Recent advances on structure-preserving algorithms

Applied and Computational Mathematics Seminar
Monday, April 25, 2022 - 2:00pm for 1 hour (actually 50 minutes)
Philippe G. LeFloch – Sorbonne Univ. and CNRS – pglefloch@gmail.com
Yingjie Liu
Structure-preserving methodologies led to interesting advances on the design of computational algorithms: one observes that an (obvious or hidden) structure is enjoyed by the problem under consideration and one then designs numerical approximations enjoying the same structure at the discrete level. For problems involving a large number of dimensions, for instance in mathematical finance and machine learning, I have introduced the 'transport-based mesh-free method' which uses a reproducing kernel and a transport mapping in a way that is reminiscent of Lagrangian methods developed in computational fluid dynamics. This method is now implemented in a Python library (CodPy) and used in industrial applications. 
In compressible fluid dynamics, astrophysics, or cosmology, one needs to compute with propagating singularities, such as shock waves, moving interfaces, or gravitational singularities, I will overview recent progress on structure-preserving algorithms in presence of small-scale dependent waves which drive the global flow dynamics. I recently introduced asymptotic-preserving or dissipation-preserving methods adapted to such problems. This lecture is based on joint collaborations with F. Beyer (Dunedin), J.-M. Mercier (Paris), S. Miryusupov (Paris), and Y. Cao (Shenzhen). Blog: