| Modern tensor numerical methods allow the efficient representation of functions and operators, as well as the solution of high-dimensional PDEs with the linear complexity scaling in the dimension. Most commonly used separable representations combine the canonical, Tucker, tensor train (TT), the quantized-TT (QTT), QTT-canonical, and Tucker-TT-QTT decompositions, as well as the general MPS-type formats in the framework of DMRG-MPS optimization methods. The novel QTT tensor format was proven to provide the remarkable approximation properties on the wide class of functions and operators [4,5], which made it possible to solve steady-state and dynamical problems in quantized tensor spaces with the log-volume complexity scaling in the size of full functional tensor. We address the basic approximation and complexity results in the quantized tensor formats applied in electronic structure calculations (on the example of two-electron integrals) and to the solution of $d$-dimensional time-dependent equations (on the example of molecular Schrödinger and Fokker-Planck models), [1]-[3], [5]-[7]. Numerical tests indicate the logarithmic computational complexity of the QTT-based tensor-structured solvers. [1] I.V. Gavrilyuk, and B.N. Khoromskij. Quantized-TT-Cayley transform to compute dynamics and spectrum of high-dimensional Hamiltonians. Comp. Meth. in Applied Math., v.11 (2011), No. 3, 273-290. [2] V. Khoromskaia, D. Andrae, and B.N. Khoromskij. Fast and Accurate Tensor Calculation of the Fock Operator in a General Basis. Preprint 4/2012, MPI MiS, Leipzig 2012 (CPC, submitted). [3] V. Khoromskaia, B.N. Khoromskij, and R. Schneider. Tensor-structured calculation of two-electron integrals in a general basis. Preprint MPI MiS, Leipzig 2012 (submitted). [4] B.N. Khoromskij. O(d\log N)-Quantics Approximation of N-d Tensors in High-Dimensional Numerical Modeling. J. Constr. Approx. v. 34(2), 257-289 (2011). [5] B.N. Khoromskij. Introduction to Tensor Numerical Methods in Scientific Computing. Lecture Notes, Preprint 06-2011, University of Zuerich, Institute of Mathematics, 2011, pp. 1 - 238. http://www.math.uzh.ch/fileadmin/math/preprints/06\_11.pdf. [6] B.N. Khoromskij, and I. Oseledets. DMRG + QTT approach to the computation of ground state for the molecular Schr\"odinger operator. Preprint 68/2010, MPI MiS, Leipzig 2010 (Numer. Math., submitted). [7] I. Oseledets, B.N. Khoromskij, and R. Schneider. Efficient time-stepping scheme for dynamics on TT-manifolds. Preprint 24/2012, MPI MIS, Leipzig 2012. |
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