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

完全非线性二阶椭圆偏微分方程（PDEs）出现在许多科学和工程应用中，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。这些偏微分方程是最难分析研究和数值求解的偏微分方程之一。应用中经常出现两种主要的完全非线性二阶偏微分方程，即monge - amere （MA）型偏微分方程和Hamilton-Jacobi-Bellman （HJB）型偏微分方程。它们具有非常不同的结构，并且产生于不同的应用领域。然而，PI研究小组最近的一项发现发现，这两类pde是密切相关的。这一发现为利用和调整相对丰富的hpb型偏微分方程的数值方法和技术来求解ma型偏微分方程打开了一扇门，并为弥合这两大类全非线性偏微分方程在数值方法上的差距提供了可能。它还提供了对两类全非线性偏微分方程的现有数值方法的优缺点有更深的了解。该研究项目的教育部分是吸引和培训两名研究生发展必要的应用和计算数学知识和技能，以便他们能够在不久的将来在学术界或工业界追求成功的职业生涯。在这个项目中，PI将开发ma型和hjb型全非线性偏微分方程的有效数值方法。PI将在本项目中实现以下目标：(1)建立一般ma型方程的等效（在粘度意义上）hjb -重新公式，特别是从最优质量输运的ma型偏微分方程和抛物型ma型偏微分方程；(2)利用宽模板有限差分法、非结构三角形有限元和不连续伽辽金（DG）方法的优势，系统地建立了hpb型和ma型偏微分方程的高阶半拉格朗日方法和框架。(3)基于一些新的广义数值单调性概念，发展了hjb型和ma型全非线性偏微分方程的收敛窄模板有限差分、有限元和DG方法和框架；(4)考虑并发展随机ma型和hjb型偏微分方程的有效数值方法，将不确定性纳入全非线性偏微分方程模型；(5)将预期的数值方法应用于数学金融学中由最优质量传递、半转流和随机最优控制引起的全非线性PDE应用问题。通过解决具有挑战性的数值偏微分方程问题，建立基本的数值全非线性偏微分方程方法和理论，本项目将对数值全非线性偏微分方程的新兴领域以及计算和应用数学产生重大的理论和实践影响。新的数值技术可用于解决微分几何、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、最优质量传递和随机最优控制等领域的各种全非线性PDE问题。

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.