Abstract In the present paper, we propose and analyse a class of tensor methods for the efficient numerical computation of the dynamics and spectrum of high-dimensional Hamiltonians. We focus on the complex-time evolution problems. We apply the quantized-TT (QTT) matrix product states type tensor approximation that allows to represent N-d tensors generated by the grid representation of d-dimensional functions and operators with log-volume complexity, O(d log N), where N is the univariate discretization parameter in space. Making use of the truncated Cayley transform method allows us to recursively separate the time and space variables and then introduce the efficient QTT representation of both the temporal and the spatial parts of the solution to the high-dimensional evolution equation. We prove the exponential convergence of the m-term time-space separation scheme and describe the efficient tensor-structured preconditioners for the arising system with multidimensional Hamiltonians. For the class of "analytic" and low QTT-rank input data, our method allows to compute the solution at a fixed point in time t=T>0 with an asymptotic complexity of order O(d log N ln^q (1/ε)), where ε>0 is the error bound and q is a fixed small number. The time-and-space separation method via the QTT-Cayley-transform enables us to construct a global m-term separable (x,t)-representation of the solution on a very fine time-space grid with complexity of order O(dm^4 log N_t log N), where N_t is the number of sampling points in time. The latter allows efficient energy spectrum calculations by FFT (or QTT-FFT) of the autocorrelation function computed on a sufficiently long time interval [0,T]. Moreover, we show that the spectrum of the Hamiltonian can also be represented by the poles of the t-Laplace transform of a solution. In particular, the approach can be an option to compute the dynamics and the spectrum in the time-dependent molecular Schrödinger equation.
Abstract In the present paper, we propose and analyse a class of tensor methods for the efficient numerical computation of the dynamics and spectrum of high-dimensional Hamiltonians. We focus on the complex-time evolution problems. We apply the quantized-TT (QTT) matrix product states type tensor approximation that allows to represent N-d tensors generated by the grid representation of d-dimensional functions and operators with log-volume complexity, O(d log N), where N is the univariate discretization parameter in space. Making use of the truncated Cayley transform method allows us to recursively separate the time and space variables and then introduce the efficient QTT representation of both the temporal and the spatial parts of the solution to the high-dimensional evolution equation. We prove the exponential convergence of the m-term time-space separation scheme and describe the efficient tensor-structured preconditioners for the arising system with multidimensional Hamiltonians. For the class of "analytic" and low QTT-rank input data, our method allows to compute the solution at a fixed point in time t=T>0 with an asymptotic complexity of order O(d log N ln^q (1/ε)), where ε>0 is the error bound and q is a fixed small number. The time-and-space separation method via the QTT-Cayley-transform enables us to construct a global m-term separable (x,t)-representation of the solution on a very fine time-space grid with complexity of order O(dm^4 log N_t log N), where N_t is the number of sampling points in time. The latter allows efficient energy spectrum calculations by FFT (or QTT-FFT) of the autocorrelation function computed on a sufficiently long time interval [0,T]. Moreover, we show that the spectrum of the Hamiltonian can also be represented by the poles of the t-Laplace transform of a solution. In particular, the approach can be an option to compute the dynamics and the spectrum in the time-dependent molecular Schrödinger equation.
In the present paper, we propose and analyse a class of tensor methods for the efficient numerical computation of dynamics and spectrum of high-dimensional Hamiltonians. The main focus is on the complex-time evolution problems. We apply the recent quantized-TT (QTT) matrix product states tensor approximation that allows to represent N -d tensors generated by grid representation of d-dimensional functions and operators with log-volume complexity, O(d logN), where N is the univariate discretization parameter in space. Making use of the truncated Cayley transform method allows us to recursively separate the time and space variables and then introduce the efficient QTT representation of both the temporal and spatial parts of solution to the high-dimensional evolution equation. We prove the exponential convergence of the m-term time-space separation scheme and describe the efficient tensor-structured preconditioners for the arising system with multidimensional Hamiltonians. For a class of “analytic” and low QTT-rank input data, our method allows to compute the solution at a fix point in time t = T > 0 with the asymptotic complexity of order O(d logN ln 1e ), where e > 0 is the error bound and q a fixed small number. The time-and-space separation method via the QTT-Cayley-transform enables us to construct the global m-term separable (x, t)-representation of a solution on very fine time-space grid with complexity of order O(dm4 logNt logN), where Nt is the number of sampling points in time. The latter allows the efficient energy spectrum calculations by FFT (or QTTFFT) of autocorrelation function computed on sufficiently long time interval [0, T ]. Moreover, we show that the spectrum of Hamiltonian can also be represented by the poles of the t-Laplace transform of a solution. In particular, the approach can be an option to compute dynamics and spectrum in the time-dependent molecular Schrodinger equation. AMS Subject Classification: 65F30, 65F50, 65N35, 65F10
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.
Abstract In this paper, the tensor-structured numerical evaluation of the Coulomb and exchange operators in the Hartree-Fock equation is supplemented by the usage of recent quantized-TT (QTT) formats. It leads to O(log n) complexity at computationally extensive stages in the rank-structured calculation with the respective 3D Hartree and exchange potentials discretized on large n×n×n Cartesian grids. The numerical examples for some volumetric organic molecules confirm that the QTT ranks of these potentials are nearly independent of the one-dimension grid size n. Thus, paradoxically, the complexity of the grid-based evaluation of the Coulumb and exchange matrices becomes almost independent of the grid size, being regulated only by the structure of a molecular system. As a result, the grid approximation of the Hartree-Fock equation allows to gain the high resolution with a guaranteed accuracy.
We introduce the “tensor-train split-operator Fourier transform” (TT-SOFT) method for simulations of multidimensional nonadiabatic quantum dynamics. TT-SOFT is essentially the grid-based SOFT method implemented in dynamically adaptive tensor-train representations. In the same spirit of all matrix product states, the tensor-train format enables the representation, propagation, and computation of observables of multidimensional wave functions in terms of the grid-based wavepacket tensor components, bypassing the need of actually computing the wave function in its full-rank tensor product grid space. We demonstrate the accuracy and efficiency of the TT-SOFT method as applied to propagation of 24-dimensional wave packets, describing the S 1 / S 2 interconversion dynamics of pyrazine after UV photoexcitation to the S 2 state. Our results show that the TT-SOFT method is a powerful computational approach for simulations of quantum dynamics of polyatomic systems since it avoids the exponential scaling problem of full-rank grid-based representations.
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.
We resume the recent successes of the grid-based tensor numerical methods and discuss their prospects in real-space electronic structure calculations. These methods, based on the low-rank representation of the multidimensional functions and integral operators, first appeared as an accurate tensor calculus for the 3D Hartree potential using 1D complexity operations, and have evolved to entirely grid-based tensor-structured 3D Hartree-Fock eigenvalue solver. It benefits from tensor calculation of the core Hamiltonian and two-electron integrals (TEI) in O(n log n) complexity using the rank-structured approximation of basis functions, electron densities and convolution integral operators all represented on 3D n × n × n Cartesian grids. The algorithm for calculating TEI tensor in a form of the Cholesky decomposition is based on multiple factorizations using algebraic 1D "density fitting" scheme, which yield an almost irreducible number of product basis functions involved in the 3D convolution integrals, depending on a threshold ε > 0. The basis functions are not restricted to separable Gaussians, since the analytical integration is substituted by high-precision tensor-structured numerical quadratures. The tensor approaches to post-Hartree-Fock calculations for the MP2 energy correction and for the Bethe-Salpeter excitation energies, based on using low-rank factorizations and the reduced basis method, were recently introduced. Another direction is towards the tensor-based Hartree-Fock numerical scheme for finite lattices, where one of the numerical challenges is the summation of electrostatic potentials of a large number of nuclei. The 3D grid-based tensor method for calculation of a potential sum on a L × L × L lattice manifests the linear in L computational work, O(L), instead of the usual O(L(3) log L) scaling by the Ewald-type approaches.