Volume 35, Issue 4

A series of problems related to the class of inverse problems of ocean hydrothermodynamics and problems of variational data assimilation are formulated in the present paper. We propose methods for solving the problems studied here and present results of numerical experiments.

Mathematical immunology is the branch of mathematics dealing with the application of mathematical methods and computational algorithms to explore the structure, dynamics, organization and regulation of the immune system in health and disease. We review the conceptual and mathematical foundation of modelling in immunology formulated by Guri I. Marchuk. The current frontier studies concerning the development of multiscale multiphysics integrative models of the immune system are presented.

For elliptic boundary value problems (the diffusion equation and elasticity theory ones) with highly varying coefficients, there are proposed iterative methods with the number of iterations independent of the coefficient jumps. In the differential case these methods take solving the boundary value problem for the Poisson equation at each step of iterations while in the finite difference (finite element) approximation it is possible to use another operator as a preconditioner.

This work presents an overview of techniques that enable the construction of collocated finite volume method for complex multi-physics models in multiple domains. Each domain is characterized by the properties of heterogeneous media and features a distinctive multi-physics model. Coupling together systems of equations, corresponding to multiple unknowns, results in a vector flux. The finite volume method requires continuity of intradomain and interdomain vector fluxes. The continuous flux is derived using an extension of the harmonic averaging point concept. Often, the collocated coupling of the equations results in a saddle-point problem subject to inf-sup stability issues. These issues are addressed by the eigen-splitting of indefinite matrix coefficients encountered in the flux expression. The application of the techniques implemented within INMOST platform to hydraulic fracturing problem is demonstrated.

Some properties of tensor ranks and the non-closeness issue of sets with given restrictions on the rank of tensors entering those sets are studied. It is proved that the rank of the d -dimensional Laplacian equals d . The following conjecture is formulated: for any tensor of non-maximal rank there exists a nonzero decomposable tensor (tensor of rank 1) such that the rank increases by one after adding this tensor. In the general case, it is proved that this property holds algebraically almost everywhere for complex tensors of fixed size whose rank is strictly less than the generic rank.

In contrast to many other heuristic and stochastic methods, the global optimization based on TT-decomposition uses the structure of the optimized functional and hence allows one to obtain the global optimum in some problem faster and more reliable. The method is based on the TT-cross method of interpolation of tensors. In this case, the global optimum can be found in practice even in the case when the approximation of the tensor does not possess a high accuracy. We present a detailed description of the method and its justification for the matrix case and rank-1 approximation.