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−λ I)−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 −i κ‖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∈ℝ3, in the case of Newton (λ=0), Yukawa (λ∈ℝ+), and Helmholtz (λ=i κ, 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 ≈(−Δ)−1, with 3≤d≤50.