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Simplified CSP analysis of a stiff stochastic ODE system

机译：刚性随机ODE系统的简化CSP分析

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We develop a simplified computational singular perturbation (CSP) analysis of a stochastic dynamical system. We focus on the case of parametric uncertainty, and rely on polynomial chaos (PC) representations to quantify its impact. We restrict our attention to a system that exhibits distinct timescales, and that tends to a deterministic steady state irrespective of the random inputs. A detailed analysis of eigenvalues and eigenvectors of the stochastic system Jacobian is conducted, which provides a relationship between the PC representation of the stochastic Jacobian and the Jacobian of the Galerkin form of the stochastic system. The analysis is then used to guide the application of a simplified CSP formalism that is based on relating the slow and fast manifolds of the uncertain system to those of a nominal deterministic system. Two approaches are specifically developed with the resulting simplified CSP framework. The first uses the stochastic eigenvectors of the uncertain system as CSP vectors, whereas the second uses the eigenvectors of the nominal system as CSP vectors. Numerical experiments are conducted to demonstrate the results of the stochastic eigenvalue and eigenvector analysis, and illustrate the effectiveness of the simrjlified CSP algorithms in addressing the stiffness of the svstem dvnamics.
机译：我们开发了一种随机动力系统的简化计算奇异摄动（CSP）分析。我们关注参数不确定性的情况，并依靠多项式混沌（PC）表示来量化其影响。我们将注意力集中在表现出不同时间尺度的系统上，并且该系统倾向于确定性稳态，而与随机输入无关。对随机系统雅可比行列式的特征值和特征向量进行了详细的分析，从而提供了随机雅可比行列式的PC表示与随机系统的Galerkin形式的雅可比行列之间的关系。然后，该分析用于指导简化的CSP形式化的应用，该形式化基于将不确定系统的慢速和快速流形与名义上确定性系统的流形联系起来。通过简化的CSP框架专门开发了两种方法。第一种将不确定系统的随机特征向量用作CSP向量，而第二种将标称系统的特征向量用作CSP向量。进行了数值实验，以证明随机特征值和特征向量分析的结果，并说明了简化的CSP算法在解决系统动态刚度方面的有效性。

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