

Preprint 82/2008
Tensor-structured Preconditioners and Approximate Inverse of Elliptic Operators in ℝd
Boris N. Khoromskij
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Submission date: 11. Nov. 2008 (revised version: March 2009)
Pages: 24
published in: Constructive approximation, 30 (2009) 3, p. 599-620
DOI number (of the published article): 10.1007/s00365-009-9068-9
Bibtex
MSC-Numbers: 65F50, 65F30, 46B28
Keywords and phrases: preconditioning, high dimensions, elliptic resolvent, tensor approximation, Green's kernels
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Abstract:
In the present paper we analyse a class of
tensor-structured preconditioners for the multidimensional second order
elliptic operators in ,
.
For equations in bounded domain the construction is based on the
rank-R tensor-product approximation
of the elliptic resolvent
,
where
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
, that is well developed in
the case of non-oscillatory potentials.
For the oscillating kernels
,
,
we constructive proof of the rank-
separable approximation.
This leads to the tensor
representation for the discretized 3D Helmholtz kernel on
grid that requires only
reals for storage.
Such representations can be applied to both the 3D
volume and boundary calculations with sublinear cost
even in the case
.
Numerical illustrations demonstrate the efficiency of
low tensor rank approximation for Green's kernels
,
, in the case
of Newton (
), Yukawa (
) and
Helmholtz (
) potentials,
as well as for the kernel functions
and
,
, 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
, with
.