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ICES Upcoming Events & Seminars

 

ICES Seminar
Thursday, May 16, 2013 from 3:30PM to 5PM

Generalization of Finite Elements
by Marc Alexander Schweitzer

Institut fuer Numerische Simulation, University of Bonn

Several generalizations of the classical finite element method via the use of non-polynomial problem-dependent enrichment functions have been proposed in the last two decades, e.g. the extended finite element method (XFEM) or the generalized finite element method (GFEM). The ultimate goal of these approaches is the development of a numerical scheme whose convergence properties is not limited by the regularity of the problem.

We review and interpret this enrichment approach in the context of the even more general framework of the partition of unity method (PUM). We present the basic construction principles and implementational issues, some results on the approximation properties and the stability of the PUM with smooth, discontinuous and singular enrichment functions as well as the construction of efficient multilevel solvers. Examples of applications from fracture mechanics, problems with microstructures and fluid flow are given.

Hosted by Ivo Babuska

 

ICES Seminar
Tuesday, May 21, 2013 from 3:30PM to 5PM

Living on the Edge: A Geometric Theory of Phase Transitions in Convex Optimization
by Joel Tropp

Department of Applied Math, Caltech

Recent empirical research indicates that many convex optimization problems with random constraints exhibit a phase transition as the number of constraints increases. For example, this phenomenon emerges in the l1 minimization method for identifying a sparse vector from random linear samples. Indeed, the l1 technique succeeds with high probability when the number of samples exceeds a threshold that depends on the sparsity level; otherwise, it fails with high probability.

This talk summarizes a rigorous analysis that explains why phase transitions are ubiquitous in random convex optimization problems. It also describes tools for making reliable predictions about the quantitative aspects of the transition, including the location and the width of the transition region. These techniques apply to regularized linear inverse problems with random measurements, to demixing problems under a random incoherence model, and also to cone programs with random affine constraints.

Joint work with D. Amelunxen, M. Lotz, and M. B. McCoy.

Hosted by Inderjit Dhillon