In this presentation, we discuss a few basic questions for PDE learning from observed solution data. Using various types of PDEs as examples, we show (1) how large the data space spanned by all snapshots along a solution trajectory is, (2) if one can construct an arbitrary solution by superposition of snapshots of a single solution, and (3) identifiability of a differential operator from a single solution data on local patches.

Adaptive mesh refinement (AMR) is one of several techniques for adapting the spatial resolution of a simulation in particular regions of the spatial domain. Block-structured AMR specifically refines the mesh by defining locally structured regions with finer spatial, and possibly temporal, resolution. This combination of locally structured meshes within an irregular global hierarchy is in some sense the best of both worlds in that it enables regular local data access while enabling greater flexibility in the overall computation.

Originally, block-structured AMR was designed for solving hyperbolic conservation laws with explicit time-stepping; in this case the changes to solution methodology in transforming a single-level solver to an AMR-based solver are relatively straightforward. AMR has come a long way, however, and the more complex the simulation, the more complex the changes to effectively use AMR. One can even consider whether to use different physical models at different levels of resolution. In this talk I will give an overview of block-structured AMR and focus on a few key exemplars for how to think about adaptivity for multiphysics simulations.

Over the last 40 years there have been great advances in computer hardware, solvers (methods for solving Ax=b), meshing algorithms, time stepping methods, adaptivity and so on. Yet accurate prediction of fluid motion (for settings where this is needed) is still elusive. This talk will review three major hurdles that remain: ensemble simulations, time accuracy and model stagnation. Three recent ideas where numerical analysis can help push forward the boundary between what can be done and what can't be done will be described. This talk is based on joint work with Catalin Trenchea, Ming Chen, Michael McLaughlin, Kiera Kean, Wenlong Pei, Nan Jiang, Joe Fiordelino, Ali Pakzad, Victor DeCaria, Yao Rong, Yi Li, Li Qin, Haiyun Zhao, Yong Li, Ahmet Guzel, Songul Kaya-Merdan, Michael Schneier and ….