Volume 33, Issue 1

The aim of the paper is the construction and numerical solution of stochastic differential equations whose trajectories are located on a given smooth manifold with probability 1. Second order cylindrical surfaces, i.e., elliptic, hyperbolic, and parabolic cylinders serve as examples of such manifolds for the tree-dimensional space (the phase space is two-dimensional). Classes of stochastic differential equations are constructed for these surfaces and linear equations with multiplicative noise are marked in these classes. The results of modelling were used to estimate the deviations of numerical solutions from the manifold. A comparative analysis of considered examples was carried out for accuracy of eight numerical solution methods for stochastic differential equations.

The block inverse iteration and block Newton’s methods proposed by the authors of this paper for computing invariant pairs of regular linear matrix pencils are generalized to the case of regular nonlinear matrix pencils. Numerical properties of the proposed methods are demonstrated with a typical quadratic eigenproblem.

The problem of variational data assimilation (estimation) for a nonlinear model is considered in general operator formulation. Hessian-based methods are presented to compute the estimation error covariances. The importance of dynamic formulation and the role of the Hessian and its inverse are discussed.

Chemical systems are often characterized by a number of peculiar properties that create serious challenges to state estimator algorithms. They may include hard nonlinear dynamics, states subject to some constraints arising from a physical nature of the process (for example, all chemical concentrations must be nonnegative), and so on. The classical Extended Kalman Filter (EKF), which is considered to be the most popular state estimator in practice, is shown to be ineffective in chemical systems with infrequent measurements. In this paper, we discuss a recently designed version of the EKF method, which is grounded in a high-order Ordinary Differential Equation (ODE) solver with automatic global error control. The implemented global error control boosts the quality of state estimation in chemical engineering and allows this newly built version of the EKF to be an accurate and efficient state estimator in chemical systems with both short and long waiting times (i.e., with frequent and infrequent measurements). So chemical systems with variable sampling periods are algorithmically admitted and can be treated as well.

The paper is focused on study of non-stationary piecewise-linear processes on Poisson point flows with independent identically distributed random variables at support points. An approach to calculate the correlation function of the process on the base of the total probability formula is considered. A general expression for the correlation function of a non-stationary process is obtained. Particular cases are considered. Using the method of direct simulation, it is shown numerically that the correlation function of the process has a point of inflection.