Lebedeva O. S. Tensor conjugate-gradient-type method for Rayleigh quotient minimization in block QTT format.Russian journal of numerical analysis and mathematical modelling, 26 (5), pp. 465-489, 2011.
The method of solution of a partial spectral problem is constructed with the use of a special tensor structure of eigenvalues: in the case where the eigenvalues of a symmetric matrix represented in the QTT-format can be approximated in the form of a QTT-decomposition with a small number of parameters the method determines such approximate decomposition. The working time of the algorithm and the required memory are proportional to the logarithm of the total number of unknowns, whereas generally this dependence has been linear in the best case. It is shown how the efficiency of the tensor representation can be improved due to the use of a special block extension of the QTT-format. Convergence rate estimates are obtained for the modified method. The tensor method has been implemented in the block QTT-format on the base of the block conjugate gradient method. Numerical experiments have been performed for the solution of problems of mathematical physics in spaces of dimensions 2, 3, and more, and also for optimization problems for many-dimensional functions.
tensor train format, quantics tensor train format, block tensor train format, low-parametric representations, block conjugate gradient method.