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

全非线性二阶椭圆和抛物线 Monge-Ampere 和 Hamilton-Jacobi-Bellman 方程的新颖数值方法

• 批准号: 1620168
• 负责人: Xiaobing Feng
• 金额: $ 27万
• 依托单位: University of Tennessee Knoxville
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2021-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1620168&HistoricalAwards=false
• 关键词: Novel; numerical; methods; fully; nonlinear

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.

完全非线性的二阶椭圆形偏微分方程（PDE）来自许多科学和工程应用，例如差异几何，天线设计，天体物理学，地球物理流体动力学，图像处理，数学融资，最佳质量传输和随机最佳控制。这些PDE是在分析和数值方面进行研究的最困难的PDE。两种主要和不同类别的完全非线性二阶PDE通常是由应用程序引起的，即Monge-Ampere（MA）型PDE和Hamilton-Jacobi-Bellman（HJB）型PDES。它们具有非常不同的结构，并由不同的应用程序字段产生。但是，PI的研究小组最近发现这两类PDE是密切相关的。这一发现为HJB型PDES的相对富裕的数值方法和技术打开了一扇门，以解决MA型PDE，并有可能在这两个完全非线性PDE的两个主要类别之间弥合数值方法的差距。它还对两种完全非线性PDE类别的现有数值方法的优势和劣势提供了更深入的了解。该研究项目的教育部分是吸引和培训两名研究生，以开发必要的应用和计算数学知识和技能，以便他们可以在不久的将来从事学术界或行业的成功职业。在该项目中，PI将开发有效的MA-TYPE和HJB类型的数值方法。 PI将在该项目中实现以下目标：（1）为一般MA型方程式建立同等（粘度意义上的）HJB改制，尤其是为MA-TYPE PDE，用于最佳的质量传输和抛物线寄生虫MA型PDES； （2）系统地开发高级半拉格朗日方法和框架，该方法和框架的优点是HJB-type和Ma-type PDE的宽模式有限差异方法以及非结构化的三角有限元和不连续的Galerkin（DG）方法。 （3）基于一些新的和广义的数值单调性概念，为HJB型和MA型完全非线性PDE开发了收敛的窄模式差异，有限元和DG方法和框架完全非线性PDE； （4）通过考虑和开发用于随机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最佳的质量传输，半充实流动和由数学金融的随机最佳控制引起的完全非线性PDE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完全非线性的PDE方法和理论，该项目将对数值完全非线性PDE，计算和应用数学的新兴领域产生重大的理论和实际影响。新的数值技术可用于解决各种由差异几何形状，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，最佳质量传输和随机最佳控制引起的各种完全非线性PDE问题。