Alternating minimal energy methods for linear systems in higher dimensions. Part II: Faster algorithm and application to nonsymmetric systems

摘要

In this paper we accomplish the development of the fast rank-adaptive solverfor tensor-structured symmetric positive definite linear systems in higherdimensions. In [arXiv:1301.6068] this problem is approached by alternatingminimization of the energy function, which we combine with steps of the basisexpansion in accordance with the steepest descent algorithm. In this paper wecombine the same steps in such a way that the resulted algorithm works with oneor two neighboring cores at a time. The recurrent interpretation of thealgorithm allows to prove the global convergence and to estimate theconvergence rate. We also propose several strategies, both rigorous andheuristic, to compute new subspaces for the basis enrichment in a moreefficient way. We test the algorithm on a number of high-dimensional problems,including the non-symmetrical Fokker-Planck and chemical master equations, forwhich the efficiency of the method is not fully supported by the theory. In allexamples we observe a convincing fast convergence and high efficiency of theproposed method.