Efficient Numerical Methods and Algorithms for Nonlinear Stochastic Partial Differential Equations

工业、工程和领域科学中的建模通常涉及不同程度的随机性和不确定性影响。一大类重要的包含不确定性的模型是随机和/或随机偏微分方程组。这项研究项目将针对几个重要的SPDEs，旨在开发稳定、准确和高效的改进的数值方法，重点关注非线性问题和适当的采样方法。由此产生的数值方法和算法有望为SPDEs从数学上描述的系统的计算建模提供急需的工具，这些系统来自许多科学、工程和工业应用，如材料科学、流体和量子力学、波散射、数学金融和随机最优控制。此外，该项目将通过参与研究来培训研究生，帮助他们发展在学术界或行业成功职业所需的应用和计算数学知识和技能。该研究项目发展了一般非线性随机和/或随机偏微分方程组(R/SPDEs)的高级数值方法和算法。当前求解R/SPDEs的方法在大范围内面临着相当大的挑战：这类系统涉及的巨大计算量阻碍了高空间和时间分辨率的使用，并且通常没有考虑解算器优化。同时，随着附加的非线性和噪声源的考虑，R/SPDEs变得更加复杂。这对数字PDE社区来说是一个巨大的挑战，但也是一个巨大的机遇。该项目致力于开发高效的数值方法和算法来求解各种科学和工程应用中的非线性SPDEs，包括随机Allen-Cahn方程和Cahn-Hilliard方程以及随机非线性波动和薛定谔方程。正在开发的数值方法将致力于在网格大小和物理参数、结构保持属性以及快速和可并行实施的适应性方面具有稳定性。该奖项反映了NSF的法定使命，并已通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.