Wick-type Stochastic Modeling: Algorithms and Applications

灯芯型随机建模：算法与应用

• 批准号: 1115632
• 负责人: Xiaoliang Wan
• 金额: $ 10.02万
• 依托单位: Louisiana State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1115632&HistoricalAwards=false
• 关键词: Wick; type; Stochastic; Modeling; Algorithms

The goal of this project is to understand mathematically the effect of Wick product, as a generalization of Ito integral, in infinite dimensional space and to develop new algorithms to quantify the uncertainty in complex dynamical systems. Many stochastic models for physical and biological applications include "noise" terms to account for the uncertainty in the parameters or interactions of the system. To eliminate the singularity induced by the randomness, regularization or renormalization approaches are often required for stochastic modeling. Originating from the Euclidean quantum field theory as a renormalization technique, the Wick product has a direct and deep mathematical connection with many modern theories of stochastic analysis, such as the white noise analysis and Malliavin calculus. Furthermore, the Wick product has many favorable numerical properties, which give it the potential to deal effectively with problems of high random dimension. Hence, the Wick product formulation provides a rigorous mathematical foundation for analysis but also a promising candidate for developing efficient numerical algorithms for uncertainty quantification. More specifically, this project includes two important issues related to Wick-type stochastic modeling: (1) Stochastic elliptic modeling based on the Wick product; (2) Random perturbations of dynamical systems. For the first problem, the PI will develop new stochastic finite element methods based on a new modeling strategy given by the Wick product; for the second problem, the PI will develop scalable parallel minimum action methods for random perturbations of high dimensional dynamical systems. The developed algorithms can be used in a wide range of physical, biological and engineering applications. The understanding of the Wick product may shed new light on modeling of porous media, and the related algorithms can be applied to engineering applications such as petroleum engineering, underground water, etc. The effect of random perturbations of dynamical systems can be rare but profound. Typical problems include chemical reactions, bistable genetic toggle switch, nucleation events during phase transitions, regime changes in climate, instability in fluid mechanics, etc. Scalable parallel minimum action methods can help people understand better high dimensional configuration space, which is crucial to study the aforementioned phenomena through large-scale simulations. The PI will disseminate the codes as open source codes via existing external open source websites as soon as the algorithms are developed and tested.

该项目的目标是在数学上理解Wick乘积的效果，作为Ito积分的推广，在无限维空间中，并开发新的算法来量化复杂动力系统中的不确定性。许多物理和生物应用的随机模型包括“噪声”项，以说明系统参数或相互作用的不确定性。为了消除随机性引起的奇异性，经常需要正则化或重整化方法进行随机建模。Wick乘积作为一种重整化技术，起源于欧几里德量子场论，与许多现代随机分析理论，如白色噪声分析和Malliavin演算，有着直接而深刻的数学联系。此外，Wick积具有许多有利的数值性质，这使它有可能有效地处理高随机维数的问题。因此，威克产品配方提供了一个严格的数学基础分析，但也是一个有前途的候选人开发高效的数值算法的不确定性量化。更具体地说，这个项目包括两个重要的问题有关的威克型随机建模：（1）随机椭圆建模的威克产品的基础上;（2）随机扰动的动力系统。对于第一个问题，PI将开发新的随机有限元方法的基础上的一个新的建模策略，由威克产品;对于第二个问题，PI将开发可扩展的并行最小行动方法的随机扰动的高维动力系统。 所开发的算法可用于广泛的物理，生物和工程应用。对Wick乘积的理解可以为多孔介质的建模提供新的思路，相关算法可以应用于石油工程、地下水等工程应用。动力系统的随机扰动的影响可以是罕见的，但意义深远。典型问题包括化学反应、相变过程中的成核事件、气候变化、流体力学不稳定性等。可扩展的并行最小作用量方法可以帮助人们更好地理解高维位形空间，这对于通过大规模模拟研究上述现象至关重要。一旦算法被开发和测试，PI将通过现有的外部开源网站将代码作为开源代码传播。