Fast solution of multi-dimensional parabolic problems in the TT/QTT-format with initial application to the Fokker-Planck equation
Sergey Dolgov, Boris N. Khoromskij,and Ivan V. Oseledets
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Submission date: 08. Dec. 2011 (revised version: February 2012)
published in: SIAM journal on scientific computing, 34 (2012) 6, p. A3016-A3038
DOI number (of the published article): 10.1137/120864210
with the following different title: Fast solution of parabolic problems in the tensor train/quantized tensor train format with initial application to the Fokker-Planck equation
MSC-Numbers: 35K20, 65F50, 15A69, 65D15, 33F05, 65F10, 35Q84, 82D60
Keywords and phrases: parabolic problems, QTT-format, DMRG, higher dimensions, tensor methods, fokker-planck equation, dumbbell model
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In this paper we propose two schemes of using the QTT tensor approximations for solution of multi-dimensional parabolic problems. First, we present a simple one-step implicit time integration scheme and modify it using the matrix multiplication and a linear ALS-type solver in the TT format. As the second approach, we propose the global space-time formulation, resulting in a large block linear system, encapsulating all time steps, and solve it at once in the QTT format. We prove the QTT-rank estimate for certain classes of multivariate potentials and respective solutions in (x,t) variables. We observe the log-linear complexity of storage and the solution algorithm in both spatial and time grid sizes, and at most cubic scaling in the QTT ranks of the discretized operator matrix and solution. The method is applied to the Fokker-Planck equation arising from the beads-springs models of polymeric liquids. For the dumbbell model numerical experiments are shown to demonstrate logarithmic behavior of computational time versus number of grid points in space and time, as well as accuracy. However, in numerical tests for the case of multispring Hookean potential we observe, that the rank properties of more general models might make the straightforward application of the tensor product approximations inefficient, requiring modifications in model descriptions and tensor discretizations.