New theory and corresponding numerical algorithms are proposed for addressing fundamental open questions in stochastic modeling of physical and biological systems, e.g., the curse-of-dimensionality, the lack of regularity and the long-time integration of stochastic systems. Such problems arise in applications involving processes with small relative correlation length or large number of random parameters, and for time-dependent nonlinear systems subject to uncertainty. The new equations are formulated in terms of the time-evolution of the joint probability density function (PDF) between the system's response and the stochastic excitation. In particular, functional integral methods are employed to determine new types of linear deterministic partial differential equations satisfied by the joint response-excitation PDF associated with the stochastic solution of nonlinear stochastic ordinary and partial differential equations. So far the theory is complete for nonlinear and for quasilinear first-order stochastic PDEs subject to random boundary conditions, random initial conditions or random forcing terms. For higher-order equations, such the stochastic wave equation or the Oberbeck-Boussinesq thermal convection equations, it is proposed to develop a new PDF method based on differential constraints for the PDF of the solution. It is proposed to investigate the theoretical and numerical effectiveness of this new approach for high-dimensional random systems, such as random flows subject to high-dimensional random boundary or initial conditions in bounded domains.
Stochastic modeling and uncertainty quantification are important new directions in computational mathematics that will enable accurate predictions of physical and biological phenomena,in critical applications such as climate, energy and the design of new products. The proposed work will have significant and broad impact as it will set new rigorous foundations in uncertainty quantification, data assimilation and sensitivity analysis for many physical and biological systems. It will affect fundamentally the way we design new experiments and the type of questions that we can address, while the interaction between simulation and experiment will become more meaningful and more dynamic. This work will also aid in educating a new cadre of simulation scientists in this metadiscipline at the interface of computational mathematics and probability theory.
