Discretization of Fully Nonlinear Elliptic PDEs

Max Jensen (homepage)

28 February - 3 March 2020

Fully nonlinear PDEs arise in optimal control, optimal transport and many other mathematical fields. Their numerical discretization is far from trivial. It requires an understanding of the PDE theory as well as of results from numerical analysis, which are not part of standard computational PDE courses.

The aim of this mini-course, which was initially presented at a Graduate School at the IISc in Bangalore, is to give an introduction for MSc and PhD students from mathematics and related fields.

Part 1: A Grand Tour of Fully Nonlinear Partial Differential Equations

At the beginning of this course we want to understand which applications lead to fully nonlinear equations and how should we interpret solutions:

Where do fully nonlinear PDEs arise?
Why are Bellman equations used in optimal control?
What are Monge-Ampère equations?
What are viscosity solutions and why are they an important concept?

In the second video of the first lecture we examine the tools to ensure existence and uniqueness:

How can comparison principles establish well-posedness?
What are second-order jets and why do we want to consider their closure?
How do viscosity solutions satisfy boundary conditions?

Links to key references: (Yong, Zhou; 1999), (Fleming, Soner; 2006), (Crandall, Ishii, Lions; 1992)

Part 2: Principles of Monotone Discretizations

The second lecture introduces key theorems which shape the design of numerical methods.

How should we formulate properties such as consistency, stability or monotonicity?
How does the Barles-Souganidis theorem prove convergence of numerical methods?
Barles and Souganidis used a different notion of viscosity solutions. How serious is this?

The consequences of the monotonicity assumption reach beyond the convergence proof:

What does the Motzkin-Wasow theorem imply for the design of numerical methods?
Why does monotonicity lead to low-order methods?
How can the existence of a unique numerical solution be verified with a global Newton method?

Links to key references: (Feng, Glowinski, Neilan; 2013), (J, Smears; 2018), (Barles, Perthame; 1988), (Ishii; 1989), (Barles, Souganidis; 1991), (Crandall, Ishii, Lions; 1992), (Crandall, Lions; 1996), (Wesseling; 2001), (Bokanowski, Maroso, Zidani; 2009)

Part 3: Monotone Semi-Lagrangian Methods

In the third lecture we finally construct concrete numerical schemes, namely the widely used semi-Lagrangian methods.

How may we combine the Eulerian and Lagrangian viewpoint to define a numerical discretization?
Why can Fokker-Planck and Kolmogorov equations guide the discretisation of second-order terms?
How does one obtain consistency in a unified manner?

While monotonicity and consistency can be introduced in some generality, many steps of a convergence analysis require detailed understanding of the concrete boundary value problem. The second half of the third lecture is a case study of a Monge-Ampère equation.

What is the relationship between Monge-Ampère equations and Hamilton-Jacobi-Bellman equations?
How can convexity be imposed in the setting of viscosity solutions?
Why can numerical schemes become inconsistent near the domain boundary and how can one control the resulting error?

Links to key references: (Kushner, Dupuis; 1992), (Falcone, Ferretti; 2013), (Debrabant, Jakobsen; 2013), (Feng, J; 2017), (Krylov; 1987), (Krylov; 2013), (J; 2018), (J, Smears; 2018), (Bokanowski, Maroso, Zidani; 2009), (Reisinger, Arto; 2017), (Nochetto, Ntogkas, Zhang; 2018), (Li, Nochetto; 2018)

Part 4: Snapshots of Alternative Numerical Schemes

In this last, much shorter lecture we discuss two further numerical methods to show other possible approaches.

What are the trade-offs when using P1 finite element methods instead of semi-Lagrangian schemes?
How can Monte-Carlo methods be used for high-dimensional equations?

We conclude the course with a short review.

Links to key references: (J, Smears; 2013), (J; 2017), (J, Majee, Prohl, Schellnegger; 2019), (Falcone, Ferretti; 2013), (Feng, Glowinski, Neilan; 2013), (Neilan, Salgado, Zhang; 2017)