Direct tensor-product solution of one-dimensional elliptic equations with parameter-dependent coefficients
Sergey Dolgov, Vladimir A. Kazeev, and Boris N. Khoromskij
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Submission date: 15. Aug. 2012 (revised version: November 2014)
published in: Mathematics and computers in simulation : transactions of IMACS, 145 (2018), p. 136-155
DOI number (of the published article): 10.1016/j.matcom.2017.10.009
MSC-Numbers: 35J15, 15A69, 65F10, 34B08, 60H15
Keywords and phrases: elliptic equations, parametric problems, iterative methods, Tensor formats, Sherman-Morrison correction, preconditioning
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We consider a one-dimensional second-order elliptic equation with a high-dimensional parameter in a hypercube as a parametric domain. Such a problem arises, for example, from the Karhunen–Loève expansion of a stochastic PDE posed in a one-dimensional physical domain. For the discretization in the parametric domain we use the collocation on a tensor-product grid. The paper is focused on the tensor-structured solution of the resulting multiparametric problem, which allows to avoid the curse of dimensionality owing to the use of the separation of parametric variables in the tensor train and quantized tensor train formats.
We suggest an efficient tensor-structured preconditioning of the entire multiparametric family of one-dimensional elliptic problems and arrive at a direct solution formula. We compare this method to a tensor-structured preconditioned GMRES solver in a series of numerical experiments.