Alternating minimal energy methods for linear systems in higher dimensions. Part II: Faster algorithm and application to nonsymmetric systems
Sergey Dolgov and Dmitry Savostyanov
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Submission date: 03. Apr. 2013 (revised version: April 2013)
Keywords and phrases: high--dimensional problems, tensor train format, ALS, steepest descent, superfast algorithms, DMRG
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In this paper we accomplish the development of the fast rank--adaptive solver for tensor--structured symmetric positive definite linear systems in higher dimensions. In the previous article this problem is approached by alternating minimization of the energy function, which we combine with steps of the basis expansion in accordance with the steepest descent algorithm. In this paper we combine the same steps in such a way that the resulted algorithm works with one or two neighboring cores at a time. The recurrent interpretation of the algorithm allows to prove the global convergence and estimate the convergence rate. We also propose several strategies, both rigorous and heuristic, to compute new subspaces for the basis enrichment in a more efficient way. We test the algorithm on a number of high--dimensional problems, including the non-symmetrical Fokker--Planck and chemical master equations, for which the efficiency of the method is not fully supported by the theory. In all examples we observe a convincing fast convergence and high efficiency of the proposed method.