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Tensor product methods in numerical simulation of high-dimensional dynamical problems

Sergey Dolgov

Published Sep 8, 2014• 12 citations
OpenAlex

Abstract

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 $10^{10}$ 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.

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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 101010^{10}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.
We present a brief survey on the modern tensor numerical methods for multidimensional stationary and time-dependent partial differential equations (PDEs). The guiding principle of the tensor approach is the rank-structured separable approximation of multivariate functions and operators represented on a grid. Recently, the traditional Tucker, canonical, and matrix product states (tensor train) tensor models have been applied to the grid-based electronic structure calculations, to parametric PDEs, and to dynamical equations arising in scientific computing. The essential progress is based on the quantics tensor approximation method proved to be capable to represent (approximate) function related d-dimensional data arrays of size Nd with log-volume complexity, O(dlog N). Combined with the traditional numerical schemes, these novel tools establish a new promising approach for solving multidimensional integral and differential equations using low-parametric rank-structured tensor formats. As the main example, we describe the grid-based tensor numerical approach for solving the 3D nonlinear Hartree-Fock eigenvalue problem, that was the starting point for the developments of tensor-structured numerical methods for large-scale computations in solving real-life multidimensional problems. We also discuss a new method for the fast 3D lattice summation of electrostatic potentials by assembled low-rank tensor approximation capable to treat the potential sum over millions of atoms in few seconds. We address new results on tensor approximation of the dynamical Fokker-Planck and master equations in many dimensions up to d = 20. Numerical tests demonstrate the benefits of the rank-structured tensor approximation on the aforementioned examples of multidimensional PDEs. In particular, the use of grid-based tensor representations in the reduced basis of atomics orbitals yields an accurate solution of the Hartree-Fock equation on large N × N × N grids with a grid size of up to N = 105.
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.
The conventional numerical methods when applied to multidimensional problems suffer from the so-called "curse of dimensionality", that cannot be eliminated by using parallel architectures and high performance computing. The novel tensor numerical methods are based on a "smart" rank-structured tensor representation of the multivariate functions and operators discretized on Cartesian grids thus reducing solution of the multidimensional integral-differential equations to 1D calculations. We explain basic tensor formats and algorithms and show how the orthogonal Tucker tensor decomposition originating from chemometrics made a revolution in numerical analysis, relying on rigorous results from approximation theory. Benefits of tensor approach are demonstrated in ab-initio electronic structure calculations. Computation of the 3D convolution integrals for functions with multiple singularities is replaced by a sequence of 1D operations, thus enabling accurate MATLAB calculations on a laptop using 3D uniform tensor grids of the size up to 10 15 . Fast tensor-based Hartree-Fock solver, incorporating the grid-based low-rank factorization of the two-electron integrals, serves as a prerequisite for economical calculation of the excitation energies of molecules. Tensor approach suggests efficient grid-based numerical treatment of the long-range electrostatic potentials on large 3D finite lattices with defects.The novel range-separated tensor format applies to interaction potentials of multi-particle systems of general type opening the new prospects for tensor methods in scientific computing. This research monograph presenting the modern tensor techniques applied to problems in quantum chemistry may be interesting for a wide audience of students and scientists working in computational chemistry, material science and scientific computing.
We discuss extended definitions of linear and multilinear operations such as Kronecker, Hadamard, and contracted products, and establish links between them for tensor calculus. Then we introduce effective low-rank tensor approximation techniques including Candecomp/Parafac (CP), Tucker, and tensor train (TT) decompositions with a number of mathematical and graphical representations. We also provide a brief review of mathematical properties of the TT decomposition as a low-rank approximation technique. With the aim of breaking the curse-of-dimensionality in large-scale numerical analysis, we describe basic operations on large-scale vectors, matrices, and high-order tensors represented by TT decomposition. The proposed representations can be used for describing numerical methods based on TT decomposition for solving large-scale optimization problems such as systems of linear equations and symmetric eigenvalue problems.
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
There has been a surge of interest in the study of multidimensional arrays, known as tensors. This is due to the fact that many real-world datasets can be represented as tensors. For example, colour images are naturally third-order tensors, which include two indices (or modes) for their spatial index, and one mode for colour. Also, a colour video is a fourth-order tensor comprised of frames, which are colour images, and an additional temporal index. Traditional tools for matrix analysis does not generalise so well in tensor analysis. The main issue is that tensors prescribe a natural structure, which is destroyed when they are vectorised. Many mathematical techniques such as principal component analysis (PCA) or linear discriminant analysis (LDA) used extensively in machine learning rely on vectorised samples of data. Additionally, since tensors may often be large in dimensionality and size, vectorising these samples and applying them to PCA or LDA may not lead to the most efficient results, and the computational time of the algorithms can increase significantly. This problem is known as the so-called curse of dimensionality. Tensor decompositions and their interesting properties are needed to circumvent this problem. The Tucker (TD) or CANDECOMP/PARAFAC (CP) decompositions have been predominantly used for tensor-based machine learning and signal processing. Both utilise common factor matrices and a core tensor, which retains the dimensionality of the original tensor. A main problem with these type of decompositions is that they essentially rely on an unbalanced matricization scheme, which potentially converts a tensor to a highly unbalanced matrix, where the row size is attributed to always one mode and the column size is the product of the remaining modes. This method is not optimal for problems that rely on retaining as much correlations within the data, which is very important for tensor-based machine learning and signal processing. In this thesis, we are interested in utilising the matrix product state (MPS) decomposition. MPS has the property that it can retain much of the correlations within a tensor because it is based on a balanced matricization scheme, which consists of permutations of matrix sizes that can investigate the different correlations amongst all modes of a tensor. Several new algorithms are proposed for tensor object classification, which demonstrate an MPS-based approach as an efficient method against other tensor-based approaches. Additionally, new methods for colour image and video completion are introduced, which outperform the current state-of-the-art tensor completion algorithms.