Novel Numerical Methods for Nonlinear Stochastic PDEs and High Dimensional Computation

许多科学、工程和工业应用都涉及随机效应。为了从生物、工程和物理应用中开发出更准确、更健壮的数学模型，将不确定性纳入数学模型变得不可或缺，这使得考虑随机偏微分方程变得必要，因为它们是应用中最常见的随机模型。为了寻求这些方程的解，需要准确、高效和健壮的计算方法和算法，对这种方法和算法的需求从未像现在这样大。现有的求解随机偏微分方程组的方法在大范围内面临着相当大的挑战，同时随着非线性和噪声的增加，这类随机模型变得更加复杂。现有的数值方法不能有效地解决这些问题，这反过来又需要新的思想和方法。随着数学金融、图像处理、经济学和数据科学等非传统应用科学的迅速发展，人们对计算积分和求解偏微分方程等具有挑战性的高维问题的高效数值方法的需求越来越大。传统的基于网格的方法受到臭名昭著的维度诅咒的阻碍，在这种情况下，求解问题所需的计算量在维度上呈指数级增长。这项研究项目的主要目标是开发新的和有效的数值方法来应对这些挑战。该项目由两个组成部分组成。第一部分着重于发展和分析求解几个非线性随机偏微分方程组的有效数值方法，这些偏微分方程组来自于各种科学和工程应用，如材料科学、流体和量子力学以及最优控制。第二部分将致力于开发用于高维计算的新的数值方法，重点是高维数值积分和高维偏微分方程组的问题。这个项目的远大愿景是开发一个框架，用于构建和分析一般非线性随机偏微分方程组的数值方法，并开发新的途径和使能方法，以克服高维计算的维度灾难挑战。该项目将包括对研究生的培训。该研究项目开发了用于非线性随机偏微分方程组和高维计算的高级数值方法。这样一个及时和先进的项目引起了STEM社区的极大兴趣，因为预期的数值方法和算法将提供亟需的使能工具，用于解决由随机偏微分方程数学描述的挑战性问题，或涉及许多科学、工程和工业应用以及人工智能、机器学习和数据科学的高维计算。这一研究项目也有望对数值随机偏微分方程组和高维计算的发展产生持久的影响。此外，该项目将提供一个宝贵的机会和资源来培训博士研究生，并帮助他们发展必要的应用数学和计算数学以及人工智能、机器学习和数据科学的知识和技能，以便他们能够在不久的将来在学术界或行业追求成功的职业生涯。该奖项反映了NSF的法定使命，并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。

Many scientific, engineering, and industrial applications involve random effects. Incorporating uncertainties into mathematical models becomes indispensable in order to develop more accurate and robust mathematical models from biological, engineering, and physical applications, which makes it necessary to consider stochastic partial differential equations as they are the most commonly encountered stochastic models from applications. To seek solutions of those equations require accurate, efficient, and robust computational methods and algorithms, the demand for such methods and algorithms has never been greater. The current approaches for solving stochastic partial differential equations face considerable challenges at large scales, in the meantime, such stochastic models become more complicated as more difficult nonlinearity and noise are considered. The existing numerical approaches are not efficient to solve those problems, which in turn calls for new ideas and approaches. With rapid developments in nontraditional applied sciences such as mathematical finance, image processing, economics, and data science, there is an ever-increasing demand for efficient numerical methods for solving challenging high-dimensional problems such as computing integration and solving partial differential equations. The traditional grid-based methods are hampered by the infamous curse of dimensionality in which the amount of required computations for solving a problem grows exponentially in the dimension. The primary goal of this research project is to develop novel and efficient numerical methods to address those challenges. The project consists of two integral parts. Part I focuses on developing and analyzing efficient numerical methods for solving several nonlinear stochastic partial differential equations which arise from various scientific and engineering applications such as materials science, fluid and quantum mechanics, and optimal control. Part II will be devoted to developing novel numerical methods for high-dimensional computation with a focus on problems of high-dimensional numerical integration and high-dimensional partial differential equations. The overreaching vision of this project is to develop a framework for constructing and analyzing numerical methods for general nonlinear stochastic partial differential equations and to develop new approaches and enabling methods for overcoming the curse of dimensionality challenge for high-dimensional computation. The project will include training of graduate students.This research project develops advanced numerical methods for nonlinear stochastic partial differential equations and high-dimensional computation. Such a timely and advanced project is of great interest to the STEM community as the anticipated numerical methods and algorithms will provide much-needed enabling tools for tackling challenging problems described mathematically by stochastic partial differential equations or involved with high-dimensional computation from many scientific, engineering, and industrial applications as well as AI, machine learning, and data science. This research project is also expected to have a lasting impact on the advancement of numerical stochastic partial differential equations and high-dimensional computation. Moreover, the project will provide a valuable opportunity and resource to train Ph.D. graduate students and to help them to develop necessary applied and computational mathematics as well as AI, machine learning, and data science knowledge and skills so that they can pursue successful careers in either academia or industry in the near future.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.