Adaptive mesh refinement is a crucial component for accurately solving partial differential equations numerically, particularly in the case of nonlinear behaviors such as shocks and singularities. One method for providing mesh adaptation is r-adaptivity, in which a fixed number of mesh points is redistributed within the domain toward high interest areas. Radaptive meshes can be generated by solving an optimal transport problem, whose solution gives a transportation plan for distributing a probability measure from one location to another with minimal cost. The optimal transport map can be calculated by solving the Monge-Ampére equation, a fully nonlinear partial differential equation. A limiting factor of using optimal transport to generate meshes for large scale problems is the computational burden of solving an additional partial differential equation for the mesh. In this presentation we describe several applications of optimal transport-based mesh refinement in science and engineering, starting with finite difference approximations of the Monge-Ampére equation applied to microelectromechanical systems. Next, we discuss a low-order mixed finite element approach to solving the Monge-Ampére equation, whose fast solution hinges on using an optimized nonlinear solver. We will show how our low order method scales up straightforwardly and outperforms previous methods. We conclude with several examples of mesh adaptivity using the Monge-Ampére equation and discuss potential applications for the methods, including image analysis and climate modeling.

Zoom Link: https://argonne.zoomgov.com/j/1609454120