Seminar| Institute of Mathematical Sciences
Time: Wednesday, June 19th, 2024 , 14:30-15:30
Speaker: Difeng Cai, Southern Methodist University
Abstract: Kernel matrices associated with a non-local kernel arise frequently in many applications, from integral equations, fluid dynamics, to statistics and machine learning. Since such matrices are dense, they provide grand computational challenges for large scale data due to the quadratic or cubic complexity for matrix operations such as multiplication, inversion, eigendecomposition, etc. In this talk, we focus on hierarchical matrix techniques to resolve the computational bottleneck. We introduce a data-driven approach to construct the hierarchical representation of the kernel matrix with linear complexity. Theoretical insights are provided for the data-driven construction. Compared to existing hierarchical techniques, the data-driven approach is able to not only achieve better efficiency, but also scale to higher dimensions. We present numerical experiments to demonstrate the advantages over existing methods and the state-of-the-art packages. Most notably, the significantly improved robustness to the well-known curse of dimensionality can be seen in the experiment.