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.