DMRG+QTT approach to computation of the ground state for the molecular Schrödinger operator

By Boris N. Khoromskij, Ivan V. Oseledets. 2010

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A literature survey of low‐rank tensor approximation techniques (2013, neutral): “…149], and the 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…”
TT-GMRES: on solution to a linear system in the structured tensor format (2012, methodology): “the preconditioner from [21], when the whole matrix MA can not be computed explicitly due to very high ranks, and multiplications are applied during several successive implicit procedures, which provide approximate products (for example, the DMRG-based TT-MatVec, see [23,18] for the DMRG schemes).”
Two-level Tucker-TT-QTT format for optimized tensor calculus (2012, background): “The orthogonality property is heavily utilized in the approximation and solution tensor routines, since it allows to control the introduced approximation error, and make orthogonal projections, see [24] [39] [41] [42].”
Fast Solution of Parabolic Problems in the Tensor Train/Quantized Tensor Train Format with Initial Application to the Fokker-Planck Equation (2012, background): “The general ALS and DMRG schemes for the typical quadratic functionals (error, residual, Rayleigh quotient) were discussed in [19], and the QTT-DMRG eigenvalue solver was presented in [28].”
On Local Convergence of Alternating Schemes for Optimization of Convex Problems in the Tensor Train Format (2013, background): “[1, 2, 22, 26, 29]; while the former approach seems to be superior on some problems in the sense that it sometimes provides lower ranks [16], the latter has advantage of being conceptually quite simple, thus often allowing for a simpler analysis from the theoretical point of view, and enabling, from”
Superfast Fourier Transform Using QTT Approximation (2012, methodology): “Some function-related examples are already mentioned, more examples of (piecewise) smooth functions and functions with singularities that have a low-rank QTT representation can be found in [14, 36, 41].”
Tensor-product approach to global time-space-parametric discretization of chemical master equation (2012, methodology): “Such a solver, the TT-Solve algorithm, was described in [29] (see also [49] for the eigenvalue solver of this type and [28], where the general ALS-type schemes are discussed).”