Novel numerical methods for fully nonlinear second order elliptic and parabolic Monge-Ampere and Hamilton-Jacobi-Bellman equations

完全非线性的二阶椭圆型偏微分方程组(PDE)广泛地应用于许多科学和工程领域，如微分几何、天线设计、天体物理学、地球物理流体动力学、图像处理、数学金融、最优质量传输和随机最优控制等。这些偏微分方程组是最难解析研究和数值求解的偏微分方程组之一。两类主要且截然不同的完全非线性二阶偏微分方程组经常出现在应用中，即Monge-Ampere(MA)型和Hamilton-Jacobi-Bellman(HJB)型偏微分方程组。它们具有非常不同的结构，来自不同的应用领域。然而，PI的研究团队最近的一项发现发现，这两类PDE是密切相关的。这一发现为利用和采用相对丰富的HJB型偏微分方程组的数值方法和技巧来求解MA型偏微分方程组打开了大门，并为弥合这两大类完全非线性偏微分方程组在数值方法上的差距提供了可能。同时也加深了对这两类完全非线性偏微分方程组现有数值方法的优缺点的理解。这项研究项目的教育部分是让两名研究生发展必要的应用和计算数学知识和技能，以便他们在不久的将来能够在学术界或工业界获得成功的职业生涯。在这个项目中，PI将为MA类型和HJB类型的完全非线性偏微分方程组开发有效的数值方法。PI将在本项目中实现以下目标：(1)建立一般MA类型方程的等价(在粘性意义上)HJB-重构式，特别是对于来自最优质量输运的MA类型偏微分方程组和抛物型MA类型偏微分方程组；(2)系统地发展高阶半拉格朗日方法和框架，该方法和框架利用宽模板有限差分方法和非结构三角有限元和间断Galerkin(DG)方法的优点，用于HJB类型和MA类型的偏微分方程组。(3)基于一些新的和广义的数值单调性概念，发展收敛的窄模板有限差分、有限元和DG方法和框架，用于HJB型和MA型完全非线性偏微分方程组；(4)通过考虑和发展随机MA型和HJB型偏微分方程组的有效数值方法，将不确定性融入到完全非线性偏微分方程组模型中；(5)将预期的数值方法应用于从数学金融学的最优质量传输、半地转流和随机最优控制中产生的完全非线性偏微分方程应用问题。通过解决具有挑战性的数值偏微分方程组问题，建立基本的数值完全非线性偏微分方程组方法和理论，该项目将对新兴的数值完全非线性偏微分方程组领域以及整个计算和应用数学产生重大的理论和实践影响。新的数值方法可用于求解各种完全非线性的偏微分方程组问题，这些问题涉及到微分几何、天线设计、天体物理学、地球物理流体动力学、图像处理、数学金融、最优质量传输和随机最优控制等领域。

Fully nonlinear second order elliptic partial differential equations (PDEs) arise from many scientific and engineering applications such as differential geometry, antenna design, astrophysics, geophysical fluid dynamics, image processing, mathematical finance, optimal mass transport, and stochastic optimal control. These PDEs are among most difficult PDEs to study analytically and to solve numerically. Two major and distinct classes of fully nonlinear second order PDEs often arise from applications, namely, the Monge-Ampere (MA) type PDEs and the Hamilton-Jacobi-Bellman (HJB) type PDEs. They have very different structures and arise from distinct application fields. However, a recent discovery by the PI's research team finds that these two classes of PDEs are intimately related. This finding opens a door for utilizing and adapting the relatively wealthy numerical methods and techniques for HJB-type PDEs to solve MA-type PDEs and enables a possibility for bridging the gap on numerical methods between those two major classes of fully nonlinear PDEs. It also provides a deeper understanding about the strength and weakness of the existing numerical methods for both classes of fully nonlinear PDEs. The education component of this research project is to engage and train two graduate students in developing necessary applied and computational mathematics knowledge and skills so that they can pursue a successful career in either academia or industry in the near future.In this project, the PI will develop efficient numerical methods for both MA-type and HJB-type fully nonlinear PDEs. The PI will achieve the following goals in this project: (1) to establish equivalent (in the viscosity sense) HJB-reformulations for general MA-type equations, in particular, for the MA-type PDEs from optimal mass transport and for parabolic MA-type PDEs; (2) to systematically develop a high order semi-Lagrangian methodology and framework, which take the advantages of wide-stencil finite difference methods and unstructured triangular finite element and discontinuous Galerkin (DG) methods, for HJB-type and MA-type PDEs. (3) to develop convergent narrow-stencil finite difference, finite element and DG methods and framework for HJB-type and MA-type fully nonlinear PDEs based on some new and generalized numerical monotonicity concept; (4) to incorporate uncertainty into fully nonlinear PDE models by considering and developing efficient numerical methods for stochastic MA-type and HJB-type PDEs; (5) to apply the anticipated numerical methods to fully nonlinear PDE application problems arising from optimal mass transport, semigeostrophic flow, and stochastic optimal control from mathematical finance. By addressing the challenging numerical PDE problems and establishing fundamental numerical fully nonlinear PDE methodologies and theories, this project will have a significant theoretical and practical impact to the emerging field of numerical fully nonlinear PDEs and to computational and applied mathematics at large. The new numerical techniques can be used to solve various fully nonlinear PDE problems arising from differential geometry, antenna design, astrophysics, geophysical fluid dynamics, image processing, mathematical finance, optimal mass transport, and stochastic optimal control.