Quantized-TT-Cayley transform to compute dynamics and spectrum of high-dimensional Hamiltonians

Ivan P. Gavrilyuk and Boris N. Khoromskij

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Submission date: 14. Jul. 2011 (revised version: September 2011)
Pages: 22
published in: Computational methods in applied mathematics, 11 (2011) 3, p. 273-290 
DOI number (of the published article): 10.2478/cmam-2011-0015
Bibtex
MSC-Numbers: 65F30, 65F50, 65N35
Keywords and phrases: high-dimensional problems, tensor approximation, quantized representation of vectors, Cayley transform, model reduction, time-dependent problems
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Abstract:
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. We mainly focus on the complex-time evolution problems. We apply the recent quantics-TT (QTT) matrix product states tensor approximation that allows to represent N-d tensors generated by d-dimensional functions and operators with log-volume complexity, O(dlog N), where N is the univariate discretization parameter in space. We apply the truncated Cayley transform method that allows 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 parabolic equation. We show the exponential convergence of the m-term time separation scheme and describe the efficient tensor-structured preconditioners for the class of multidimensional Hamiltonians. For a class of “analytic” input data, the asymptotic numerical complexity of our method can be estimated by (dmlog N ln^q), where ε > 0 is the error bound. The QTT-Cayley-transform time-space separation method 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(dmlog N_t log N), where N_t is the number of sampling points in time. The latter allows the efficient energy spectrum calculations by FFT (or QTT-FFT) transform 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 a norm of the t-Laplace transform of a solution. In particular, our approach can be an option to compute dynamics and spectrum of time-dependent molecular Schödinger equations.