Tensors-structured Numerical Methods in Scientific Computing: Survey on Recent Advances
Boris N. Khoromskij
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Submission date: 02. May. 2010 (revised version: July 2011)
published in: Chemometrics and intelligent laboratory systems, 110 (2012) 1, p. 1-19
DOI number (of the published article): 10.1016/j.chemolab.2011.09.001
MSC-Numbers: 65F30, 65F50, 65N35
Keywords and phrases: high-dimensional problems, rank structured tensor approximation, quantics folding of vectors, 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 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, O(dlog 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.