# Math Colloquia: Spring 2006

#### Friday January 27

11:00–noon

 Title Adaptive Discontinuous Galerkin Methods for Reactive Transport Problems Speaker Shuyu Sun University of Texas at Austin

Abstract:
Discontinuous Galerkin (DG) methods are specialized finite element methods that utilize discontinuous piecewise polynomial spaces to approximate the solutions of differential equations, with boundary conditions and inter-element continuity weakly imposed through bilinear forms. DG methods have recently gained popularity for many attractive properties. First of all, the methods are locally mass conservative and have small numerical diffusion. In addition, they support general nonconforming spaces including unstructured meshes, nonmatching grids and variable degrees of local approximations, thus allowing efficient h-, p-, and hp-adaptivities. Moreover, DG algorithms treat rough coefficient problems and effectively capture discontinuities in solutions. They have excellent parallel efficiency since data communications are relatively local. For time-dependent problems in particular, their mass matrices are block diagonal, providing a substantial computational advantage if explicit time integrations are used.

In this talk, we consider a family of discontinuous Galerkin (DG) applied to reactive transport problems. They are the four primal discontinuous Galerkin schemes for the space discretization: Symmetric Interior Penalty Galerkin (SIPG), Oden-Baumann-Babuska DG formulation (OBB-DG), Nonsymmetric Interior Penalty Galerkin (NIPG) and Incomplete Interior Penalty Galerkin (IIPG) methods. We address a priori error bounds in the L2(H1), L2(L2) and negative norms, and a posteriori error estimates in L2(H1) and L2(L2). Efficient implementation issues are discussed with emphasis on dynamic mesh adaptation strategies. A number of numerical examples are presented to illustrate various features of DG methods including their sharp a posteriori error indicators and effective adaptivities.

#### Wednesday February 01

12:00–1:00pm

 Title Optimization Techniques for Mesh Warping and the Geometry of Materials Speaker Suzanne Shontz University of Minnesota

Abstract:
This talk will address two recent themes in my research: mesh warping and the geometry of materials.

In the first part of my talk, I will describe a mesh warping problem of interest. The process of warping a mesh from a source to a target domain can potentially drastically alter the quality of the mesh from step to step. One problem that can occur is element reversal, which is when an element changes its orientation. We consider an algorithm called FEMWARP for warping tetrahedral finite element meshes that computes the warping using the finite element method itself. Our main concern is when this algorithm reverses elements. We analyze the causes for this undesirable behavior and propose techniques to make the method more robust against reversals. Among the methods includes combining FEMWARP with an optimization-based untangler. We will also demonstrate the applicability of FEMWARP to cardiology.

In the second part of my talk, I will describe my work on quasi-Newton and nonlinear conjugate gradient methods for the geometry optimization of materials.

Finally, I will conclude with upcoming plans for research in the areas of meshing and optimization as applied to bioengineering and materials modeling.

Part of this talk represents joint work with S. Vavasis of Cornell University.

#### Wednesday February 08

12:00–1:00pm

 Title Optimal control as a regularization method for ill-posed problems Speaker Carmeliza Navasca Department of Mathematics University of California, Los Angeles http://www.math.ucla.edu/~navasca

Abstract:
We describe two regularization techniques based on optimal control for solving two types of ill-posed problems. The two ill-posed problems we consider are in signal processing and parameter identification. Our new mathematical formulations of these ill-posed problems lead to new efficient numerical methods. We compare our numerical results to the same examples using the well-known Tikhonov regularization.

#### Friday February 10

 Title Inverse backscattering for the wave equation in heterogenous media Speaker William W. Symes Department of Computational and Applied Mathematics Rice University http://www.trip.caam.rice.edu/txt/bios/symes/william_symes.html

Abstract:
Active source (“reflection”) seismology leads to a version of the inverse scattering problem for the wave equation (and more general hyperbolic systems). The mechanical properties of the earth are heterogeneous on all scales, so techniques developed for detection of objects in free space have little direct relevance. Most of the useful work to date nonetheless concerns linearization of the coefficient – solution relation about smooth reference coefficients. The nature of this linearized relation is by now understood, up to a point. Perturbations in the coefficients map to perturbations in the solutions via operators which in many practically useful cases belong to well-behaved classes of Fourier Integral Operators, and this fact underlies the construction, and explains the behaviour, of operational seismic imaging algorithms.

Reference coefficients must also be estimated for use in these algorithms, however, as these are no more known a priorin are any other aspects of the earth model. Practical algorithms for estimating reference coefficients rely on extensions of the coefficient – solution map. There exist at least two inequivalent extensions, one of which has much better global properties than the other. I will describe these extended coefficent – solution maps, explain how they are used to estimate the reference coefficients in linearized scattering, sketch a possible generalization to nonlinear inverse scattering, and mention a few of the mathematical puzzles that abound in this subject.

#### Tuesday February 14

10:00–11:00am

 Title The Spectral Volume Reconstruction on Simplex Speaker Qian-Yong Chen Institute for Mathematics and its Applications http://www.ima.umn.edu/~qchen

Abstract:
The spectral volume method (a discontinuous Petrov-Galerkin method) was developed by ZJ Wang and Y Liu in 2002 to resolve the difficulty related to the reconstruction stencil in standard high-order finite volume schemes for hyperbolic conservation laws. A key element of the new method is the spectral volume reconstruction, for which a good partition of the simplex (triangle in 2D) is required.

In this talk, I present several systematic techniques, based on the Voronoi diagram, to partition the one-, two-, and three-dimensional simplex. Almost optimal polynomial interpolation points are used as the input. The resultant partitions are very accurate (have small Lebesgue constants), the number of edges (roughly proportional to computational cost) for 2D partitions is shown to at most twice the minimum number of edges for the same order reconstruction. Certain optimizations are also made to those partitions. The optimized partitions have the smallest Lebesgue constant among currently available partitions.

DATE Thursday February 16
11:00–noon

 Title Computing Accurate Eigenvalues—From Electrical Impedance Tomography to 3D Target Recognition Speaker Plamen Koev Department of Mathematics Massachusetts Institute of Technology

Abstract:
Many practical problems in elasticity, medical imaging, etc., when modeled mathematically, reduce to the matrix eigenvalue problem. Rounding errors severely limit the accuracy of the computed eigenvalues– the conventional matrix algorithms (e.g., the ones employed by MATLAB) usually compute only the largest eigenvalues accurately. The tiny ones are lost to roundoff even though often they are accurately determined by the data and of most physical significance.

The obvious remedy in this situation is to increase the precision, but it often comes with a prohibitively high computational cost.

In contrast, for many classes of structured matrices we developed algorithms that compute the full spectrum to full machine precision without the need for extra precision or extra computational time. I will present the key techniques in the design of these algorithms.

The picture is different and much more challenging with (the distributions of) the eigenvalues of random matrices. These distributions are critical in many multivariate statistical tests and a variety of applications– 3D target recognition, wireless comminications, genomics, etc.

Interestingly, there are explicit formulas for these eigenvalue distributions, but only in terms of the hypergeometric function of a matrix argument– a notoriously slowly converging series of (generalized) Schur functions whose efficient computation had eluded researchers for over 40 years.

It is the combinatorial properties of the Schur function that ultimately allowed us to develop the first efficient algorithm for computing the hypergeometric function of a matrix argument. As a result, 3D target classification is now possible and efficient, as are multivariate statistical methods in genomics, wireless communications, etc. I will present the key ideas in the development of our algorithm as well as the impact it has had on the above applications.

DATE Thursday February 16
2:30–3:30pm

 Title The coarsening dynamics of dewetting fluid films Speaker Thomas P. Witelski Department of Mathematics Duke University http://www.math.duke.edu/~witelski/

Abstract:
The study of instabilities of thin fluid films on solid surfaces is of great importance in understanding coating flows. These instabilities lead to rupture, the formation of dry spots, and further morphological changes that promote non-uniformity of coatings; these behaviors in unstable thin films are generally called dewetting. Following rupture and subsequent transient behavior, the long-time structure of films takes the form of an array of droplets. The evolution of this system can be represented in terms of coupled ODEs for the masses and positions of the droplets. Regimes where droplet coarsening by each of two mechanisms (collision and collapse) are identified, and power laws for the statistics of the coarsening processes are explained. This is joint work with Karl Glasner, University of Arizona.

#### Friday February 24

 Title System and Control Issues of Complementarity Systems Speaker Jinglai Shen Department of Mathematical Sciences Rensselaer Polytechnic Institute http://www.rpi.edu/~shenj2/

Abstract:
This talk is concerned about system and control issues of complementarity systems that find a wide range of applications in engineering fields and dynamic optimization. We will talk about solution existence and uniqueness of low index complementarity systems. We then focus on the Zeno issue of complementarity systems, which plays a fundamental role in system/control analysis. We introduce the concept of strong non-Zenoness and show it for strongly regular complementarity systems. Finally, we discuss the observability issue and show local observability conditions derived based upon the non-Zeno results.

#### Wednesday March 01

noon–1:00pm

 Title Analysis of some interface and free boundary problems Speaker Daniel Coutand Department of Mathematics University of California Davis http://www.math.ucdavis.edu/~coutand/

Abstract:
The main focus of my talk shall be on the well-posedness for the interface problem between a viscous fluid and an elastic solid. This is a two-phases problem, where each phase satisfies its own natural equation of evolution, and where the interaction between the two phases comes from the natural continuity of velocity field and normal stress across the unknown moving interface. The methods known in fluid moving boundary problems (viscous or inviscid) cannot handle the apparent incompatibility between the regularity of the two phases, which has led previous authors to consider the case where the solid satisfies a simplified law where the difficulties are not present. I shall present the new methods that were required in order to allow the treatment of classical elasticity laws in this moving interface problem.

I shall then briefly explain how some of these ideas and some new tools preserving the transport structure of the Euler equations can provide the well-posedness for the free surface Euler equations with (or without) surface tension, without any restriction on the curl of the initial velocity.

#### Friday March 03

 Title Efficiency considerations for the numerical integration of stochastic differential equations Speaker Habir Lamba Department of Mathematical Sciences George Mason University http://math.gmu.edu/www/people/Lamba.htm

Abstract:
In recent years there has been an explosion of interest in the modelling of stochastic phenomena, and stochastic differential equations (SDEs) are now routinely used to describe systems throughout the physical and social sciences. However, from a numerical point of view, there is a distinct lack of both theoretical results and sophisticated, general-purpose, numerical integration software that can safely be used by ‘non-experts’. This is especially apparent when the current state of affairs is compared with the vast and highly successful body of knowledge that exists for ODEs.

This seminar, which assumes no prior knowledge of SDEs or stochastic calculus, will compare and contrast the fundamental problems inherent in the efficient and reliable numerical integration of SDEs versus those of ODEs. Of particular interest is the idea of adaptive timestepping, a strategy that has been very successful for ODEs where it is capable of impressive efficiency gains, but which is both more difficult to implement and less rewarding in the SDE case. Some recent approaches to this problem will be described, together with a brief discussion of other related issues such as numerical stability.

#### Saturday March 04

1:00–2:00pm

 Title Multigrid methods as efficient solvers for ill-posed problems Speaker Andrei Draganescu Sandia National Laboratories http://www.cs.sandia.gov/optimization/draganescu/

Abstract:
Scenario: An air contamination event takes place in a heavily populated area. A chemical agent is being diffused in the air, and, at the same time, is moved by the winds. Sensors monitoring air quality detect increased concentrations of the pollutant. At what location was the pollutant released in the air? What areas will be affected over the next few minutes, hours, days, and and to what degree?

Fast answers to these questions are critical for hazard containment and assessment, and for evacuation procedures. An efficient response to the above scenario requires the solution of a problem that is ill-posed from a mathematical point of view: the backward-in-time simulation of a physical process that is essentially irreversible, namely a time dependent advection-reaction-diffusion process. In this talk I will present a method for efficiently solving the regularized inverse problem of identifying initial conditions given the end-time state for an equation of parabolic type. The method is a new embodiment of the well-known multigrid paradigm, which consists in taking advantage of several discretization levels for the same continuous problem in order to speed up the solution process. I will discuss mathematical results for both the linear and semilinear cases, and I will present supporting computations. In the end I will show the result of applying this algorithm to the scenario above, which proves that the associated inverse problem can be resolved in a reasonable time-frame.

#### Friday March 10

 Title Who Wrote this Document? Speaker Charles Nicholas Department of Computer Science and Electrical Engineering UMBC http://www.umbc.edu/engineering/csee/faculty/nicholas.html

Abstract:
Questions of authorship have fascinated historians, theologians, and other scholars for centuries. In recent years statisticians and now at last computer scientists are also addressing these issues. We’ll present an overview of the study of authorship attribution, including famous examples such as “The Federalist Papers”, the various “Wizard of Oz” books, as well as the Hebrew Bible and the Christian New Testament. We’ll present results from our own work in applying latent semantic analysis (LSA), a well-known technique in information retrieval, to the authorship attribution problem.

Speaker bio: Dr. Nicholas is currently Professor and Chair of the Department of Computer Science and Electrical Engineering at UMBC, where he has been on the faculty since 1988. He received the B.S. degree from the University of Michigan-Flint in 1979, and the M.S. and Ph.D. degrees from The Ohio State University in 1982 and 1988, respectively. In addition to his appointment at UMBC, Dr. Nicholas has held appointments at the National Institute of Standards and Technology (NIST), and the NASA Goddard Space Flight Center. He spent academic year 1996–97 on sabbatical at the National Security Agency.

Dr. Nicholas’ research interests include electronic document processing, information retrieval, and software engineering. His work has been funded by a number of agencies, including NASA, Maryland Industrial Partnerships, DARPA, AFOSR, and the Department of Defense. Dr. Nicholas has served five times as the General Chair of the ACM Conference on Information and Knowledge Management (CIKM). He also twice chaired the Workshop on Digital Document Processing. Dr. Nicholas is a member of the Board of Directors of UMBC Training Centers.

#### Friday March 17

 Title Conjugate Points for Neumann boundary conditions Speaker Robert Manning Mathematics Department Haverford College http://www.haverford.edu/math/rmanning.html

Abstract:
The classic theory of the calculus of variations lays out a simple and elegant procedure for analyzing a functional $J[y] = \int_0^1 W(y'(s), y(s),s) ds$ subject to Dirichlet boundary conditions: solve the Euler-Lagrange equations to find critical points $y_0$ of $J$, and then find conjugate points, which classify $y_0$ because of the fact that the number of conjugate points equals the number of negative eigenvalues (the “index”) of the second derivative (variation) of $J$ at $y_0$. When we change the boundary conditions to Neumann, however, the story changes: the number of conjugate points need no longer equal the index. We will discuss why this is, and how to rescue the classic theory by associating a sign to each conjugate point, so that the index equals the signed sum of the conjugate points. We apply this theory to the buckling of a rod into a wall subject to pinned-pinned boundary conditions, and find that the index results match the hysteresis effects observed in experiments.

Spring Break

#### Friday March 31

No colloquium today due to Finite Element Circus

DATE Thursday April 6
11:00–noon

 Title Modeling: Heart, Mind, and… Bacteria Speaker Brad Peercy Department of Computational & Applied Mathematics Rice University http://www.caam.rice.edu/~bpeercy

Abstract:
In this talk I discuss dynamics of three prototypical cellular constituents: transmembrane potential, calcium, and extracellular signal transduction with gene regulation in the context of cardiac arrhythmias, neuronal modulation, and oxygen sensing in E. coli.

During a heart attack ischemic inhomogeneity in space generates a difference in potential driving a “current of injury” leading to anomalous electrical activity. We develop a model for coupled cell experiments and theory for 1) predicting when an ischemic region will force a normal region into oscillations for varying, degrees of ischemia, size of affected/normal region, and coupling strength, 2) showing that certain forcing of a normal region is equivalent to coupling a normal region to an ischemic region in certain limits. To model strip of tissue experiments we construct a reaction-diffusion system and mention results on existence and uniqueness of steady state solutions using geometrical and sub/super solution techniques. We analyze the bifurcation to oscillatory dynamics.

Calcium, The second messenger in the central nervous system, initiates a number of biochemical cascades that modulate the cell’s firing properties. As many of these cascades involve gene transcription there is great interest in recent experimental detection of intracellular calcium waves traveling from dendrite to cell body. We construct a model aimed at understanding the initiation and propagation of cytosolic calcium in a CA1 pyramidal neuron. We find mechanisms of 1) focal initiation, 2) directionality of propagation, and 3) conditions for wave entry into the soma. Results on timing between synaptic events and depolarization agree well with experiments.

Ecoli does an efficient job of survival in oxygen deprivation. The result is the production of byproduct metabolites such as succinate useful in many pharmaceuticals (e.g. Toprol-XL (metoprolol succinate) – a recently developed cardiac beta receptor blocker). Understanding the transition from aerobic through microaerobic to anaerobic conditions may lead to novel bioengineering techniques that maximize metabolite output. We construct a model of the oxygen sensitive machinery in the electron transport chain and two primary transcription factors ArcA and Fnr. ArcA and Fnr act to regulate terminal electron acceptors leading to feedback to the sensing components as well as to down stream metabolism enzymes. The kinetic model reproduces steady state stationary phase experiments for two cytochrome populations and exhibits oscillations and hysteresis in certain parameter regions suggesting experiments.

DATE Thursday April 6
2:30–3:30pm

 Title Viscoelastic Fluids (Polymers and Wormlike Micelles): Properties and Mathematical Models Speaker Pamela Cook Department of Mathematical Sciences University of Delaware http://www.math.udel.edu/~cook/

Abstract:
Complex fluids are encountered in a variety of industries and personal uses. The polymer industry alone accounts for well over \$100 billion in annual revenues in the US. Micellar solutions are used as rheological modifiers in consumer products (paints, detergents, pharmceuticals, oil recovery). Thus, understanding the flow behavior and properties of these complex fluids is of critical importance. This talk will concentrate on mathematical modeling of flows of dilute and concentrated mixtures of polymers and of wormlike micellar solutions.

Physical properties of various non-Newtonian fluids will be discussed, and specifically the effect of elasticity in the fluid. Then, starting from “simple” mathematical models the talk will progress to more involved models and their predictions. Mathematically the techniques involve modeling using kinetic theory and network theory, with multiple scales (from the microscopic to the macroscopic) involved. Ultimately the solution of a nonlinear coupled system of partial differential equations must be found. The solutions are examined both asymptotically and numerically.

This talks reports on joint work with Gareth McKinley (MIT), and Lou Rossi, Paula Vasquez, Lin Zhou (University of Delaware). This work was supported by the NSF-DMS.

#### Friday April 14

 Title Iterative Clustering: Discrete and Continuous Optimization Problems Speaker Jacob Kogan Department of Mathematics and Statistics UMBC http://www.math.umbc.edu/~kogan/

Abstract:
We consider a class of iterative clustering algorithms traditionally presented as solutions for a discrete optimization problem. We restate this problem as a continuous optimization problem. A solution involving smoothing technique is suggested. Numerical results illustrating the accuracy and speed of convergence of the corresponding “discrete” and “continuous” algorithms are provided.

No talk today

#### Friday April 28

 Title Finite Element Partition of Unity Method for Nonmatching Grids Speaker Constantin Bacuta Department of Mathematical Sciences University of Delaware http://www.math.udel.edu/~bacuta/

Abstract:
We give a general framework on applying the partition of unity method for nonmatching grids. Numerical aspects of discretization by partition of unity finite elements are considered for a model problem. We then present recent results on applying the method to Stokes equations discretized on nonmatching grids. The approach is based on creating global discrete spaces for velocity and pressure by gluing subspaces associated with subdomains. The global discrete spaces for pressure and velocity are build using the a-priori knowledge of the solution on subdomains and the local approximation properties are preserved. In addition to good approximation properties, the global spaces for velocity and pressure, satisfy the Babuska-Brezzi’s inf-sup condition.

This work is supported by UDRF.

#### Friday May 05

 Title Schottky’s storms: Classical function theory and ideal fluid mechanics Speaker Darren Crowdy Imperial College, London http://www.ma.imperial.ac.uk/~dgcrowdy/index.html

Abstract:
This talk will describe novel applications of ideas from classical function theory (due to Poincare, Schottky, Schwarz and Klein, among others) to problems arising in ideal fluid mechanics. New solutions, only recently reported in the literature, to some important basic problems will be presented and the constructive analytical techniques outlined. The problems considered arise in a variety of applications ranging from geophysical situations involving the dynamics of oceanic eddies around islands and headlands, aerodynamical problems involving the computation of lift forces on multi-component aerofoils, through to astrophysical applications requiring the study of vortex motion on spherical shells.