Wick-type Stochastic Modeling: Algorithms and Applications

本项目的目标是从数学上理解无限维空间中Wick积的影响，作为伊藤积分的推广，并开发新的算法来量化复杂动力系统中的不确定性。许多用于物理和生物应用的随机模型包括“噪声”术语，以解释系统参数或相互作用中的不确定性。为了消除随机性引起的奇异性，通常需要正则化或重整化方法来进行随机建模。Wick积作为一种重正化技术起源于欧几里得量子场论，它与白噪声分析、Malliavin微积分等许多现代随机分析理论有着直接而深刻的数学联系。此外，Wick积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，Wick产品公式为分析提供了严格的数学基础，同时也为开发有效的不确定性量化数值算法提供了有希望的候选者。具体而言，本项目包括与Wick型随机建模相关的两个重要问题：(1)基于Wick积的随机椭圆建模；(2)动力系统的随机摄动。对于第一个问题，PI将基于Wick产品给出的新建模策略开发新的随机有限元方法；对于第二个问题，PI将开发高维动态系统随机扰动的可扩展并行最小作用方法。所开发的算法可用于广泛的物理、生物和工程应用。对Wick产品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能是罕见的，但深刻的。典型问题包括化学反应、双稳态遗传切换、相变成核事件、气候状态变化、流体力学不稳定性等。可扩展的并行最小作用方法可以帮助人们更好地理解高维构型空间，这对于通过大规模仿真研究上述现象至关重要。一旦算法开发和测试完成，PI将通过现有的外部开源网站将代码作为开源代码传播。

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.