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We represent the solution u ⁢ ( t ) {u(t)} of an initial value problem (IVP) for the first-order differential equation with an operator coefficient as a series using the Cayley transform of the corresponding operator coefficient and the Laguerre polynomials. In the case of a boundary value problem (BVP) for the second-order differential equation with an operator coefficient, we represent its solution using the Cayley transform and the Meixner-type polynomials. The approximate solution is the truncated sum of N (the discretization parameter) summands. We give the error estimate of these approximations depending on N and the distance of t to the initial point of the time interval or of the spatial argument x to the boundary of the spatial domain.

In the present paper we propose and analyze a class of tensor approaches for the efficient numerical solution of a first order differential equation ψ ′ ⁢ ( t ) + A ⁢ ψ = f ⁢ ( t ) {\psi^{\prime}(t)+A\psi=f(t)} with an unbounded operator coefficient A. These techniques are based on a Laguerre polynomial expansions with coefficients which are powers of the Cayley transform of the operator A. The Cayley transform under consideration is a useful tool to arrive at the following aims: (1) to separate time and spatial variables, (2) to switch from the continuous “time variable” to “the discrete time variable” and from the study of functions of an unbounded operator to the ones of a bounded operator, (3) to obtain exponentially accurate approximations. In the earlier papers of the authors some approximations on the basis of the Cayley transform and the N-term Laguerre expansions of the accuracy order 𝒪 ⁢ ( e - N ) {\mathcal{O}(e^{-N})} were proposed and justified provided that the initial value is analytical for A. In the present paper we combine the Cayley transform and the Chebyshev–Gauss–Lobatto interpolation and arrive at an approximation of the accuracy order 𝒪 ⁢ ( e - N ) {\mathcal{O}(e^{-N})} without restrictions on the input data. The use of the Laguerre expansion or the Chebyshev–Gauss–Lobatto interpolation allows to separate the time and space variables. The separation of the multidimensional spatial variable can be achieved by the use of low-rank approximation to the Cayley transform of the Laplace-like operator that is spectrally close to A. As a result a quasi-optimal numerical algorithm can be designed.

Most important computational problems nowadays are those related to processing of the large data sets and to numerical solution of the high-dimensional integral-differential equations. These problems arise in numerical modeling in quantum chemistry, material science, and multiparticle dynamics, as well as in machine learning, computer simulation of stochastic processes and many other applications related to big data analysis. Modern tensor numerical methods enable solution of the multidimensional partial differential equations (PDE) in ℝ d {\mathbb{R}^{d}} by reducing them to one-dimensional calculations. Thus, they allow to avoid the so-called “curse of dimensionality”, i.e. exponential growth of the computational complexity in the dimension size d, in the course of numerical solution of high-dimensional problems. At present, both tensor numerical methods and multilinear algebra of big data continue to expand actively to further theoretical and applied research topics. This issue of CMAM is devoted to the recent developments in the theory of tensor numerical methods and their applications in scientific computing and data analysis. Current activities in this emerging field on the effective numerical modeling of temporal and stationary multidimensional PDEs and beyond are presented in the following ten articles, and some future trends are highlighted therein.

In the present paper, we propose and analyse a class of tensor methods for the efficient numerical computation of the dynamics and spectrum of high-dimensional Hamiltonians. We focus on the complex-time evolution problems. We apply the quantized-TT (QTT) matrix product states type tensor approximation that allows to represent N-d tensors generated by the grid representation of d-dimensional functions and operators with log-volume complexity, O(d log N), where N is the univariate discretization parameter in space. Making use of the truncated Cayley transform method allows us to recursively separate the time and space variables and then introduce the efficient QTT representation of both the temporal and the spatial parts of the solution to the high-dimensional evolution equation. We prove the exponential convergence of the m-term time-space separation scheme and describe the efficient tensor-structured preconditioners for the arising system with multidimensional Hamiltonians. For the class of "analytic" and low QTT-rank input data, our method allows to compute the solution at a fixed point in time t=T>0 with an asymptotic complexity of order O(d log N ln^q (1/ε)), where ε>0 is the error bound and q is a fixed small number. The time-and-space separation method via the QTT-Cayley-transform enables us to construct a global m-term separable (x,t)-representation of the solution on a very fine time-space grid with complexity of order O(dm^4 log N_t log N), where N_t is the number of sampling points in time. The latter allows efficient energy spectrum calculations by FFT (or QTT-FFT) of the autocorrelation function computed on a sufficiently long time interval [0,T]. Moreover, we show that the spectrum of the Hamiltonian can also be represented by the poles of the t-Laplace transform of a solution. In particular, the approach can be an option to compute the dynamics and the spectrum in the time-dependent molecular Schrödinger equation.

A mathematical model of the linear sloshing in an infinite chute with a rectangular section filled by an inviscid incompressible liquid is considered. This model is an abstract second order dierential equation in time with a singular integral operator coecient on a free surface. In the case of a chute with a rib this operator coecient is defined semi-explicitely through an auxiliary singular integral equation. Mathematical properties of the solution of the model are studied. The fundamental frequencies of the liquid as functions of the width and depth of the rib are investigated. The mathematical model is discretized by using the collocation quadrature method for the operator on the free surface and the Cayley transform method for the time derivative. This fully discrete model is investigated and the error estimates are obtained showing the spectral accuracy with respect to both the spatial and the time discretization parameters. 2000 Mathematics Subject Classification: 65M15, 65M70, 65R20, 76B07, 76M22.