Sergey V. Dolgov

Sergey V. Dolgov, Dmitry V. Savostyanov

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 [arXiv:1301.6068] 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 to 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.

Sergey V. Dolgov, Dmitry V. Savostyanov

We introduce a family of numerical algorithms for the solution of linear system in higher dimensions with the matrix and right hand side given and the solution sought in the tensor train format. The proposed methods are rank--adaptive and follow the alternating directions framework, but in contrast to ALS methods, in each iteration a tensor subspace is enlarged by a set of vectors chosen similarly to the steepest descent algorithm. The convergence is analyzed in the presence of approximation errors and the geometrical convergence rate is estimated and related to the one of the steepest descent. The complexity of the presented algorithms is linear in the mode size and dimension and the convergence demonstrated in the numerical experiments is comparable to the one of the DMRG--type algorithm.

Sergey V. Dolgov

A adapted tensor-structured GMRES method for the TT format is proposed and investigated. The Tensor Train (TT) approximation is a robust approach to high-dimensional problems. One class of problems is solution of a linear system. In this work we study the convergence of the GMRES method in the presence of tensor approximations and provide relaxation techniques to improve its performance. Several numerical examples are presented. The method is also compared with a projection TT linear solver based on the ALS and DMRG methods. On a particular sPDE (high-dimensional parametric) problem, these methods manifest comparable performance, with a good preconditioner the TT-GMRES overcomes the ALS solver.

Sergey V. Dolgov

We propose an algorithm for solution of high-dimensional evolutionary equations (ODEs and discretized time-dependent PDEs) in the Tensor Train (TT) decomposition, assuming that the solution and the right-hand side of the ODE admit such a decomposition with a low storage. A linear ODE, discretized via one-step or Chebyshev differentiation schemes, turns into a large linear system. The tensor decomposition allows to solve this system for several time points simultaneously using an extension of the Alternating Least Squares algorithm. This method computes the TT approximation of the solution directly, without ever solving the original large problem, and encapsulates the Galerkin model reduction of the ODE. This allows an efficient estimation of the time discretization error, and hence provides a way to adapt the time steps. Besides, conservation laws can be preserved exactly in the reduced model by expanding the approximation subspace with the generating vectors of the linear invariants and correction of the euclidean norm. In numerical experiments with the transport and the chemical master equations, we demonstrate that the new method is faster than traditional time stepping and stochastic simulation algorithms, whereas the invariants are preserved up to the machine precision irrespectively of the TT approximation accuracy.