Principles of stochastic geometric numerical integration

Abstract: This talk is devoting to sharing recent advances in the numerical preservation of invariant laws characterizing the underlying dynamics of stochastic problems, following the spirit of the so-called stochastic geometric numerical integration. We first address stochastic Hamiltonian problems, in order to obtain long-term energy conservation. Specifically, we study the behaviour of stochastic Runge-Kutta methods arising as stochastic perturbation of symplectic Runge-Kutta methods. The analysis is provided through epsilon-expansions of the solutions (where epsilon is the amplitude of the stochastic fluctuation) and shows the presence of secular terms destroying the long-term preservation of the expected Hamiltonian. Then, an energy-preserving scheme is developed and analyzed to fill this gap in. We finally consider the nonlinear stability properties of stochastic theta-methods with respect to mean-square dissipative nonlinear test problems, generating a mean-square contractive behaviour. The pursued aim is that of making the same property visible also along the numerical discretization via stochastic theta–methods: this issue is translated into sharp stepsize restrictions depending on some parameters of the problem, accurately estimated. A selection of numerical tests confirming the effectiveness of the analysis and its sharpness is also provided.
References
[1] C. Chen, D. Cohen, R. D’Ambrosio, A. Lang, "Drift-preserving numerical inte-grators for stochastic Hamiltonian systems", Adv. Comput. Math. 46, article number 27 (2020).
[2] R. D’Ambrosio, "Numerical approximation of differential problems", Springer (toappear).
[3] R. D’Ambrosio, S. Di Giovacchino, "Mean-square contractivity of stochastic theta-methods", Comm. Nonlin. Sci. Numer. Simul. 96, article number 105671 (2021).
[4] R. D’Ambrosio, S. Di Giovacchino, "Nonlinear stability issues for stochastic Runge-Kutta methods", Comm. Nonlin. Sci. Numer. Simul. 94, article number 105549 (2021).
[5] R. D’Ambrosio, G. Giordano, B. Paternoster, A. Ventola, "Perturbative analysis of stochastic Hamiltonian problems under time discretizations", Appl. Math. Lett. 120, article number 107223 (2021).

numerical analysis

Audience: researchers in the topic

Seminars on Numerics and Applications