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…

Guided waves are widely utilized in the fields of nondestructive testing and geophysical inversion, to estimate the medium properties through inversion of the dispersion curves. In this talk, we present improved methodologies for computing both dispersion curves and their derivatives, the two main ingredients of guided wave inversion. Specifically, a novel discretization approach based on Pade Approximants, named complex-length finite element…

Sums-of-squares certificates define a hierarchy of relaxations for polynomial optimization problems which are parametrized with the degree of the polynomials in the sums-of-squares representation. Each level of the hierarchy generates a lower bound on the true optimal value, which can be computed in polynomial time via semidefinite programming, and these lower bounds converge to the…

Numerical simulation of large-scale dynamical systems plays a crucial role and may be the only possibility in studying a great variety of complex physical phenomena with applications ranging from heat transfer to fluid dynamics, to signal propagation and interference in electronic circuits, and many more. However, these large-scale dynamical systems present significant computational difficulties when…

I will discuss formulations and algorithms for computing sparse optimal controls in systems governed by PDEs. These sparse solutions can guide the placement of control devices in applications.  After reviewing results for elliptic and parabolic PDEs, I will focus on recent work on sparse optimal control governed by linear PDEs with uncertain coefficients. Here, we aim at finding stochastic controls that…

An increasing number of scientific and enterprise data sets are multidimensional, where data is gathered for every configuration of three or more parameters. For example, physical simulations often track a set of variables in two or three spatial dimensions over time, yielding 4D or 5D data sets. Tensor decompositions are structured representations of multidimensional data…

The numerical solution of large-scale risk-averse PDE-constrained optimization problems requires substantial computational effort due to the discretization in physical and stochastic dimensions. Managing the cost is essential to tackle such problems with high dimensional uncertainties. In this work, we combine an inexact trust-region (TR) algorithm from with a local, reduced basis (RB) approximation to efficiently solve risk-averse optimization problems…

Semidefinite programs (SDPs) -- optimization problems with linear constraints, linear objective, and semidefinite matrix variables --  are some of the most useful, versatile, and pervasive optimization problems to emerge in the last 30 years. They find applications in combinatorial optimization, machine learning, and statistics, to name just a few areas. Unfortunately, SDPs often behave pathologically: the optimal values of the primal…

In general, solutions to the Laplacian equation enjoy relatively high smoothness. However, they can exhibit singular behaviors at domain corners or points where boundary conditions change type. In this talk, I will focus on the mixed Dirichlet-Neumann boundary conditions for Laplacian equation, and discuss how singularities in this case adversely affect the accuracy and convergence…

Poroelastic models have been widely used in Biomechanics. For example, modeling brain edema and cancellous bones. We aim at solving the Biot model under the MAC Finite Difference discretization and the stabilized finite element discretizations. To solve the resulting saddle point linear systems, some iterative methods are proposed and compared. In these methods, the outer iteration solver can be the GMRES…

Asynchronous methods refer to parallel iterative procedures where each process performs its task without waiting for other processes to be completed, i.e., with whatever information it has locally available and with no synchronizations with other processes. In this talk, an asynchronous version of the optimized Schwarz method is presented for the solution of differential equations on a parallel computational environment. Convergence is…