Multidimensional quantum dynamics : MCTDH theory and applications

By Hans-Dieter Meyer, F. Gatti, G. Worth. 2009

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A literature survey of low‐rank tensor approximation techniques (2013, neutral): “…chemical master / Fokker-Planck equations [4,6,61,64,86,122,133,135,196] • the solution of high-dimensional Schr¨odinger equations [46,160,172,184,192,193]; • computational tasks in micromagnetics [80,81]; • rational approximation problems [242]; • computational homogenization [42,96,176]; •…”
TT-GMRES: on solution to a linear system in the structured tensor format (2012, unknown): “Such problems are posed, for example, in quantum chemistry [1,2], ﬁnancial mathematics [3,4] and many others.”
Efficient time-stepping scheme for dynamics on TT-manifolds (2012, unknown): “The analogous procedure is well-established for the case of the Tucker format which is in fact used in the MCTDH method, where the equations for the factors and for the core are derived and then solved numerically 1 .”
A projector-splitting integrator for dynamical low-rank approximation (2013, methodology): “In quantum dynamics, such an approach was used previously in the multiconfiguration time-dependent Hartree (MCTDH) method [11, 12] to determine an approximate solution to the time-dependent multi-particle Schrödinger equation, and the above minimization process is known there as the Dirac–Frenkel ti”
Quantized-TT-Cayley Transform for Computing the Dynamics and the Spectrum of High-Dimensional Hamiltonians (2011, methodology): “An important example in molecular dynamics is given by the Schrödinger equation for the motion of d nuclei obtained from the Born-Oppenheimer approximation (see [4, 31, 32] for more detail),”
Dynamical Tensor Approximation (2010, background): “Problem (1.2) yields an initial value problem of nonlinear ordinary differential equations onMr, which becomes numerically efficiently ∗Vienna University of Technology, Institute for Analysis and Scientific Computing, Wiedner Hauptstrasse 8–10, A-1040 Wien, Austria.”
DMRG+QTT approach to computation of the ground state for the molecular Schrödinger operator (2010, background): “However, the complicated hierarchical structure of such methods limits their wide applicability, and the original MCTDH method, which has exponential scaling in the dimension is still preferable for systems with moderate number of degrees of freedom [35].”
Two-level Tucker-TT-QTT format for optimized tensor calculus (2012, methodology): “During the history of tensor methods, different formats were proposed and analyzed , such as canonical [8] [9] [10] [11], Tucker [12] [13] [14] [15], Matrix Product States [16] [17] [18] [19] and MCTDH [20] in quantum physics community, HT (Hierarchical Tucker) [21] [22] and TT (Tensor Train) in num”
Quantics-TT Collocation Approximation of Parameter-Dependent and Stochastic Elliptic PDEs (2010, unknown): “AMS Subject Classification: 65F30, 65F50, 65N35, 65F10 Key words: elliptic equations, stochastic PDEs, the Karhunen-Loève expansion, separable approximation, quantics-TT tensors, preconditioners, tensor-truncated iteration.”
Fast Solution of Parabolic Problems in the Tensor Train/Quantized Tensor Train Format with Initial Application to the Fokker-Planck Equation (2012, methodology): “In quantum molecular dynamics simulation the Tucker format was successfully used in the MCTDH framework (see the book [33]).”
DMRG Approach to Fast Linear Algebra in the TT-Format (2011, background): “Tensors arise naturally in high-dimensional problems, for example, in quantum chemistry [1, 2], financial mathematics [3, 4], and in many other areas.”
Quantized-TT-Cayley transform to compute dynamics and spectrum of high-dimensional Hamiltonians (2011, methodology): “Important example in molecular dynamics is given by the Schrödinger equation for the motion of d nuclei obtained from the BornOppenheimer approximation (see [4, 31, 32] for more detail),”