The Kaczmarz method is a numerical method to solve systems of linear equations and compute minimum-norm solutions of underdetermined systems. Because the method has very low memory requirements it has gained new attention in recent years. In this talk we propose a flexible algorithmic framework that extends the Kaczmarz method such that it also can…

This talk will introduce a primal-dual finite element method for variational problems where the trial and test spaces are different. The essential idea behind the primal-dual method is to formulate the original problem as a constrained minimization problem. The corresponding Euler-Lagrange formulation then involves the primal (original) equation and its dual with homogeneous data. The…

In  June 23, 2006's "Science" magazine, Pendry et al and Leonhardt independently published their papers on electromagnetic cloaking. In Nov.10, 2006's Science magazine, Pendry et al demonstrated the first practical realization of such a cloak with the use of artificially constructed metamaterials. Since then, there is a growing interest in using metamaterials to design invisibility cloaks.…

Clustering is a fundamental unsupervised learning technique that aims to discover groups of objects in a dataset. Biclustering extends clustering to two dimensions where both observations and variables are grouped simultaneously, such as clustering both cancerous tumors and genes or both documents and words. In this work, we develop and study a convex formulation of the…

Optimization models involving a polynomial objective function and multiple polynomial constraints with discrete variables are often encountered in engineering, management and systems. Treating the non-convex cross-product terms is the key. State-of- the-art methods usually convert such a problem into a 0-1 mixed integer linear programming problem, and, then adopt a branch-and- bound scheme to find…

Starting with an introduction to variational/PDE based image processing, this talk will focus on new developments of fast algorithms for higher order variational imaging models.  For example, recent developments of fast algorithms, based on operator splitting, augmented Lagrangian, and alternating minimization, enabled us to revisit some of the variational image models, such as Euler's Elastica…