Algorithm of dynamical input reconstruction for a stochastic differential equation: tuning of parameters and numerical experiments

The problem of reconstructing unknown inputs in a stochastic differential equation is investigated by means of the approach of the theory of dynamic inversion. The statement when the simultaneous reconstruction of disturbances in the deterministic and stochastic terms of the equation is performed from the discrete information on a number of realizations of the stochastic process is considered. The problem is reduced to an inverse problem for ordinary differential equations describing the mathematical expectation and covariance matrix of the process. A software-oriented solving algorithm based on constructions of the theory of positional control with a model is designed; an estimate for its convergence rate with respect to the number of measurable realizations is obtained. A program procedure for the automatic tuning of the algorithm’s parameters in order to have the best approximation results for different disturbances satisfying a priori constraints in a specific dynamical system is proposed. Desired dependencies of the algorithm’s parameters on the number of measured realizations are determined empirically via solving a specific extremal problem, where the deviation of the algorithm’s output from some test function is minimized. To optimize the time-taking adaptation process assuming the simulation of a large number of independent trajectories of the stochastic process, the parallelization of calculations is applied. A model example illustrating the method proposed is given. A system approximately describing the population dynamics of two interacting biological species is considered. Calculation results and parallelization efficiency characteristics are presented.