Past Seminars: Spring 2014
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February 27, 2014
Dongbin Xiu, University of Utah
Multi-dimensional polynomial interpolation on arbitrary nodes
Abstract:
Polynomial interpolation is well understood on the real line. In multi-dimensional spaces, one often adopts a well-established one-dimensional method and fills up the space using certain tensor product rule. Examples like this include full tensor construction and sparse grids construction. This approach typically results in fast growth of the total number of interpolation nodes and certain fixed geometrical structure of the nodal sets. This imposes difficulties for practical applications, where obtaining function values at a large number of nodes is infeasible. Also, one often has function data from nodal locations that are not by "mathematical design" and are ``unstructured''.
In this talk, we present a mathematical framework for conducting polynomial interpolation in multiple dimensions using arbitrary set of unstructured nodes. The resulting method, least orthogonal interpolation, is rigorous and has a straightforward numerical implementation. It can faithfully interpolate any function data on any nodal sets, even on those that are considered degenerate by the traditional methods. We also present a strategy to choose ``optimal'' nodes that result in robust interpolation. The strategy is based on optimization of Lebesgue function and has certain highly desirable mathematical properties.
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TUESDAY March 11, 2014, 5:00pm, ROOM 6417 (Note special time, and room!)
Friedemann Schuricht, TU-Dresden
Non-smooth variational problems
Abstract:
Several problems arising in physics lead to natural lack of differentiability in the models used to describe them. Appropriate tools are needed to handle the
questions of existence of solutions in such framework. We present some fundamental ideas that allow to treat some convex functional that are not necessarily differentiable.
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March 13, 2014
Edger Sterjo, Graduate Center at CUNY
An introduction to Morse Theory and its applications
Abstract:
Given a function f defined on a manifold M, both sufficiently differentiable, what is the topological significance of the critical points of f? In one dimension, for example, the level sets of the parabola f(x) = x^2 all have 2 components, except at the critical value y=0, at which the level set is a point.
Similarly on R^2, the function f(x,y) = x^2 - y^2 has level sets which are hyperbolas, except at the critical value z=0, where the level set is the union of
two diagonals. The idea of Morse theory is to relate the critical point structure of f to the topological characteristics of it's (sub)level sets.
In all of this compactness properties play a big role. However, in the infinite dimensional setting, where our manifold is no longer locally compact,
we must make up for this lack of compactness by imposing a further condition on our function.
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March 20, 2014
Alessandro Ottazzi, CIRM, Bruno Kessler Foundation, Trento, Italy
Maps on stratified Lie groups
Abstract:
In this talk I discuss some new results concerning the study of special maps
on Carnot-Carathéodory spaces, obtained in different collaborations.
I shall begin with the definition of Carnot-Carathéodory space and that of
Carnot group. Then I describe the classes of maps in which I am interested: isometries, conformal maps, quasiconformal maps. Finally, I will briefly comment on the techniques
that are used in the proofs, with particular attention on Tanaka prolongation theory.
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March 27, 2014
Cyril Joel Batkam, University of Sherbrooke (Canada)
Multiple solutions to some differential systems with strongly indefinite variational structure
Abstract
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April 10, 2014
Michael McCourt, University of Colorado, Denver
Positive Definite Kernels, Opportunities and Challenges
Abstract:
Scattered data approximation, the process of fitting a function to given data, is an important tool in applications including spatial statistics, econometrics, machine learning, computer graphics and others. In one dimension, polynomials and splines are often used; however, when higher dimensions are considered, possibly with oddly-shaped domains, polynomials and splines can prove problematic.
Positive definite kernels (sometimes called radial basis functions) provide a mechanism for conducting mesh free approximation in higher dimensions and complicated domains without the fear of non-uniqueness that accompanies polynomials. In this talk we will introduce positive definite kernels and focus on the properties of the Gaussian, long the preferred kernel of many applications because of its spectrally accurate convergence properties. We will explain how interpolation with a Gaussian basis has the potential to outperform polynomial interpolation, even in one dimension, and also show how numerical instabilities can sabotage this superior performance. We will then describe a new technique to stably evaluate Gaussian interpolants, allowing for their theoretically optimal behavior to emerge.
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April 24, 2014
Dhanya Rajendran, Tata Institute, Bangalore (India)
Critical growth elliptic problem with singular discontinuous nonlinearity in R2
Abstract:
Elliptic problems with discontinuous nonlinearity has its own
difficulties due to the non-differentiability of the associated
functional. Hence, a generalized gradient approach developed by Chang has
been used to solve such problems if the associated functional is known to
be Lipchitz continuous. In this talk, we will consider critical elliptic
problem in a bounded domain in R^2 with the simultaneous
presence of a Heaviside type discontinuity and a power-law type
singularity and investigate the existence of multiple positive solutions.
Here discontinuity coupled with singularity does not fit into any of the
known framework and we will discuss our approach employed to obtain
multiple positive solutions.
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May 1, 2014
One day symposium in Room 4102 from 9:30am-4:00pm
Hyperbolic Conservation Laws: Recent Progress
Speakers:
Alberto Bressan, Geng Chen, Sebastian Noelle, Ronghua Pan
An event sponsored by the Initiative for the Theoretical Sciences at CUNY
(ITS)
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May 8, 2014
No seminar: Undergraduate Lecture at CSI
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May 15, 2014 (!! Meeting at 3:00pm !!)
Abbas Bahri, Rutgers University
Periodic orbits of Reeb vector fields in dimension three
Abstract:
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May 22, 2014
Mythily Ramaswamy, Tata Institute, Bangalore
Control aspects of Navier-Stokes equations
Abstract:
We recall the concepts of contrability and stabilization for PDEs'
and apply it for the compressible Navier-Stokes system in one dimension.
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