numerical analysis

Monday, October 27, 2008 - 12:15pm - 1:05pm
Partha Niyogi (University of Chicago)
Increasingly, we face machine learning problems in very high
dimensional spaces. We proceed with the intuition that although
natural data lives in very high dimensions, they have relatively few
degrees of freedom. One way to formalize this intuition is to model
the data as lying on or near a low dimensional manifold embedded in
the high dimensional space. This point of view leads to a new class of
algorithms that are manifold motivated and a new set of theoretical
questions that surround their analysis. A central construction in
Wednesday, October 22, 2014 - 3:40pm - 4:25pm
Jinchao Xu (The Pennsylvania State University)
In this talk, I will report some recent results on the qualitative and numerical analysis of structure-preserving discretizations and robust preconditioners for several mathematical models that couple with the Navier-Stokes equations. I will study the Navier-Stokes equations coupled with elasticity equations, the Maxwell's equations, and the Poisson-Nernst-Planck system. These systems model fluid-structure interaction, magnetohydrodynamics, and electrokinetic phenomena, respectively, and are widely used in engineering applications and other scientific disciplines.
Monday, October 28, 2013 - 11:30am - 12:20pm
Douglas Arnold (University of Minnesota, Twin Cities)
This talk will discuss a substantial interplay of algebraic topology with numerical analysis which has developed over the last decade. During this period, de Rham cohomology and the Hodge theory of Riemannian manifolds have come to play a crucial role in the development and understanding of computational algorithms for the solution of problems in partial differential equations.
Monday, October 28, 2013 - 3:15pm - 4:05pm
Ari Stern (Washington University)
In the numerical analysis of elliptic PDEs, much attention has been given (quite rightly) to the discretization of the Laplace operator and other second-order Laplace-type operators, e.g., the Hodge--Laplace operator on differential $k$-forms. By comparison, Dirac-type operators have received little attention from the perspective of numerical PDEs---despite being, in many ways, just as fundamental. Informally, a Dirac-type operator is a square root of some Laplace-type operator, and is therefore a first-order (rather than second-order) differential operator.
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