Numerical methods for random dynamical systems

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Uncertainty quantification methods are an important part of the verification and validation process of computational simulation results used in modern design of engineering systems. My proposal includes two themes regarding numerical methods for non-linear random and stochastic differential equations. The first theme included is stochastic dynamics for an aeroelastic system. Recently we have proposed computationally efficient stochastic collocation methods for approximating the solutions of a system of random differential equations with polynomial non-linearities representing a two-degrees-of-freedom aeroelastic model for the oscillations of an aircraft wing structure. Moreover, for a system perturbed by multiplicative noise, we have applied the stochastic normal form to study the stochastic Hopf bifurcation. I will extend this study to the secondary bifurcation. For a similar system, but perturbed by additive noise, I will study theoretically the existence of random attractors using Lyapunov functions. The main difficulty is to develop effective methods for high dimensional random dynamical systems, and, eventually, to extend this study for a three-degrees-of–freedom aeroelastic model. An asymptotic study is of crucial importance in many practical applications, such as finding the flutter speed, at which the vibrations of the aircraft wings become unstable. The second theme concerns symplectic and quasi-symplectic numerical methods. Mathematical models with parameter uncertainties are often expressed by stochastic differential equations, so accurate numerical methods are essential to approximate the solutions of these equations. We have developed a systematic method, based on generating functions, to construct new symplectic finite difference numerical schemes for stochastic Hamiltonian systems. For stochastic models, numerical simulations show that symplectic difference schemes are more accurate than non-symplectic methods in long-time simulations. However, no theoretical proof of this fact was done in the stochastic case, and an extension of the approach used in the deterministic case is challenging. I intend to study the error associated with symplectic and quasi-symplectic numerical schemes for stochastic Hamiltonian systems and Langevin type equations with multiplicative noise. The approach will include a backward error analysis for weak schemes, and techniques from the theory of Markov chains on general state spaces to study the ergodic properties of the numerical approximations. The most difficult part is to use methods from Malliavin calculus to extend this study of the error to equations with multiplicative noise and non-globally Lipschitz coefficients. This proposal includes both challenging theoretical analyses and applications of random dynamical systems, so it will contribute to the knowledge transfer from academia to industry.