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Abstract
We consider a onedimensional secondorder elliptic equation with a highdimensional 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 onedimensional physical domain. For the discretization in the parametric domain we use the collocation on a tensorproduct grid. The paper is focused on the tensorstructured 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 tensorstructured preconditioning of the entire multiparametric family of onedimensional elliptic problems and arrive at a direct solution formula. We compare this method to a tensorstructured preconditioned GMRES solver in a series of numerical experiments.
Original language  English 

Pages (fromto)  136155 
Number of pages  20 
Journal  Mathematics and Computers in Simulation 
Volume  145 
Early online date  27 Oct 2017 
DOIs  
Publication status  Published  1 Mar 2018 
Keywords
 Elliptic equations
 Parametric problems
 Preconditioning
 Sherman–Morrison correction
 Tensor formats
ASJC Scopus subject areas
 Theoretical Computer Science
 General Computer Science
 Numerical Analysis
 Modelling and Simulation
 Applied Mathematics
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Dive into the research topics of 'Direct tensorproduct solution of onedimensional elliptic equations with parameterdependent coefficients'. Together they form a unique fingerprint.Projects
 1 Finished

Sergey Dolgov Fellowship  Tensor Product Numerical Methods for HighDimensional Problems in Probablility and Quantum Calculations
Scheichl, R.
Engineering and Physical Sciences Research Council
1/01/16 → 31/12/18
Project: Research council