MiS Preprint Repository

In the present paper we analyse a class of tensor-structured preconditioners for the multidimensional second order elliptic operators in ${\mathbb{R}}^d$, $d\ge 2$.

For equations in bounded domain the construction is based on the rank-$R$ tensor-product approximation of the elliptic resolvent ${\mathcal{B}}_R\approx (\mathcal{L}-\lambda I{)}^{-1}$, where $\mathcal{L}$ 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 unbounded domain one can utilise the tensor-structured approximation of Green's kernel for the shifted Laplacian in ${\mathbb{R}}^d$, that is well developed in the case of non-oscillatory potentials. For the oscillating kernels $\frac{e^{-i\kappa \Vert x\Vert }}{\Vert x\Vert }$, $x\in {\mathbb{R}}^d$, $\kappa \in {\mathbb{R}}_{+}$ we constructive proof of the rank-$O(\kappa )$ separable approximation.

This leads to the tensor representation for the discretized 3D Helmholtz kernel on $n\times n\times n$ grid that requires only $O(\kappa |\log \epsilon {|}^{2}n)$ 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 $\kappa =O(n)$.

Numerical illustrations demonstrate the efficiency of low tensor rank approximation for Green's kernels $\frac{e^{-\lambda \Vert x\Vert }}{\Vert x\Vert }$, $x\in {\mathbb{R}}^3$, in the case of Newton ($\lambda =0$), Yukawa ($\lambda \in {\mathbb{R}}_{+}$) and Helmholtz ($\lambda =i\kappa ,\kappa \in {\mathbb{R}}_{+}$) potentials, as well as for the kernel functions $1/\Vert x\Vert$ and $1/\Vert x{\Vert }^{d-2}$, $x\in {\mathbb{R}}^d$, 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 power method with the rank truncation and based on the approximate inverse ${\mathcal{B}}_R\approx (-\mathrm{\Delta }{)}^{-1}$, with $3\le d\le 50$.