Quantized-TT-Cayley Transform for Computing the Dynamics and the Spectrum of High-Dimensional Hamiltonians

By Ivan P. Gavrilyuk, Boris N. Khoromskij. 2011

20 citations recorded by the source corpus.

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.

Subfields: Computational Mathematics, Computational Theory and Mathematics, Hardware and Architecture, Numerical Methods & Scientific Computing, AI for Physical Sciences

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Citations in the corpus

DMRG+QTT approach to computation of the ground state for the molecular Schrödinger operator (2010, methodology): “It is worth to note, that the QTT-format was already successfully applied to solving highdimensional elliptic equations [28], multiparametric SPDEs [27], and in electronic structure calculation [18] and the log-volume complexity was con rmed numerically and theoretically [9], so it is a natural idea”
Fast Solution of Parabolic Problems in the Tensor Train/Quantized Tensor Train Format with Initial Application to the Fokker-Planck Equation (2012, methodology): “On the other hand, the advantages of the QTT-format motivate us to treat the problem from a new point of view: we consider the time as an independent dimension and introduce a discretization of the whole differential equation with all spatial and time dimension connected in one large linear system.”
Tensor-product approach to global time-space-parametric discretization of chemical master equation (2012, methodology): “For example, one may use the global space-time discretization of a differential equation, using the QTT at least in the time variable [23, 24].”
Efficient time-stepping scheme for dynamics on TT-manifolds (2012, unknown): “Note that in some cases there are theoretical estimates on the structure of the solution [10].”
Low-Rank Explicit QTT Representation of the Laplace Operator and Its Inverse (2012, methodology): “Note that the straightforward discretizations of the Laplace operator we consider turn out to be very efficient in tensor-structured solvers based on the TT, QTT, and hierarchical tensor [15, 13] formats and yield convincing experimental results when constructed on fine meshes; see the papers [24, 8”