Journal
CHEMOMETRICS AND INTELLIGENT LABORATORY SYSTEMS
Volume 110, Issue 1, Pages 1-19Publisher
ELSEVIER
DOI: 10.1016/j.chemolab.2011.09.001
Keywords
High-dimensional problems; Rank structured tensor approximation; Quantics folding of vectors; Matrix valued functions; FEM/BEM; Computational quantum chemistry; Stochastic PDEs
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In the present paper, we give a survey of the recent results and outline future prospects of the tensor-structured numerical methods in applications to multidimensional problems in scientific computing. The guiding principle of the tensor methods is an approximation of multivariate functions and operators relying on a certain separation of variables. Along with the traditional canonical and Tucker models, we focus on the recent quantics-TT tensor approximation method that allows to represent N-d tensors with log-volume complexity, 0(d log N). We outline how these methods can be applied in the framework of tensor truncated iteration for the solution of the high-dimensional elliptic/parabolic equations and parametric PDEs. Numerical examples demonstrate that the tensor-structured methods have proved their value in application to various computational problems arising in quantum chemistry and in the multi-dimensional/parametric FEM/BEM modeling the tool apparently works and gives the promise for future use in challenging high-dimensional applications. (C) 2011 Elsevier B.V. All rights reserved.
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