The most difficult computational problems nowadays are those of
higher dimensions. This research monograph offers an introduction
to tensor numerical methods designed for the solution of the
multidimensional problems in scientific computing. These methods
are based on the rank-structured approximation of multivariate
functions and operators by using the appropriate tensor formats.
The old and new rank-structured tensor formats are investigated. We
discuss in detail the novel quantized tensor approximation method
(QTT) which provides function-operator calculus in higher
dimensions in logarithmic complexity rendering super-fast
convolution, FFT and wavelet transforms. This book suggests the
constructive recipes and computational schemes for a number of real
life problems described by the multidimensional partial
differential equations. We present the theory and algorithms for
the sinc-based separable approximation of the analytic radial basis
functions including Green's and Helmholtz kernels. The efficient
tensor-based techniques for computational problems in electronic
structure calculations and for the grid-based evaluation of
long-range interaction potentials in multi-particle systems are
considered. We also discuss the QTT numerical approach in
many-particle dynamics, tensor techniques for stochastic/parametric
PDEs as well as for the solution and homogenization of the elliptic
equations with highly-oscillating coefficients. Contents Theory on
separable approximation of multivariate functions Multilinear
algebra and nonlinear tensor approximation Superfast computations
via quantized tensor approximation Tensor approach to
multidimensional integrodifferential equations
General
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