Volume 40, Issue 3

The current state of research in the field of variational assimilation of observational data in models of ocean dynamics developed at the INM RAS is presented. The developed technology of four-dimensional variational data assimilation (4D-Var) is based on the method of multicomponent splitting of the mathematical model of ocean dynamics and minimization of the cost functional associated with observational data by solving an optimality system that includes adjoint equations and covariance matrices of observation errors and errors of initial approximation. Effective algorithms for solving variational problems of data assimilation based on iterative processes using direct and adjoint equations with a special choice of iterative parameters are proposed, as well as algorithms for studying the sensitivity of model characteristics to errors in observational data. The methodology is illustrated for a model of the Black Sea hydrothermodynamics with variational data assimilation to restore heat fluxes on the sea surface.

We analyzed the fundamental issues related to the development of mathematical models in immunology, i.e., the structural and practical identifiability of the models in mathematical immunology. To this end, the differential algebraic techniques and Bayesian approach implemented in StructuralIdentifiability.jl and DynamicHMC.jl Julia-based packages, respectively, are used. The experimental data on kinetics of viral load and cytotoxic T lymphocyte (CTL) response characterizing an acute lymphocytic choriomeningitis virus (LCMV) infection in mice were considered. Although the models differ in terms of one to three parameters, the structural identifiability strongly depends on details of observability and initial determination of the state variables. The estimated via a Bayesian approach posterior distributions for model parameter characterize the rates of interactions underlying the acute infection development. The results of the data assimilation on LCMV-CTL kinetics suggest that a bilinear-type description of the virus-induced CTL expansion and the CTL-driven virus elimination need to be refined to a bounded-rate (e.g., Michaelis–Menten) type parameterizations.

First, we give a short history of four algorithmic approaches in computational mathematics based on the analysis and use of the asymptotic behavior of the error of an approximate solution where the mesh size of a difference grid tends to zero. These approaches are Richardson’s ‘extrapolation to the limit’, Runge’s accuracy rule, Romberg’s rule for calculating integrals, and improving the grid solutions by high-order differences. The first two approaches were initially developed based on the intuitive conclusion about the asymptotic behavior of the error back in the early 20th century. The last two algorithms, at the time of their appearance in the second half of the 20th century, already used theoretical results on the special asymptotic behavior of the error of quadrature rules or difference solutions. The latter approach, despite its good computational efficiency for ordinary differential equations, has not yet been properly developed for solving multidimensional difference problems. Therefore, as an introductory illustration, we present a method for increasing the order of convergence in time for the solution of an initial boundary value problem for a parabolic equation by correcting the right-hand side with differences of a higher order. The increase in accuracy order is justified theoretically and demonstrated by a numerical example.

The present work introduces a numerical method for mathematical modelling of the hydraulic fracture propagation process within a naturally fractured reservoir. We consider a poroelastoplasticity model with rock failure described by an advanced constitutive material model representing the tensile, shear, and compressional failure of the formation. We propose a discretization method for the elastoplastic model coupled to the equations for two-phase fluid flow in porous media. The discretization is based on a collocated cell-centered finite volume method that uses an advanced approximation of the traction vector for mechanics and the conventional single-point upstream-weighted two-point approximation of the Darcy flux. The Biot term coupling of the mechanical and flow models is approximated assuming piecewise constant pore pressure, leading to an inf-sup stable method. To solve the plasticity problem, we use the cutting plane algorithm. The plastic strain tensor derivatives are obtained in the solution process, thus bypassing the necessity of the consistent tangent elastoplastic stiffness tensor. We study the grid convergence for the discrete solution of poroelasticity equations on a set of problems with analytical solutions and demonstrate the application of proposed methods for poroplasticity to tensile fractures near a borehole and hydraulic fracturing experiments.

The method of adjoint equations proposed by G. I. Marchuk is considered for studying climate variability problems. Adjoint equations of atmospheric and ocean dynamics are most actively used to solve 4DVAR problems of observational data assimilation. Together with direct equations, they are an integral part of optimality systems allowing efficient calculation of the gradient of the cost function. In this paper, the method of adjoint equation is used to solve quasi-geostrophic equations of ocean dynamics in a periodic channel. The two-connectivity of the domain requires that an additional integral relation should be satisfied in order to obtain the stream function on the second contour. This integral relation is the basis of the minimized cost function in numerical solution of the optimality system. As well as for 4DVAR problems, adjoint equations are used to find the gradient of a functional. The algorithm is used to solve the problem in a channel simulating the Antarctic Circumpolar Current.