Efficient time-stepping scheme for dynamics on TT-manifolds
Boris N. Khoromskij, Ivan V. Oseledets, and Reinhold Schneider
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Submission date: 20. Apr. 2012
MSC-Numbers: 15A69, 65N22, 33F05
Keywords and phrases: TT-format, Dirac-Frenkel, Splitting scheme, higher dimensions
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Solution of non-stationary problems in high dimensions is feasible only if certain low-parametric nonlinear approximation to the solution is used. Thus, even if the initial system is linear, defining equations for the model parameters are nonlinear. The general concept is to use the Dirac-Frenkel principle to obtain the approximate trajectory on the manifold. In this paper, this approach is analyzed for the dynamical approximation of high-dimensional tensors in the so-called Tensor Train (TT)-format. We obtain an explicit system of ODEs describing the evolution of parameters, defining the solution, and propose an efficient and stable numerical scheme to solve this system. The efficiency of the proposed method is illustrated by the numerical examples.