We propose algorithms for the solution of high-dimensional symmetrical positive definite (SPD) linear systems with the matrix and the right-hand side given and the solution sought in a low-rank format. Similarly to density matrix renormalization group (DMRG) algorithms, our methods optimize the components of the tensor product format subsequently. To improve the convergence, we expand the search space by an inexact gradient direction. We prove the geometrical convergence and estimate the convergence rate of the proposed methods utilizing the analysis of the steepest descent algorithm. The complexity of the presented algorithms is linear in the mode size and dimension, and the demonstrated convergence is comparable to or even better than the one of the DMRG algorithm. In the numerical experiment we show that the proposed methods are also efficient for non-SPD systems, for example, those arising from the chemical master equation describing the gene regulatory model at the mesoscopic scale

Pseudoskeleton approximation and some other problems require the knowledge of sufficiently well-conditioned submatrix in a large-scale matrix. The quality of a submatrix can be measured by modulus of its determinant, also known as volume. In this paper we discuss a search algorithm for the maximum-volume submatrix which already proved to be useful in several matrix and tensor approximation algorithms. We investigate the behavior of this algorithm on random matrices and present some its applications, including maximization of a bivariate functional.

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We consider an approximate computation of several minimal eigenpairs of large Hermitian matrices which come from high-dimensional problems. We use the tensor train format (TT) for vectors and matrices to overcome the curse of dimensionality and make storage and computational cost feasible. Applying a block version of the TT format to several vectors simultaneously, we compute the low-lying eigenstates of a system by minimization of a block Rayleigh quotient performed in an alternating fashion for all dimensions. For several numerical examples, we compare the proposed method with the deflation approach when the low-lying eigenstates are computed one-by-one, and also with the variational algorithms used in quantum physics.

We propose Fourier transform algorithms using QTT format for datasparse approximate representation of one-and multi-dimensional vectors (m-tensors). Although the Fourier matrix itself does not have a low-rank QTT representation, it can be efficiently applied to a vector in the QTT format exploiting the multilevel structure of the Cooley-Tukey algorithm. The m-dimensional Fourier transform of an n×. . .×n vector with n = 2 d has O(md 2 R 3 ) complexity, where R is the maximum QTT-rank of input, output and all intermediate vectors in the procedure. For the vectors with moderate R and large n and m the proposed algorithm outperforms the O(n m log n) fast Fourier transform (FFT) algorithm and has asymptotically the same log-squared complexity as the superfast quantum Fourier transform (QFT) algorithm. By numerical experiments we demonstrate the examples of problems for which the use of QTT format relaxes the grid size constrains and allows the high-resolution computations of Fourier images and convolutions in higher dimensions without the 'curse of dimensionality'. We compare the proposed method with Sparse Fourier transform algorithms and show that our approach is competitive for signals with small number of randomly distributed frequencies and signals with limited bandwidth.

Nuclear magnetic resonance spectroscopy is one of the few remaining areas of physical chemistry for which polynomially scaling quantum mechanical simulation methods have not so far been available. In this communication we adapt the restricted state space approximation to protein NMR spectroscopy and illustrate its performance by simulating common 2D and 3D liquid state NMR experiments (including accurate description of relaxation processes using Bloch-Redfield-Wangsness theory) on isotopically enriched human ubiquitin - a protein containing over a thousand nuclear spins forming an irregular polycyclic three-dimensional coupling lattice. The algorithm uses careful tailoring of the density operator space to only include nuclear spin states that are populated to a significant extent. The reduced state space is generated by analysing spin connectivity and decoherence properties: rapidly relaxing states as well as correlations between topologically remote spins are dropped from the basis set.

We consider a cross interpolation of high-dimensional arrays in the tensor train format. We prove that the maximum-volume choice of the interpolation sets provides the quasioptimal interpolation accuracy, that differs from the best possible accuracy by the factor which does not grow exponentially with dimension. For nested interpolation sets we prove the interpolation property and propose greedy cross interpolation algorithms. We justify the theoretical results and measure speed and accuracy of the proposed algorithm with numerical experiments

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