Finite element methods for non-divergence form partial differential equations and the Hamilton-Jacobi-Bellman equation

科学和工程中的许多模型都是用计算方法近似求解的，有必要从理论上证明计算近似的可靠性。除了提供数值方法的合理性外，理论分析还经常为改进效率、精度和可行性的新方法的开发提供洞察力。在这个项目中，研究人员和一名研究生将构建、分析和实现在随机金融模型、随机微分博弈和其他金融和工程应用中出现的线性和非线性偏微分方程类的数值方法。该项目的总体目标是开发可使用当前计算软件实施的方法，并得出近似解的显式估计。本项目的主要目的是对一类完全非线性的Hamilton-Jacobi-Bellman(HJB)方程提出稳健的有限元离散，并发展一个全面的收敛理论。该项目由两部分组成：(1)非光滑系数无散度形式二阶椭圆型方程的有限元方法的发展；(2)HJB问题的实用有限元离散格式的构造、实现和收敛分析。这项工作将把有限元方法的数学理论扩展到数值社区中相对鲜为人知的问题。该项目的成功将对随机最优控制在金融和工程中的数学和计算应用产生广泛的影响。此外，通过对下一代计算科学家的培训、课程开发以及向数学和科学计算界的传播，可以感受到该项目的影响。

Many models in the sciences and engineering are solved approximately using computational methods, and it is necessary to theoretically justify the reliability of the computed approximations. In addition to providing justification of the numerical methods, the theoretical analysis often provides insight for the development of new methods with improved efficiency, accuracy, and viability. In this project, the investigator and a graduate student will construct, analyze and implement numerical methods for classes of linear and nonlinear partial differential equations arising in stochastic financial models, stochastic differential games, and other applications in finance and engineering. The overall aim of the project is to develop methods that can be implemented using current computational software and to derive explicit estimates of the approximate solutions. The main goal of this project is to propose robust finite element discretizations for a class of fully nonlinear Hamilton-Jacobi-Bellman (HJB) equations and to develop a comprehensive convergence theory. The project consists of two integrated components: (1) The development of finite element methods for second order elliptic equation in non-divergence form with non-smooth coefficients; the building-blocks of the HJB problem, (2) The construction, implementation and convergence analysis of practical finite element discretizations for the HJB problem. The work will broaden the mathematical theory of the finite element method to problems that have been relatively untouched in the numerical community. The success of this project will have broad impacts in the mathematical and computational applications of stochastic optimal control in finance and engineering. In addition, the impacts of this project are felt though the training of the next generation of computational scientists, course development, and dissemination to the mathematical and scientific computing communities.