Linear algebra, data sparsity, factorization, exponential kernels, hierarchical matrix, optimality, complexity, algorithms
As simulation and analytics enter the exascale era, numerical algorithms, particularly implicit solvers that couple vast numbers of degrees of freedom, must span a widening gap between ambitious applications and austere architectures to support them. We present fifteen universals for researchers in scalable solvers: imperatives from computer architecture that scalable solvers must respect, strategies towards achieving them that are currently well established, and additional strategies currently being developed for an effective and efficient exascale software ecosystem. We consider recent generalizations of what it means to “solve” a computational problem, which suggest that we have often been “oversolving” them at the smaller scales of the past because we could afford to do so. We present innovations that allow to approach lin-log complexity in storage and operation count in many important algorithmic kernels and thus create an opportunity for full applications with optimal scalability.
Keyes, D. (2021). Lecture 03: Hierarchically Low Rank Methods and Applications. Mathematical Sciences Spring Lecture Series. Retrieved from https://scholarworks.uark.edu/mascsls/3