Volume 20, Issue 3

Some new domain decomposition methods (DDM) based on optimal control approach are introduced for the coupling of first- and second-order equations on overlapping subdomains. Several cost functionals and control functions are proposed. Uniqueness and existence results are proved for the coupled problem and the convergence of iterative processes is analyzed.

In his book published in English translation in 1983, Marchuk proposed a set of evolutionary equations incorporating delay-differential equations, and the corresponding initial conditions as a model ('Marchuk's model') for infectious diseases. The parameters in this model (and its subsequent extensions) represent scientifically meaningful characteristics. For a given infection, the parameters can be estimated using observational data on the course of the infection. Sensitivity analysis is an important tool for understanding a particular model; this can be viewed as an issue of stability with respect to structural perturbations in the model. Examining the sensitivity of the models based on delay differential equations leads to systems of neutral delay differential equations. Below we formulate a general set of equations for the sensitivity coefficients for models comprising neutral delay differential equations. We discuss computational approaches to the sensitivity of solutions — (i) sensitivity to the choice of model, in particular, to the lag parameter τ > 0 and (ii) sensitivity to the initial function — of dynamical systems with time lag and illustrate them by considering the sensitivity of solutions of time-lag models of Marchuk type.

In this paper we present two recent results for matrices arising in the mixed finite element method on triangular meshes. The first result is an algebraic proof for the discrete LBB condition. The second one is a new algorithm for the construction of a two-level spectrally equivalent preconditioner for the condensed matrix.

We study the optimization of the Crank–Nicolson method, also known as the Euler second-order trapezoidal rule [5] for ordinary differential equations. The Crank–Nicolson method for the numerical integration of the first-order ordinary differential equations is A -stable, but it is not L -stable. This implies that the stability region coincides exactly with the negative half-plane z : ℜz ≤ 0, but the stability function | R (z)| tends to 1 rather than zero as ℜz→ –∞. This causes the unexpected oscillatory behaviour of the numerical solution of stiff differential equations. In order to avoid this problem we optimize the stability property of the stability function. Variable steps within the sequence of steps by the Crank–Nicolson method allow us to obtain different stability functions and formulate an optimization problem for roots and poles of the stability function. The optimal solution of this problem is the classical rational Zolotarev function. The appropriate selection of the sequence of step sizes eliminates the oscillatory behaviour of the numerical solution.

We derive theoretical and numerical estimates for the spectral radius of a matrix integral operator specifying the estimate variance of radiation polarization by the Monte Carlo method. It is shown that for real problems of aerosol scattering the variances of these estimates are finite with a high degree of reliability.