Room 2-449 (unless otherwise noted)
Wednesday 4:30 PM - 5:30 PM (unless otherwise noted)
The NMPDE seminar covers numerical and data-driven methods for solving differential equations and modeling physical systems. To receive seminar announcements and zoom links, please write to daisymae@math.mit.edu.
March 6 (*2pm in 2-255*): Pengning Chao (MIT Math)
A general framework for computing fundamental limits in photonics inverse design
Advances in computing power and numerical algorithms has lead to a paradigm shift in photonics engineering towards inverse design: achieving desired functional characteristics by solving a PDE-constrained optimization problem over a large number of structural parameters. This approach has been very fruitful, with increases in device performance often measured by orders of magnitude. However, the resulting structures can be highly complex, and a natural question is to what extent further improvement is possible. Unfortunately, the high-dimensional, non-convex nature of the PDE-constrained optimization precludes the exact determination of its global optimum. To address this issue, this talk presents a general framework for computing fundamental limits to photonics device performance based on a systematic convex relaxation of the original optimization. The efficacy of the framework is demonstrated on canonical problems such as maximizing scattering cross-sections and manipulating the photonic local density of states. Further extensions and improvements will also be discussed. Based on joint work with the groups of Sean Molesky and Alejandro Rodriguez.
April 3: Chris Rackauckus (Julia Computing)
TBD
May 1: Oswald Knuth (Leibniz Institute for Tropospheric Research)
Finite element and finite volume discretization of the shallow water equation on the sphere
There is an ongoing research in the design of numerical methods for numerical weather prediction. This is connected with an increase in spatial resolution and the intensive use of graphic processor units (GPU). The shallow water equation is a prototype for testing numerical methods in atmospheric and ocean sciences. I will describe different numerical schemes for solving this equation on the sphere with different grid types from triangular to fully unstructured polygonal grids. The focus lies on finite element and finite volume discretizations with a staggered or collocated arrangement of the unknowns. The implementation is done in the Julia language whereby the different grids are described within the same data structure.
In more details I will outline the implementation of a spectral continuous Galerkin method on conforming quad grids. The implementation follows the HOMME DyCore and uses the packages MPI.jl and KernelAbstractions.jl for running the code parallel on GPU’s.
Research Scientist
Graduate Student
Professor of Applied Mathematics