Volume 28, Issue 2

The paper studies a method for solving elliptic partial differential equations posed on hypersurfaces in ℝN, N = 2; 3. The method builds upon the formulation introduced in [7], where a surface equation is extended to a neighbourhood of the surface. The resulting degenerate PDE is then solved in one dimension higher, but can be solved on a mesh that is unaligned to the surface. We introduce another extended formulation, which leads to uniformly elliptic (non-degenerate) equations in a bulk domain containing the surface.We apply a finite element method to solve this extended PDE and prove the convergence of the finite element solutions restricted to the surface to the solution of the original surface problem. Several numerical examples illustrate the properties of the method.

A laminar flow of a viscous incompressible fluid with a constant pressure gradient (Poiseuille flow) is considered in a rectangular duct for different values of cross-sectional aspect ratio. A new method, significantly more efficient than the known ones, is proposed and justified for computing the critical Reynolds number of such a flow. The dependence of the critical Reynolds number on the cross-sectional aspect ratio is numerically studied. A theoretical justification of the obtained dependence is proposed

An adapted tensor-structured GMRES method for the TT format is proposed and investigated. The Tensor Train (TT) approximation is a robust approach to highdimensional problems. One class of such problems involves the solution of a linear system. In this work we study the convergence of the GMRES method in the presence of tensor approximations and provide relaxation techniques to improve its performance. Several numerical examples are presented. The method is also compared with a projection TT linear solver based on the ALS and DMRG methods. On a particular SPDE (high-dimensional parametric) problem these methods manifest comparable performance, with a good preconditioner the TT-GMRES overcomes the ALS solver.

Various models of exponentially correlated random media related to Poisson point ensembles are considered in the paper. Asymptotic estimates of the mean probability of a particle (radiation quantum) passing through media of such type are constructed on the base of the Poisson property of the flow of the trajectory intersections with constancy domains of random density and with the help of the central limit theorem for the corresponding‘optical’ length. The possibility to reconstruct the parameters of a radiation model from the averaged passage probability is also studied.

A numerical stochastic model of spatial fields of daily sums of liquid precipitation constructed with the use of real observation data is proposed in the paper. Based on this model, simulation algorithms for conditionally distributed fields are constructed and certain examples of the use of this model in studying extreme properties of liquid precipitation fields are considered.

Algorithms of the ‘method of similar trajectories’ (MST) are implemented by a numerical-statistical simulation of the trajectories of particles, which are radiation ‘quanta’, according to an auxiliary radiation model and construction of weighted estimates of the functionals simultaneously for different values of ‘physical’ parameters. This paper is focused on the theoretical and numerical solution of the problem of choosing this auxiliary model with the aim to minimize the parametric maximum of the mean-square error of weighted estimates. We formulate new assertions concerning minimax MST algorithms and refine some previously known statements.