Some Aspects of Numerical Methods for Kinetic Equations and High Dimensional Scientific Computing: III

In those introductory lectures, I plan to cover some aspects of numerical methods for kinetic equations and dimension reduction techniques. Designing computational methods for kinetic problems presents significant challenges, primarily due to the curse of dimensionality and the need to accurately capture physical laws across multiple scales. While there are mature numerical methods available, the field continues to evolve rapidly.

We will focus on selected topics in kinetic simulations in the talks. In the first part,  we focus on high order deterministic solvers. In particular, we will review the discontinuous Galerkin (DG) finite element method, which is a well-known technique for computing PDEs. It is especially suited for designing structure-preserving deterministic methods for transport dominated problems. In the second part, we will present some work on the design of sparse grid DG methods for computing high dimensional differential equations. We show incorporating ideas of sparse grid and multiresolution can help alleviate the curse of dimensions. In the final part, we present several approaches of dimension reduction techniques based on low-rank methods, reduced basis and machine learning.

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