Efficient Numerical Methods and Algorithms for Nonlinear Stochastic Partial Differential Equations

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Modeling in industry, engineering, and domain sciences often involves various degrees of randomness and uncertainty effects. A large and important class of models incorporating uncertainty are random and/or stochastic partial differential equations (SPDEs). This research project will address several important SPDEs and aims to develop improved numerical methods that are stable, accurate, and efficient, with focus on nonlinear problems and adequate sampling methods. The resulting numerical methods and algorithms are anticipated to provide much-needed tools for computational modeling of systems described mathematically by SPDEs from many scientific, engineering, and industry applications such as materials science, fluid and quantum mechanics, wave scattering, mathematical finance, and stochastic optimal control. Moreover, the project will train graduate students through involvement in the research, helping them develop applied and computational mathematics knowledge and skills needed for successful careers in either academia or industry. This research project develops advanced numerical methods and algorithms for general nonlinear random and/or stochastic partial differential equations (R/SPDEs). Current approaches for solving R/SPDEs face considerable challenges at large scales: the sheer amount of computation involved in such systems prevents the use of high spatial and temporal resolutions, and solver optimization is often not considered. In the meantime, R/SPDEs become more complex as additional nonlinearities and sources of noise are considered. This presents a big challenge but also a great opportunity to the numerical PDE community. The project focuses on developing efficient numerical methods and algorithms for solving nonlinear SPDEs that arise from various scientific and engineering applications, including stochastic Allen-Cahn and Cahn-Hilliard equations and stochastic nonlinear wave and Schrodinger equations. The numerical methods under development will aim to feature stability with respect to mesh sizes and physical parameters, structure-preserving properties, and amenability to fast and parallelizable implementation.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.