Sergey Dolgov

Solution of Linear Systems and Matrix Inversion in the TT-Format

Tensors arise naturally in high-dimensional problems in chemistry, financial mathematics, and many other areas. The numerical treatment of such problems is difficult due to the curse of dimensionality: the number of unknowns and the computational complexity grow exponentially with the dimension of the problem. To break the curse of dimensionality, low-parametric representations, or formats, have to be used. In this paper we make use of the TT-format (tensor-train format) which is one of the most effective stable representations of high-dimensional tensors. Basic linear algebra operations in the TT-format are now well developed. Our goal is to provide a “black-box” type of solver for linear systems where both the matrix and the right-hand side are in the TT-format. An efficient DMRG (density matrix renormalization group) method is proposed, and several tricks are employed to make it work. The numerical experiments confirm the effectiveness of our approach.

Alternating Minimal Energy Methods for Linear Systems in Higher Dimensions

We propose algorithms for the solution of high-dimensional symmetrical positive definite (SPD) linear systems with the matrix and the right-hand side given and the solution sought in a low-rank format. Similarly to density matrix renormalization group (DMRG) algorithms, our methods optimize the components of the tensor product format subsequently. To improve the convergence, we expand the search space by an inexact gradient direction. We prove the geometrical convergence and estimate the convergence rate of the proposed methods utilizing the analysis of the steepest descent algorithm. The complexity of the presented algorithms is linear in the mode size and dimension, and the demonstrated convergence is comparable to or even better than the one of the DMRG algorithm. In the numerical experiment we show that the proposed methods are also efficient for non-SPD systems, for example, those arising from the chemical master equation describing the gene regulatory model at the mesoscopic scale.

Computation of extreme eigenvalues in higher dimensions using block tensor train format

We consider approximate computation of several minimal eigenpairs of large Hermitian matrices which come from high-dimensional problems. We use the tensor train (TT) format for vectors and matrices to overcome the curse of dimensionality and make storage and computational cost feasible. We approximate several low-lying eigenvectors simultaneously in the block version of the TT format. The computation is done by the alternating minimization of the block Rayleigh quotient sequentially for all TT cores. The proposed method combines the advances of the density matrix renormalization group (DMRG) and the variational numerical renormalization group (vNRG) methods. We compare the performance of the proposed method with several versions of the DMRG codes, and show that it may be preferable for systems with large dimension and/or mode size, or when a large number of eigenstates is sought.

Fast Solution of Parabolic Problems in the Tensor Train/Quantized Tensor Train Format with Initial Application to the Fokker--Planck Equation

In this paper we propose two schemes of using the so-called quantized tensor train (QTT)-approximation for the solution of multidimensional parabolic problems. First, we present a simple one-step implicit time integration scheme using a solver in the QTT-format of the alternating linear scheme (ALS) type. As the second approach, we use 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

Simultaneous state‐time approximation of the chemical master equation using tensor product formats

(x,t)

TT-GMRES: solution to a linear system in the structured tensor format

variables. The log-linear complexity of storage and the solution time is observed in both spatial and time grid sizes. The method is applied to the Fokker--Planck equation arising from the beads-springs models of polymeric liquids.

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

This is an essentially improved version of the preprint [12]. This manuscript contains all the same numerical experiments, but some inaccuracies in the description of the modeling equations are corrected. Besides, more detailed introduction to the tensor methods is presented. We study the application of the novel tensor formats (TT, QTT, QTT-Tucker) to the solution of d-dimensional chemical master equations, applied mostly to gene regulating networks (signaling cascades, toggle switches, phage- ). For some important cases, e.g. signaling cascade models, we prove good separability properties of the system operator. The Quantized tensor representations (QTT, QTT-Tucker) are employed in both state space and time, and the global state-time (d +1)-dimensional system is solved in the structured form by using the ALS-type iteration. This approach leads to the logarithmic dependence of the computational complexity on the system size. When possible, we compare our approach with the direct CME solution and some previously known approximate schemes, and observe a good potential of the newer tensor methods in simulation of relevant biological systems.

Plantlet regeneration from subculturable nodular callus of Pinus radiata

Abstract An adapted tensor-structured GMRES method for the TT format is proposed and investigated. The Tensor Train (TT) approximation is a robust approach to highdimensional problems. One class of such problems involves the 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.

Alternating minimal energy methods for linear systems in higher dimensions. Part I: SPD systems *

We propose algorithms for the solution of high-dimensional symmetrical positive definite (SPD) linear systems with the matrix and the right-hand side given and the solution sought in a low-rank format. Similarly to density matrix renormalization group (DMRG) algorithms, our methods optimize the components of the tensor product format subsequently. To improve the convergence, we expand the search space by an inexact gradient direction. We prove the geometrical convergence and estimate the convergence rate of the proposed methods utilizing the analysis of the steepest descent algorithm. The complexity of the presented algorithms is linear in the mode size and dimension, and the demonstrated convergence is comparable to or even better than the one of the DMRG algorithm. In the numerical experiment we show that the proposed methods are also efficient for non-SPD systems, for example, those arising from the chemical master equation describing the gene regulatory model at the mesoscopic scale.

Two-Level QTT-Tucker Format for Optimized Tensor Calculus

AbstractAn efficient planlet regeneration system via nodular callus formation is described for Pinus radiata. Subculturable nodular callus was induced at its highest frequency (93%) on embryonic explants excised from seeds at an early stage of germination (radicle length 2–5 mm). The optimal medium for nodular callus tissue proliferation was