arXiv subjects

Sergey V. Dolgov

Publications and source records attributed to Sergey V. Dolgov.

6 recordsLinked to original sources

arXiv2022

Tensor product approach to modelling epidemics on networks

To improve mathematical models of epidemics it is essential to move beyond the traditional assumption of homogeneous well--mixed population and involve more precise information on the network of contacts and transport links by which a stochastic process of the epidemics spreads. In general, the number of states of the network grows exponentially with its size, and a master equation description suffers from the curse of dimensionality. Almost all methods widely used in practice are versions of the stochastic simulation algorithm (SSA), which is notoriously known for its slow convergence. In this paper we numerically solve the chemical master equation for an SIR model on a general network using recently proposed tensor product algorithms. In numerical experiments we show that tensor product algorithms converge much faster than SSA and deliver more accurate results, which becomes particularly important for uncovering the probabilities of rare events, e.g. for number of infected people to exceed a (high) threshold.

physics.soc-ph↗

arXiv2017

A tensor decomposition algorithm for large ODEs with conservation laws

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.

math.NA↗

arXiv2013

One-site density matrix renormalization group and alternating minimum energy algorithm

Given in the title are two algorithms to compute the extreme eigenstate of a high-dimensional Hermitian matrix using the tensor train (TT) / matrix product states (MPS) representation. Both methods empower the traditional alternating direction scheme with the auxiliary (e.g. gradient) information, which substantially improves the convergence in many difficult cases. Being conceptually close, these methods have different derivation, implementation, theoretical and practical properties. We emphasize the differences, and reproduce the numerical example to compare the performance of two algorithms.

math.NA↗

arXiv2013

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

We consider an approximate computation of several minimal eigenpairs of large Hermitian matrices which come from high--dimensional problems. We use the tensor train format (TT) for vectors and matrices to overcome the curse of dimensionality and make storage and computational cost feasible. Applying a block version of the TT format to several vectors simultaneously, we compute the low--lying eigenstates of a system by minimization of a block Rayleigh quotient performed in an alternating fashion for all dimensions. For several numerical examples, we compare the proposed method with the deflation approach when the low--lying eigenstates are computed one-by-one, and also with the variational algorithms used in quantum physics.

math.NA↗

arXiv2013

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

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.

math.NA↗

arXiv2012

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

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.

math.NA↗