Numerical methods for random dynamical systems

不确定性量化方法是现代工程系统设计中计算仿真结果验证和验证过程的重要组成部分。我的建议包括两个关于非线性随机和随机微分方程的数值方法的主题。第一个主题是气动弹性系统的随机动力学。最近，我们提出了一种计算高效的随机配置法，用于逼近具有多项式非线性的随机微分方程组的解，该随机微分方程组代表了飞机机翼结构振动的两自由度气动弹性模型。此外，对于受乘性噪声扰动的系统，我们还应用随机范式研究了随机Hopf分叉。我将把这一研究扩展到二次分叉。对于一个被加性噪声扰动的类似系统，我将利用Lyapunov函数从理论上研究随机吸引子的存在性。主要的困难是开发高维随机动力系统的有效方法，并最终将这一研究扩展到三自由度气动弹性模型。渐近研究在许多实际应用中是至关重要的，例如求出飞机机翼振动变得不稳定的颤振速度。第二个主题是辛和准辛数值方法。具有参数不确定性的数学模型通常用随机微分方程组来表示，因此精确的数值方法对于逼近这些方程的解是必不可少的。我们发展了一种基于母函数的系统方法来构造随机哈密顿系统的新的辛差分格式。对于随机模型，数值模拟表明，在长时间的模拟中，辛差分格式比非辛方法具有更高的精度。然而，在随机情况下没有对这一事实进行理论证明，而在确定性情况下所使用的方法的扩展是具有挑战性的。我打算研究随机哈密顿系统和带有乘性噪声的朗之万型方程的辛数值格式和准辛数值格式的误差。该方法将包括对弱格式的向后误差分析，以及从一般状态空间上的马尔可夫链理论来研究数值逼近的遍历性质的技巧。最困难的部分是使用Malliavin微积分的方法将这种误差研究推广到具有乘性噪声和非全局Lipschitz系数的方程。这一提议既包括具有挑战性的随机动力系统的理论分析，也包括随机动力系统的应用，因此它将有助于知识从学术界向产业的转移。

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.