Abstract. A robust and efficient time integrator for dynamical tensor approximation in the tensor train or matrix product state format is presented. The method is based on splitting the projector onto the tangent space of the tensor manifold. The algorithm can be used for updating time-dependent tensors in the given data-sparse tensor train / matrix product state format and for computing an approximate solution to high-dimensional tensor differential equations within this data-sparse format. The formulation, implementation and theoretical properties of the proposed integrator are studied, and numerical experiments with problems from quantum molecular dynamics and with iterative processes in the tensor train format are included. Key words. Tensor train, matrix product state, low-rank approximation, time-varying tensors, tensor differential equations, splitting integrator. AMS subject classifications. 15A18,15A69,65F99,65L05 1. Introduction. In
A numerical integrator is proposed for solving the multiconfiguration time-dependent Hartree (MCTDH) equations of motion, which are widely used in computations of molecular quantum dynamics. In contrast to existing integrators, the proposed algorithm does not require inverses of ill-conditioned density matrices and obviates the need for their regularization, allowing for large stepsizes also in the case of near-singular density matrices. The nonlinear MCTDH equations are split into a chain of linear differential equations that can all be efficiently solved by Lanczos approximations to the action of Hermitian-matrix exponentials, alternating with orthogonal matrix decompositions. The integrator is an extension to the Tucker tensor format of recently proposed projector-splitting integrators for the dynamical low-rank approximation by matrices and tensor trains (or matrix product states). The integrator is time-reversible and preserves both the norm and the total energy.
We develop a dynamical tensor-train method for the Wigner--Fokker--Planck equation arising in continuous-variable open quantum systems. The Wigner function is evolved directly in tensor-train form using a projector-splitting integrator, while the nonlocal Wigner pseudo-differential operator is treated through low-rank representations of the potential difference and matrix-free exponential actions. The resulting method avoids full tensor-product computations in high-dimensional phase space. Numerical experiments in one, two, and three spatial dimensions demonstrate the accuracy and scalability of the method, including simulations in a six-dimensional phase space.
Dynamical low-rank approximation in the Tucker tensor format of given large time-dependent tensors and of tensor differential equations is the subject of this paper. In particular, a discrete time integration method for rank-constrained Tucker tensors is presented and analyzed. It extends the known projector-splitting integrator for dynamical low-rank approximation of matrices to Tucker tensors and is shown to inherit the same favorable properties. The integrator is based on iteratively applying the matrix projector-splitting integrator to tensor unfoldings but with inexact solution in a substep. It has the property that it reconstructs time-dependent Tucker tensors of the given rank exactly. The integrator is also shown to be robust to the presence of small singular values in the tensor unfoldings. Numerical examples with time-dependent problems from quantum physics and tensor optimization methods illustrate our theoretical results.
Quantification of stochastic or quantum systems by a joint probability density or wave function is a notoriously difficult computational problem, since the solution depends on all possible states (or realizations) of the system. Due to this combinatorial flavor, even a system containing as few as ten particles may yield as many as
1010 discretized states. None of even modern supercomputers are capable to cope with this curse of dimensionality straightforwardly, when the amount of quantum particles, for example, grows up to more or less interesting order of hundreds. A traditional approach for a long time was to avoid models formulated in terms of probabilistic functions, and simulate particular system realizations in a randomized process. Since different times in different communities, data-sparse methods came into play. Generally, they aim to define all data points indirectly, by a map from a low amount of representers, and recast all operations (e.g. linear system solution) from the initial data to the effective parameters. The most advanced techniques can be applied (at least, tried) to any given array, and do not rely explicitly on its origin. The current work contributes further progress to this area in the particular direction: tensor product methods for separation of variables. The separation of variables has a long history, and is based on the following elementary concept: a function of many variables may be expanded as a product of univariate functions. On the discrete level, a function is encoded by an array of its values, or a tensor. Therefore, instead of a huge initial array, the separation of variables allows to work with univariate factors with much less efforts. The dissertation contains a short overview of existing tensor representations: canonical PARAFAC, Hierarchical Tucker, Tensor Train (TT) formats, as well as the artificial tensorisation, resulting in the Quantized Tensor Train (QTT) approximation method. The contribution of the dissertation consists in both theoretical constructions and practical numerical algorithms for high-dimensional models, illustrated on the examples of the Fokker-Planck and the chemical master equations. Both arise from stochastic dynamical processes in multiconfigurational systems, and govern the evolution of the probability function in time. A special focus is put on time propagation schemes and their properties related to tensor product methods. We show that these applications yield large-scale systems of linear equations, and prove analytical separable representations of the involved functions and operators. We propose a new combined tensor format (QTT-Tucker), which descends from the TT format (hence TT algorithms may be generalized smoothly), but provides complexity reduction by an order of magnitude. We develop a robust iterative solution algorithm, constituting most advantageous properties of the classical iterative methods from numerical analysis and alternating density matrix renormalization group (DMRG) techniques from quantum physics. Numerical experiments confirm that the new method is preferable to DMRG algorithms. It is as fast as the simplest alternating schemes, but as reliable and accurate as the Krylov methods in linear algebra.
The simulation and analysis of high-dimensional problems is often infeasible due to the curse of dimensionality. In this thesis, we investigate the potential of tensor decompositions for mitigating this curse when considering systems from several application areas. Using tensor-based solvers, we directly compute numerical solutions of master equations associated with Markov processes on extremely large state spaces. Furthermore, we exploit the tensor-train format to approximate eigenvalues and corresponding eigentensors of linear tensor operators. In order to analyze the dominant dynamics of high- dimensional stochastic processes, we propose several decomposition techniques for highly diverse problems. These include tensor representations for operators based on nearest-neighbor interactions, construction of pseudoinverses for tensor-based reformulations of dimensionality reduction methods, and the approximation of transfer operators of dynamical systems. The results show that the tensor-train format enables us to compute low-rank approximations for various numerical problems as well as to reduce the memory consumption and the computational costs compared to classical approaches significantly. We demonstrate that tensor decompositions are a powerful tool for solving high-dimensional problems from various application areas.