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

全非线性二阶椭圆偏微分方程 (PDE) 源自许多科学和工程应用，例如微分几何、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、最优质量传递和随机最优控制。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。应用中经常出现两种主要且不同类别的完全非线性二阶偏微分方程，即 Monge-Ampere (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来自不同的应用领域。然而，PI 研究小组最近的一项发现发现，这两类偏微分方程密切相关。这一发现为利用和调整 HJB 型偏微分方程相对丰富的数值方法和技术来求解 MA 型偏微分方程打开了一扇大门，并为弥合这两类主要的完全非线性偏微分方程之间数值方法的差距提供了可能。它还可以更深入地了解两类完全非线性偏微分方程的现有数值方法的优缺点。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。在该项目中，PI 将为 MA 型和 HJB 型全非线性偏微分方程开发有效的数值方法。 PI 将在该项目中实现以下目标：（1）为一般 MA 型方程建立等效（在粘度意义上）HJB 重构，特别是针对最优传质的 MA 型偏微分方程和抛物线 MA 型偏微分方程； (2) 系统地开发了适用于 HJB 型和 MA 型偏微分方程的高阶半拉格朗日方法和框架，该方法和框架利用了宽模板有限差分法、非结构三角有限元和间断伽辽金 (DG) 方法的优点。 (3) 基于一些新的广义数值单调性概念，开发收敛窄模板有限差分、有限元和DG方法以及HJB型和MA型全非线性偏微分方程的框架； (4) 通过考虑和开发随机 MA 型和 HJB 型 PDE 的有效数值方法，将不确定性纳入完全非线性 PDE 模型； (5) 将预期的数值方法应用于数学金融中的最优质量传输、半地转流和随机最优控制所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 PDE 问题并建立基本的数值全非线性 PDE 方法和理论，该项目将对数值全非线性 PDE 的新兴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。新的数值技术可用于解决由微分几何、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引起的各种完全非线性偏微分方程问题。

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.