Low-rank Tensor Structure of Solutions to Elliptic Problems with Jumping Coefficients
Sergey Dolgov, Boris N. Khoromskij, Ivan V. Oseledets, and Eugene E. Tyrtyshnikov
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Submission date: 31. Mar. 2011
published in: Journal of computational mathematics, 30 (2012) 1, p. 14-23
DOI number (of the published article): 10.4208/jcm.1110-m11si08
MSC-Numbers: 65F30, 65F50, 65N35, 65N30, 65F10
Keywords and phrases: structured matrices, elliptic operators, Poisson equation, low-rank matrices, matrix approximations, tensors, canonical decomposition, finite elements
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We study the separability properties of solutions to elliptic equations with a piecewise constant diffusion coefficient in ℝd, d ≥ 2. It is proved that the solution can be approximated with a sum of O(Md-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.