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Ordinary Differential Equations and Applied Dynamics Commons™
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Articles 1 - 6 of 6
Full-Text Articles in Ordinary Differential Equations and Applied Dynamics
Lecture 14: Randomized Algorithms For Least Squares Problems, Ilse C.F. Ipsen
Lecture 14: Randomized Algorithms For Least Squares Problems, Ilse C.F. Ipsen
Mathematical Sciences Spring Lecture Series
The emergence of massive data sets, over the past twenty or so years, has lead to the development of Randomized Numerical Linear Algebra. Randomized matrix algorithms perform random sketching and sampling of rows or columns, in order to reduce the problem dimension or compute low-rank approximations. We review randomized algorithms for the solution of least squares/regression problems, based on row sketching from the left, or column sketching from the right. These algorithms tend to be efficient and accurate on matrices that have many more rows than columns. We present probabilistic bounds for the amount of sampling required to achieve a …
Lecture 13: A Low-Rank Factorization Framework For Building Scalable Algebraic Solvers And Preconditioners, X. Sherry Li
Lecture 13: A Low-Rank Factorization Framework For Building Scalable Algebraic Solvers And Preconditioners, X. Sherry Li
Mathematical Sciences Spring Lecture Series
Factorization based preconditioning algorithms, most notably incomplete LU (ILU) factorization, have been shown to be robust and applicable to wide ranges of problems. However, traditional ILU algorithms are not amenable to scalable implementation. In recent years, we have seen a lot of investigations using low-rank compression techniques to build approximate factorizations.
A key to achieving lower complexity is the use of hierarchical matrix algebra, stemming from the H-matrix research. In addition, the multilevel algorithm paradigm provides a good vehicle for a scalable implementation. The goal of this lecture is to give an overview of the various hierarchical matrix formats, such …
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 …
Lecture 11: The Road To Exascale And Legacy Software For Dense Linear Algebra, Jack Dongarra
Lecture 11: The Road To Exascale And Legacy Software For Dense Linear Algebra, Jack Dongarra
Mathematical Sciences Spring Lecture Series
In this talk, we will look at the current state of high performance computing and look at the next stage of extreme computing. With extreme computing, there will be fundamental changes in the character of floating point arithmetic and data movement. In this talk, we will look at how extreme-scale computing has caused algorithm and software developers to change their way of thinking on implementing and program-specific applications.
Lecture 01: Scalable Solvers: Universals And Innovations, David Keyes
Lecture 01: Scalable Solvers: Universals And Innovations, 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 …