Efficient Numerical Methods and Algorithms for Nonlinear Stochastic Partial Differential Equations

非线性随机偏微分方程的高效数值方法和算法

• 批准号: 2012414
• 负责人: Xiaobing Feng
• 金额: $ 27.5万
• 依托单位: University of Tennessee Knoxville
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-08-01 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2012414&HistoricalAwards=false
• 关键词: Efficient; Numerical; Methods; Algorithms; Nonlinear

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.

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效应。 包含不确定性的一类重要模型是随机和/或随机偏微分方程（SPDE）。该研究项目将解决几个重要的SPDE，旨在开发稳定，准确和高效的改进数值方法，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预计将提供急需的工具，从许多科学，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的SPDE数学描述的系统的计算建模。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工业界成功职业所需的应用和计算数学知识和技能。该研究项目为一般非线性随机和/或随机偏微分方程（R/SPDEs）开发先进的数值方法和算法。目前解决R/SPDE的方法在大规模上面临相当大的挑战：此类系统中涉及的计算量庞大，无法使用高空间和时间分辨率，并且通常不考虑求解器优化。与此同时，R/SPDE变得更加复杂，因为考虑了额外的非线性和噪声源。这对数值PDE社区来说是一个巨大的挑战，但也是一个巨大的机会。该项目的重点是开发有效的数值方法和算法，用于解决各种科学和工程应用中出现的非线性SPDE，包括随机Allen-Cahn和Cahn-Hilliard方程以及随机非线性波和薛定谔方程。正在开发的数值方法将致力于网格尺寸和物理参数的稳定性、结构保持特性以及快速和并行实施的适应性。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估而被认为值得支持。

项目成果

A new theory of fractional differential calculus

• DOI: 10.1142/s0219530521500019
• 发表时间: 2020-07
• 期刊: arXiv: Functional Analysis
• 作者: Xiaobing H. Feng;Mitchell Sutton

An Efficient Iterative Method for Solving Parameter-Dependent and Random Convection–Diffusion Problems

解决参数相关和随机对流扩散问题的有效迭代方法

• DOI: 10.1007/s10915-021-01737-z
• 发表时间: 2021-05
• 期刊: Journal of Scientific Computing
• 影响因子: 2.5
• 作者: Xiaobing Feng;Yan Luo;Liet Vo;Zhu Wang
• 通讯作者: Zhu Wang

Analysis of Fully Discrete Mixed Finite Element Methods for Time-dependent Stochastic Stokes Equations with Multiplicative Noise

具有乘性噪声的时变随机斯托克斯方程的全离散混合有限元方法分析

• DOI: 10.1007/s10915-021-01546-4
• 发表时间: 2021
• 期刊: Journal of Scientific Computing
• 影响因子: 2.5
• 作者: Feng, Xiaobing;Qiu, Hailong
• 通讯作者: Qiu, Hailong

Stable numerical methods for a stochastic nonlinear Schrödinger equation with linear multiplicative noise

具有线性乘性噪声的随机非线性薛定谔方程的稳定数值方法

• DOI: 10.3934/dcdss.2021071
• 发表时间: 2022
• 期刊: Discrete & Continuous Dynamical Systems - S
• 作者: Feng, Xiaobing;Ma, Shu
• 通讯作者: Ma, Shu

Strong Convergence of a Fully Discrete Finite Element Method for a Class of Semilinear Stochastic Partial Differential Equations with Multiplicative Noise

一类带有乘性噪声的半线性随机偏微分方程的全离散有限元方法的强收敛性

• DOI: 10.4208/jcm.2003-m2019-0250
• 发表时间: 2021
• 期刊: Journal of Computational Mathematics
• 影响因子: 0.9
• 作者: sci, global