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Full-Text Articles in Non-linear Dynamics
Lecture 09: Hierarchically Low Rank And Kronecker Methods, Rio Yokota
Lecture 09: Hierarchically Low Rank And Kronecker Methods, Rio Yokota
Mathematical Sciences Spring Lecture Series
Exploiting structures of matrices goes beyond identifying their non-zero patterns. In many cases, dense full-rank matrices have low-rank submatrices that can be exploited to construct fast approximate algorithms. In other cases, dense matrices can be decomposed into Kronecker factors that are much smaller than the original matrix. Sparsity is a consequence of the connectivity of the underlying geometry (mesh, graph, interaction list, etc.), whereas the rank-deficiency of submatrices is closely related to the distance within this underlying geometry. For high dimensional geometry encountered in data science applications, the curse of dimensionality poses a challenge for rank-structured approaches. On the other …
Lecture 07: Nonlinear Preconditioning Methods And Applications, Xiao-Chuan Cai
Lecture 07: Nonlinear Preconditioning Methods And Applications, Xiao-Chuan Cai
Mathematical Sciences Spring Lecture Series
We consider solving system of nonlinear algebraic equations arising from the discretization of partial differential equations. Inexact Newton is a popular technique for such problems. When the nonlinearities in the system are well-balanced, Newton's method works well, but when a small number of nonlinear functions in the system are much more nonlinear than the others, Newton may converge slowly or even stagnate. In such a situation, we introduce some nonlinear preconditioners to balance the nonlinearities in the system. The preconditioners are often constructed using a combination of some domain decomposition methods and nonlinear elimination methods. For the nonlinearly preconditioned problem, …
Lecture 02: Tile Low-Rank Methods And Applications (W/Review), David Keyes
Lecture 02: Tile Low-Rank Methods And Applications (W/Review), David Keyes
Mathematical Sciences Spring Lecture Series
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 …