Keywords. multiscale numerics, advection-dominated problems, finite elements, Galerkin methods, inverse problems, basis reconstruction, numerics of atmosphere and ocean, Lagrangian fluid dynamics, semi-Lagrangian methods, data-driven simulation, finite element exterior calculus, stability, deep neural nets
Deep Neural Nets in Upscaling Processes (joint with Yumeng Chen)
Machine learning techniques such as neural nets are not new. Indeed they are known since the 1950s but have become quite popular recently in computer vision around 2011/12 since neural nets with many layers (deep neural networks – DNN) triggered a large performance gain of around 30% to 40 % in, for example, recognition tasks. The strength of DNNs is that they allow an approximation of many functions provided the net is well-designed.
We explore the usage of DNNs to learn corrector operators that allow to correct the otherwise wrong system matrices for the simulation of multiscale processes (think of climate) on coarse grids.
Simulation of Urban Climate and Flows over Canopies (joint with Heena Patel)
Simulating flows in urban areas is important since this is the habitat in which we feel climate and climate change directly as a consequence on our daily life. Urban areas are characterized by rough surfaces that make it difficult to resolve all effects on flows on numerical meshes.
We plan to simulate canopy flows using our newly developed multiscale methods for avection dominated flows. First for locally emitted and transported trace gases such as NOx and later for the driving flows themselves.
Data-driven Multiscale Methods for Advection-dominated Problems
My goal is to improve the mathematical consistency of coupling different scales for transient systems with advection-dominated behavior as encountered in climate simulations. Current multiscale methods have problems with advection-dominated systems since these break essential assumptions on locality. The goal is to come up with a new framework of methods that retains this locality and yet does not loose any of the power of “standard” multiscale methods (which themselves are still subject to current research). It is our believe that this can be achieved by “doing something physical”, i.e., a numerical method connecting multiple scales must take into account (possibly different) physical behavior on all of its scales.
Stable Mixed Multiscale Elements in Various Spaces
We investigate the applicability to systems of multiscale PDEs. The problem here is that one usually has many different quantities that live in different function spaces like H(curl) or H(div). Coming up with stable pairings of finite elements for such problems is far from trivial. We combine ideas from differential geometry and multiscale finite elements to construct stable pairings by constructing a multiscale de Rham complex.
Long-Term Research Goals
- A consistent data-driven framework of Galerkin methods (finite elements, discontinuous Galerkin, etc.) for complex advection-dominated physical systems
- A consistent and stable framework for scalar and vector multiscale elements
- An improved understanding of (numerical) scale interactions
- Implementing the method on near-spherical geometries
- Applying these methods to real world problems (dynamical cores of earth system models, urban climate etc)