Department of Mathematics
Applied Math. Seminar, Fall 2006

August 30: SPECIAL TIME, DATE, and LOCATION, 2:15 - 3:05 PM, LCB 225
Speaker: Amy Shen, Washington University in St. Louis - Dept. of Mechanical and Aerospace Engineering
Title: Hydrodynamics of complex fluids at small length-scales
Abstract: Understanding fluid transport and interfacial phenomena of complex fluids at small length-scales is crucial to understanding how to design and exploit of micro- and nano-fluidic devices. I will present two examples. The first studies evaporation driven self-assembly to synthesize nanoporous thin films. A combination of experimental measurement and modeling using lubrication theory shows how self-assembly influences coating film thickness. The second example studies how length-scale and fluid elasticity affect droplet pinch-off of "simple" polymeric liquids in microfluidic environments. Boger fluids (viscoelastic liquids with nearly constant shear viscosity) are pumped into microchannels and pinched off to form droplets in an immiscible oil phase. We find a power law relation between the dimensionless capillary pinch-off time and the so-called elasticity number, E, of the fluid. Theoretical models that neglect the extensional viscosity of the fluid become increasingly more inaccurate as the fluid elasticity increases.

September 5: SPECIAL TIME, DATE, and LOCATION, 3:05 - 3:55 PM, LCB 115
Joint with the Approximation Theory Seminar
Speaker: Peter Alfeld, University of Utah
Title: The Bernstein Bézier Form of a Multivariate Polynomial
Abstract: The Bernstein-Bézier form (or just B-form) of a polynomial is a highly successful and widely used way of representing polynomials, particularly polynomials in more than one variable. Its power stems from the fact that algebraic issue, such as two polynomials joining smoothly, can be studied and interpreted geometrically. There is also a close geometric connection between the coefficients of a polynomial and the shape of its graph. In this talk I will define the B-form of a polynomial and discuss some of its properties. This will serve as the foundation for several future talks this semester. The talk will include computer demonstrations.

September 11: - Joint with the Stochastics Seminar
Speaker: Eulalia Nualart, University of Paris 13
Title: Potential theory for non-linear stochastic heat equations
Abstract: In this talk we develop potential theory for a system of non-linear stochastic heat equations in spatial dimension one and driven by a space-time white noise. In particular, we prove upper and lower bounds on hitting probabilities of the process which is solution of this system of equations, in terms of respectively Hausdorff measure and Newtonian capacity. These estimates make it possible to discuss polarity for points and to compute the Hausdorff dimension of the range and the level sets of this process. In order to prove the hitting probabilities estimates, we need to establish Gaussian type bounds for the bivariate density of the process in order to quantify its degenerance. For this, we use techniques of Malliavin calculus.

September 18:
Speaker: Joel Tropp, University of Michigan at Ann Arbor
Title: Sparse solutions to underdetermined linear systems
Abstract: A central problem in electrical engineering, statistics, and applied mathematics is to solve ill-conditioned systems of linear equations. Basic linear algebra forbids this possibility in the general case. But a recent strand of research has established that certain ill-conditioned systems can be solved robustly with efficient algorithms, provided that the solution is sparse (i.e., has many zero entries). This talk describes a popular method, called l1 minimization or Basis Pursuit, for finding sparse solutions to linear systems. It details situations where the algorithm is guaranteed to succeed. In particular, it describes some new work on the case where the matrix is deterministic and the sparsity pattern is random. These results are currently the strongest available for general linear systems.

September 25: - Joint with the Approximation Theory Seminar
Speaker: Yuliya Babenko, Sam Houston State University
Title: On asymptotically optimal methods of adaptive spline interpolation
Abstract: In this talk we shall present the exact asymptotics of the optimal error in the weighted $L_p$-norm, $1\leq p \leq \infty$, of linear spline interpolation of an arbitrary function $f \in C^2([0,1]^2)$. The connections with the problem of approximating the convex bodies by polytopes and the problem of adaptive mesh generation for finite element methods will also be discussed. We shall present review of existing results as well as a series of new ones. Proofs of these results lead to algorithms for construction of asymptotically optimal sequences of triangulations for linear interpolation. Similar results are obtained for some other classes of splines. We shall also discuss the analogous multivariate results as well.

October 2:
Speaker: Andrej Cherkaev, University of Utah
Title: New bounds for multiphase conducting composites
Abstract: New bounds for effective properties tensors of multimaterial composites are suggested. These bounds complement the translation bounds or Hashin-Shtikman bounds. We show that the bounds are exact for three-material composites and determine optimal microstructurs of them. The bounds are obtained using the theory of "localized polyconvexity" which will be also discussed.

October 9: SPECIAL TIME and LOCATION, 12:55 - 1:45 PM, LCB 225
Joint with the Stochastics Seminar

Speaker: Pierre Seppecher, University of Toulon
Title: A closed notion of locality for Dirichlet forms in the one dimensional case
Abstract: If the notion of locality is well known in the case of regular Dirichlet form, it is is not straightforward to extend it to non-regular forms. We compare different possible definitions and characterize a notion of locality which is closed with respect to Mosco or $\Gamma$-convergence. This enable us to characterize the closure of the set of diffusion functionals in the one-dimensional case.

October 16:
Speaker: Graeme Milton, University of Utah
Title: Cloaking: a New Phenomena in Electromagnetism and Elasticity
Abstract: Since my talk last semester, there have been quite a number of developments with regards to the theory of cloaking (making an object invisible). Not only developments with respect to cloaking associated with superlenses, as I had discussed, but also with proposals by Pendry, Schurig and Smith and Leonhardt, for designing a shield which cloaks objects to electromagnetic waves. This work is related to the earlier work of Greenleaf, Lassas and Uhlmann, on cloaking for conductivity. Here we will review these developments and also discuss how cloaking might be extended to elasticity using these ideas. This requires new materials, in particular materials with anisotropic density. We show how such materials can be made.

October 23:
Speaker: Jorge Balbas, University of Michigan at Ann Arbor - Dept. of Mathematics
Title: Non-oscillatory Central Schemes for One-dimensional Shallow Water Flows along Channels with Non-uniform Rectangular Cross-sections
Abstract: We present a new high-resolution, non-oscillatory semi-discrete central scheme for one-dimensional shallow water flows along channels with non-uniform rectangular cross sections. The scheme extends existing central semi-discrete schemes for hyperbolic conservation laws by incorporating a discretization of the source terms appearing in shallow water equations so that nonlinear fluxes are balanced for steady-state solutions. We also incorporate exact information in the polynomial reconstruction of the wet area, improving the control of spurious oscillations. Along with the scheme, we present a systematic approach to calculate exact steady-state solutions for the balance law. This allows us to validate the scheme by comparing the approximate numerical solutions to the exact ones.

October 30:
Speaker: Adam Oberman, Simon Fraser University - Dept. of Mathematics
Title: Fully nonlinear elliptic PDEs: models, applications, and solution methods.
Abstract: This will be an accessible talk about modeling using fully nonlinear elliptic PDEs. Modern applications of these PDEs are to Image Processing and Math Finance. As well as the Level Set Method for curve evolution, Optimal Control and Stochastic Control. I'll discuss some interesting models, overview the relevant theory, and then show how to solve these equations. Examples will include: level set motion by mean curvature, the convex hull, the infinity Laplacian, as well as examples from math finance and control theory. We will present results which allow schemes to be built for a wide class of equations.

November 6:
Speaker: Jonathan Kaplan, Stanford University - Dept. of Mathematics
Title: The Morphlet Transform: A Multiscale Transform for Diffeomorphisms
Abstract: Diffeomorphisms are a classical tool and object of study in theoretical mathematics. Recently, there has been an increase in the use and study of diffeomorphisms in applied mathematics. In particular, diffeomorphisms have appeared as a new and potent tool in image analysis. There is a growing interest in understanding computationally efficient mechanisms for representing and manipulating diffeomorphisms. Inspired by the success of wavelets in signal processing, we describe a multiscale transform acting on diffeomorphisms. This transform is defined on dyadic samples and is nonlinear. Its design draws from the theory of interpolating wavelet transforms and nonlinear subdivision schemes. We call this transform the morphlet transform.

November 13:
Speaker: Coralia Cartis, Rutherford Appleton Laboratory - Computational Science & Engineering Dept.
Title: Some challenges in interior point methods for linear programming
Abstract: Through the depth of their theory and the span of their successful applications, interior point methods have sparked a veritable "revolution" in convex optimization. Now, fifteen years after their landmark discovery, interior point methods have become highly successful at solving (very) large-scale linear programming problems, with millions of variables and constraints not uncommon. Nonetheless, some important questions at the interface of theory and practice remain open and I will address three such topics in this talk. In particular, I will present a new way of initializing these algorithms which overcomes the fundamental assumptions underlying interior point methods theory that require the set of admissible solutions to be full-dimensional and that are rarely satisfied by real-life problems (this is joint work with Nick Gould, Oxford University). Furthermore, addressing the lack of theoretical reliability of the interior point algorithm implemented in most commercial and public software, I show on an example what may go wrong and then describe a theoretically reliable alternative. As interior point methods have made linear programming solvable in polynomial time, complexity is a crucial aspect of this area. We set up a new general framework in which we perform such a complexity analysis, that attempts to be more practical and insightful than existing, highly-constructive, techniques by employing stiffness analysis of vector fields, a concept traditionally associated with ordinary differential equations (this is joint work with Raphael Hauser, Oxford University).

November 14: SPECIAL TIME, DATE, and LOCATION, 3:00 - 4:00 PM LCB 215
Special Seminar - Joint with Bio-Math
Speaker: Doron Levy, Stanford University - Mathematics Dept.
Title: Modeling the Dynamics of the Immune Response to Chronic Myelogenous Leukemia
Abstract: Chronic Mylogenous Leukemia (CML) is a blood cancer with a common acquired genetic defect resulting in the overproduction of malformed white blood cells. The cause of CML is an acquired genetic abnormality in hematopoietic stem cells in which a reciprocal translocation between chromosomes 9 and 22 occurs. It is this abnormality that leads to dysfunctional regulation of cell growth and survival, and consequently to cancer. Treatment and control of CML underwent a dramatic change with the introduction of the new drug, Gleevec, which was shown to be an effective treatment available to nearly all CML patient. Nevertheless, by now it is widely agreed that Gleevec does not represent a true cure for CML, with many patients beginning to relapse despite of continued therapy. The only known treatment that can potentially cure CML is a bone-marrow (or stem-cell) transplant.
In this talk we will describe our recent works in modeling the interaction between the immune system and cancer cells in CML patients. One model follows this dynamics after a stem-cell transplant. A second model follows the immune-cancer dynamics in patients treated with Gleevec. Related mathematical questions and possible exciting applications of the models will be discussed. This is a joint work with Peter Kim and Peter Lee (Hematology, Stanford Medical School).

November 17: SPECIAL TIME, DATE, and LOCATION, 3:30 - 4:20 PM LCB 121
Special Seminar - Joint with the Approximation Theory Seminar
Speaker: Frank Stenger, U. of Utah - Computer Science Dept.
Title: SINC-PACK Enables Separation of Variables
Abstract: This talk is mainly for mathematicians. It consists of a "proof-part" of Stenger's SINC-PACK computer package (an approx. 400-page tutorial + about 250 Matlab programs) that one can always achieve separation of variables when solving linear elliptic, parabolic, and hyperbolic PDE (partial differential equations) via use of Sinc methods.
Some examples illustrating computer solutions via Sinc-Pack will nevertheless be given in the talk. For example, in one dimension, Sinc-Pack enables the following, over finite, semi--infinite, infinite intervals or arcs: interpolation, differentiation, definite and indefinite integration, definite and indefinite convolution, Hilbert and Cauchy transforms, inversion of Laplace transforms, solution of ordinary differential equation initial value problems, and solution of convolution-type integral equations. The methods of the package are especially effective for problems with (known or unknown - type) singularities, for problems over infinite regions, and for PDE problems.
In more than one dimension, the package enables solution of linear and nonlinear elliptic, hyperbolic, and parabolic partial differential equations, as well as integral equations and conformal map problems, in relatively short programs that use the above one-dimensional methods. The regions for these problems can be curvilinear, finite, or infinite. Solutions are uniformly accurate, and the rates of convergence of the programs of SINC-PACK are exponential.
In Vol 1. of their 1953 text, Morse and Feshbach prove for the case of 3-dimensional Poisson and Helmholtz PDE that separation of variables is possible for essentially 13 different types of coordinate systems. A few of these (rectangular, cylindrical, spherical) are taught in our undergraduate engineering-math courses. We prove in the talk that one can ALWAYS achieve separation of variables via use of Sinc-Pack, under the assumption that calculus is used to model the PDE.

November 20: - Joint with the Approximation Theory Seminar
Speaker: Tatyana Sorokina, University of Georgia
Title: Quasi-Interpolation by Multivariate Piecewise Polynomials.
Abstract: Quasi-interpolants provide an alternative to finite elements in multivariate approximation. While there are reliable tools for studying classical finite elements, there is no theory of quasi-interpolation in several variables. We will discuss some theoretical aspects of quasi-interpolation, consider explicit bivariate and trivariate schemes, and state open problems.

November 27:
Speaker: Paul Fife, University of Utah
Title: The structure of turbulent flow near boundaries
Abstract: The problem of deriving key features of steady turbulent flow adjacent to a wall has drawn the attention of some of the most noted fluid dynamicists of all time. Standard examples of such features are found in the mean velocity profiles of turbulent flow in channels, pipes or boundary layers. Possibly the best known of the elementary theoretical efforts along this line, and certainly the simplest, is the argument (obtained independently) by Izakson (1937) and Millikan (1939) regarding the mean velocity profile. They showed that if an inner scaling and an outer scaling for the profile are valid near the wall and near the center of the flow (or the edge of the boundary layer), respectively, and if there is an overlap region where both scalings are valid, then the profile must be logarithmic in that common region. That piece of theory has been used and expanded upon by innumerable authors for over 60 years, and at the present time is still rightly enjoying popularity.
The main foci of the talk will be on (i) a careful examination of the Izakson-Millikan argument, and (ii) an account of a newer approach to gaining theoretical understanding of the mean velocity and Reynolds stress profiles, due to Klewicki, McMurtry, Metzger, Wei and myself. It has similar goals but entirely different methods. The question will be how, and how well, do these arguments supply insight into the structure of the mean flow profiles? Answering the question WHY? is even more important than WHAT?

December 4:
Speaker: Patrick J. Wolfe, Harvard University - Department of Statistics
Title: Time-Frequency Representations and Statistical Models for Speech:
Exploring the Space of Acoustic Waveform Variation
Abstract: Time-frequency representations are ubiquitous in audio signal processing, their use being motivated by both auditory physiology and the mathematics of Fourier analysis. Indeed, information-carrying natural sound signals can often be meaningfully represented as a superposition of translated, modulated versions of a simple window function exhibiting good time-frequency concentration. In combination with statistical models formulated in the space of time-frequency coefficients, such an approach provides a principled way of decomposing sounds into their constituent parts, as well as an effective means of exploiting the local correlation present in the time-frequency structure of natural sound signals such as speech. In addition to applications involving the reconstruction of noisy and missing audio data, this talk will describe the ways in which generative statistical models provide a means of exploring the space of acoustic waveform variation, and how in doing so they point toward a new way forward in a variety of speech processing and discrimination tasks.

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