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MiS Preprint

Low-rank Tensor Structure of Solutions to Elliptic Problems with Jumping Coefficients

Sergey Dolgov, Boris N. Khoromskij, Ivan V. Oseledets and Eugene E. Tyrtyshnikov


We study the separability properties of solutions to elliptic equations with a piecewise constant diffusion coefficient in $\mathbb{R}^d$, $d \ge 2$. It is proved that the solution can be approximated with a sum of $O(M^{d-1})$ products of univariate functions, where $M$ is a number of cells with constant coefficient in each direction. For discrete solutions in the 2D case the better estimate was obtained in series of numerical experiments: the separation rank of the solution is only proportional to the separation rank of the coefficient instead of the number of cells.

MSC Codes:
65F30, 65F50, 65N35, 65N30, 65F10
structured matrices, elliptic operators, Poisson equation, low-rank matrices, matrix approximations, tensors, canonical decomposition, finite elements

Related publications

2012 Repository Open Access
Sergey Dolgov, Boris N. Khoromskij, Ivan V. Oseledets and Eugene E. Tyrtyshnikov

Low-rank tensor structure of solutions to elliptic problems with jumping coefficients

In: Journal of computational mathematics, 30 (2012) 1, pp. 14-23