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

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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.