid, by tensor truncated iteration of complexity O(N log N ), was recently presented in [58]. Other successful applications to high-dimensional eigenvalue problems [37,54] and to stochastic PDEs [64,62] are reported. A class of low tensor rank preconditioners for the multidimensional elliptic problems with jumping coefficients in R d is proposed in [18].…”

“…The preconditioner B = B M can be chosen as inverse of the shifted Laplacian (see [37,54] for more details).…”

“…In the case of unbounded domains, the low tensor rank preconditioner can be chosen as inverse of the translated Laplacian with positive shift (separable approximation to the Yukawa type Green kernel), that is proven to have the low canonical rank approximation [53,54]. In the case of finite Example 4.7 (Direct tensor approximation of the solution operator).…”

Computation of the Hartree-Fock Exchange by the Tensor-Structured Methods

“…Theoretical analysis of the multilinear tensor product approaches for the treatment of some multivariate operators and functions arising in scientific computing was performed in [5,6,8,12]. The application of tensor decomposition algorithms to discretized multivariate functions and operators [10,14,13,11] showed that methods of multi-way analysis can be applied to the numerical solution of basic equations of mathematical physics placing stringent requirements upon the accuracy of results.…”