# Math Colloquia: Spring 2005

#### Friday February 04

No talk today

#### Friday February 11

Title |
Primal-dual algorithms for smooth nonlinear constrained optimization |

Speaker |
Igor Griva |

Department of Mathematical Sciences | |

George Mason University | |

http://mason.gmu.edu/~igriva/ |

**Abstract:**

We consider two primal-dual algorithms: interior and exterior point methods. The latter is also known as the primal-dual nonlinear rescaling method. At each iteration the algorithms solve corresponding primal-dual linear systems for finding Newton directions. We discuss the linear systems’ properties and emphasize their role in global and asymptotic convergence properties of both methods. We formulate convergence results for both algorithms and demonstrate their numerical behavior.

#### Friday February 18

Title |
Edge detections, hierarchical decompositions and multiscale nonlinear dynamics |

Speaker |
Eitan Tadmor |

Center for Scientific Computation and Mathematical Modeling (CSCAMM), Department of Mathematics, Institute for Physical Science & Technology | |

University of Maryland, College Park | |

http://www.cscamm.umd.edu/~tadmor/ |

**Abstract:**

I will discuss three prototype problems which are dominated by the presence of different scales. The first deals with data processing involving different regions of smoothness. The interfaces of such regions are detected from their noisy spectral data using separation of scales. The second problem continues with image processing, where I will present a novel hierarchical decomposition of texture into different scales of edges. I will conclude with a discussion on nonlinear dynamics, involving the passage from kinetic to macroscopic scales. Regularizing effect quantified by the averaging lemma demonstrate multiscale dispersive effects.

#### Friday February 25

Title |
Convergence rates of adaptive algorithms for ordinary and stochastic differential equations |

Speaker |
Kyoung-Sook Moon |

Department of Mathematics | |

University of Maryland College Park | |

http://www.nada.kth.se/~moon/ |

**Abstract:**

The theory of a posteriori error estimates suitable for adaptive refinement is well established. But the issue of convergence rates of adaptive algorithms is less studied. I will present a simple and general adaptive algorithm applied to ordinary and stochastic differential equations with proven convergence rates. The presentation has two parts: The error approximations used to build error indicators for the adaptive algorithm are based on error expansions with computable leading order terms. The adaptive algorithm, performing successive mesh refinements, either reduces the maximal error indicator by a factor or stops with the error asymptotically bounded by the prescribed accuracy requirement. Furthermore, the algorithm stops using the optimal number of degrees of freedom, up to a problem independent factor. Similar results on convergence rates of adaptive algorithms for weak approximations of Ito stochastic differential equations are proved for the Monte Carlo Euler method.

Finally I will show some numerical examples which illustrate the efficiency to use the adaptive methods compared to a uniform step size or a method based only on the local error.

#### Friday March 04

Title |
Motion and shape of a falling polymer drop |

Speaker |
Michael Sostarecz |

Department of Mathematics | |

University of Delaware | |

http://www.nada.kth.se/~moon/ |

**Abstract:**

The motion and shape of a drop of dilute polymer solution translating through a quiescent viscous Newtonian fluid is investigated. Experimentally, the drop may exhibit a stable dimple at its trailing edge. At higher volumes the dimple extends far into the interior of the drop, and pinches off via a Rayleigh-type instability, injecting droplets of the bulk fluid into the drop. At even larger volumes, a toroidal shape develops. A perturbation analysis is used to reproduce the dimpled shape and to calculate the drag on the drop.

#### Friday March 11

Title |
Constraint Reduction for Linear Programs with Many Constraints |

Speaker |
Andre Tits |

Electrical Engineering and the Institute for Systems Research | |

University of Maryland College Park | |

http://www.isr.umd.edu/~andre/ |

**Abstract:**

It is now well established that, especially on large linear programming problems, the simplex method typically takes up a number of iterations considerably larger than recent interior-points methods in order to reach a solution. On the other hand, at each iteration, the size of the linear system of equations solved by the former can be significantly lower than that of the linear system solved by the latter.

The approach proposed in this paper can be thought of as a simple-minded attempt at a compromise between the two extremes: viewed from the dual framework standpoint, it makes use of interior-point search directions computed by solving a reduced-size Newton-KKT system, where only constraints in a “critical set”, updated at each iteration, are taken into account. Two algorithms are considered. Promising numerical results are reported on randomly generated problems. Global and local quadratic convergence are proved in the case of one of them (of the primal-dual affine scaling type).

#### Friday March 18

Title |
Stochastic Variational Inequalities and Neumann Problem on Unbounded Convex Domains |

Speaker |
Viorel Barbu |

University of Iasi (Romania), Visiting University Of Virginia | |

http://www.acad.ro/pag_cv/cv_vbarbu.htm |

**Abstract:**

#### Friday March 25

Spring Break

#### Friday April 01

Title |
Modeling Nonlinear Waves of Spreading Depression |

Speaker |
Robert M. Miura |

Department of Mathematical Sciences | |

New Jersey Institute of Technology | |

http://m.njit.edu/~miura/ |

**Abstract:**

Slow chemical waves of spreading cortical depression (SD) have been observed during experiments in a variety of brain structures in different animals. Several mechanisms that are believed to be important in modeling SD will be described, including ion diffusion, the spatial buffer mechanism, membrane ionic currents, osmotic effects, neurotransmitter substances, gap junctions, metabolic pumps, and synaptic connections. Ion diffusion and spatial buffering have been treated both theoretically and numerically, in simplified geometries, and several of the other mechanisms have been investigated numerically. In this talk, I will describe continuum models that consist of coupled nonlinear diffusion equations for the ion concentrations, and a discrete model that corresponds to treating the brain-cell microenvironment using a lattice Boltzmann method.

#### Friday April 08

No talk today

#### Friday April 15

Title |
Algebraic Topology and the Evolution of Complex Patterns |

Speaker |
Thomas Wanner |

Department of Mathematical Sciences | |

George Mason University | |

http://math.gmu.edu/www/people/Wanner.htm |

**Abstract:**

Phase separation processes in binary alloys can produce intriguing and complicated patterns. Yet, characterizing the geometry of these patterns quantitatively can be quite challenging. In the first part of the talk I will demonstrate how methods from computational algebraic topology can be used to obtain such a characterization for the complex microstructures observed during spinodal decomposition and early coarsening in both deterministic and stochastic Cahn-Hilliard models. While these models produce evolving patterns which seem to be qualitatively similar, the topological characterization uncovers significant differences. In the second part of the talk I will briefly address another phase separation phenomenon called nucleation. Its explanation is tied to the long-term dynamics on an underlying global attractor—and also in this case algebraic topology can provide crucial insights.

#### Friday April 22

Title |
The representation of 3-d vector fields and the solvability of div-curl boundary value problems |

Speaker |
Giles Auchmuty |

University of Houston and National Science Foundation |

**Abstract:**

Physicists and engineers often use scalar and vector potentials to solve boundary value problems. In this talk I will first describe how to define *L ^{2}*-orthogonal decompositions of vector fields involving these potentials. There are many such decompositions including the Helmholtz, Hodge-Weyl and other representations.

With the right choice of such decompositions, necessary and sufficient conditions for the finite energy solvability of div-curl boundary value problems can be found. These conditions depend not only on the boundary data but also on the differential topology of the domain. This provides a mathematical explanation of a number of features of electric and magnetic fields that are described in a first course in electromagnetism.

These results are based on joint work with J.C. Alexander.

#### Friday April 29

No talk today

#### Friday May 06

Title |
Falling Leaves, Flapping Flight, and Making a Virtual Insect |

Speaker |
Z. Jane Wang |

Department of Theoretical and Applied Mechanics and Center for Applied Mathematics | |

Cornell University | |

http://tam.cornell.edu/Wang.html |

**Abstract:**

Insects are fascinating to watch but difficult to catch, so are falling leaves. The diverse maneuver executed by insects and the flutter and tumbling motion of leaves are manifestations of complex interactions between the moving surfaces and the surrounding unsteady air. In this talk, I will describe some of the lessons we learned from analyzing them using computers, theoretical models and simple experiments. In particular, I will show 1) a basic two dimensional mechanism of insect hovering and the associated vortical flow and forces, 2) the role of drag in insect hovering, 3) the rise of falling leaves and the lift mechanism which is responsible for the center of mass elevation, 4) models of circulation and fluid forces for falling objects in fluid or an acclerating plate, and 5) the take-off of a pair of three dimensional elastic flapping wings driven by muscles (on computer).

#### Friday May 13

No talk today