CCMA Seminars on PDE & Numerical Methods Fall 2006 

DATE  SPEAKER  TITLE 
09/11/06  Amy Shen
Washington University (Joint with Pritchard Seminar) 
TITLE: Hydrodynamics of complex fluids at small lengthscales
Abstract: Understanding fluid transport and interfacial phenomena of complex fluids at small lengthscales is crucial to understanding how to design and exploit of micro and nanofluidic devices. I will present two examples. The first studies evaporation driven selfassembly to synthesize nanoporous thin films. A combination of experimental measurement and modeling using lubrication theory shows how selfassembly influences coating film thickness. The second example studies how lengthscale and fluid elasticity affect droplet pinchoff 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 pinchoff time and the socalled 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. 
09/18/06  Zhongxuan Luo
Dalian Inst of Tech 
TITLE: Intrinsic Propertics and Invariant of Planar Algebraic Curves
and their Applications.
Abstract 
10/02/06  Dima Burago
Penn State 
Title: Knocking and listening: what is inside?
Abstract: The goal of the talk is to present a math result that suggests a challenging problem in computational math. Imagine that one wants to find out what the Earth is made of. More generally, one wants to find out what is inside a solid body made of different materials (in other words, properties of the material change from point to point). The speed of sound depends on the material. One can "tap" at some points of the surface of the body and "listen when the sound gets to other points". The question is whether this information is enough to determine what is inside. More formally, we have a region and a (Riemannian) metric inside it. We can measure distances between boundary points: they form the boundary distance function. One says that it is rigid if it uniquely determines the metric inside the region. In a joint work with S. Ivanov we prove that if a metric has small second derivatives then it is boundary rigid. This is the first result proving that dim>2 metrics other than extremely special ones (of constant curvature) are rigid. However, we only prove that there exists at most one metric with a given boundary distance function: we do not give a construction for finding ("computing") it. Still the proof might give some hints for developing an algorithm for this (classic) inverse problem. 
10/09/06  Tom Beale
Duke University 
TITLE: Computing with Singular and Nearly Singular Integrals
Abstract: We will describe a simple, direct approach to computing a singular or nearly singular integral, such as a harmonic function given by a layer potential on a curve in 2D or a surface in 3D. The value is found by a standard quadrature, using a regularized form of the singularity, with correction terms added for the errors due to regularization and discretization. These corrections are found by local analysis near the singularity. This technique might be useful in fluid calculations with moving interfaces, since a pressure term due to a boundary force can be written as a layer potential. The accurate evaluation of a layer potential near the curve or surface on which it is defined is not routine, since the integral is nearly singular. In work with M.C. Lai, we solve boundary value problems in 2D by computing the integral at grid points near the curve as described and using these values to find those at all points. A similar approach works in 3D, with the surface integrals computed in overlapping coordinate grids on the surface. To solve a boundary value problem, we first need to solve an integral equation for the strength of a dipole layer on the surface. We have proved that the solution of the discrete integral equation converges to the exact solution. In related work with G. Baker, we use a special choice of regularization in a boundary integral calculation of an unstable interface in 2D inviscid flow, such as a RayleighTaylor flow with a heavy fluid over a lighter fluid. 
10/16/06  Greg Lyng
University of Wyoming 
TITLE: The Secondary Caustic in the Semiclassical Limit for the
Focusing Nonlinear Schroedinger Equation
Abstract: We consider the cubic focusing nonlinear Schroedinger equation in one space dimension, with fixed initial data, in the semiclassical limit when a dispersion parameter analogous to Planck's constant tends to zero. This problem is relevant in the theory of "supercontinuum generation" in which coherent white light is produced from a monochromatic source by propagation in an optical fiber with small dispersion. This is a highly unstable problem with limiting "dynamics" valid for analytic initial data being described by an initialvalue problem for a nonlinear system of elliptic PDEs. Nonetheless, the assumption of analyticity of the initial data allows for detailed asymptotics to be obtained with the help of the solution of the nonlinear Schroedinger equation via the inversescattering transform. The solutions display remarkable structure consisting of regions of smoothly modulated quasiperiodic oscillations separated by asymptotically sharp "caustic" curves in the space/time plane. The first "primary" caustic curve has been explained by passage to an appropriate continuum limit of a dense distribution of discrete eigenvalues of an associated linear operator. This talk will describe recent joint work with Peter Miller (University of Michigan) in which the "secondary" caustic curve is studied, and a new mechanism is found to explain it that depends essentially on the discrete nature of the spectrum and (unlike the case of the primary caustic) cannot be obtained from a naive continuum limit. 
10/23/06  Robert Hardt
Rice University 
TITLE: TBA
Abstract: 
11/06/06  Jiequan Li
PSU 
TITLE: Interaction of Rarefaction Waves of
the TwoDimensional SelfSimilar Compressible Euler Equations
Abstract: I will talk about classical selfsimilar solutions to the interaction of two arbitrary planar rarefaction waves for the polytropic Euler equations in two space dimensions that we have constructed. The binary interaction represents a major type of interaction in the twodimensional Riemann problems, and includes in particular the classical problem of the expansion of a wedge of gas into vacuum. Based on the hodograph transformation, the method involves the phase space analysis of a secondorder equation and the inversion back to (or development onto) the physical space. I will add some basic information to make the talk accessible to nonexperts. 
11/20/06  Hengguang Li
Penn State University 
TITLE: Weighted Sobolev Spaces and Cornerlike Singularities for Elliptic
PDEs
Abstract: Consider the secondorder elliptic equations with certain boundary conditions on a bounded domain with angular or conical points in Rd, d = 2 or 3. It is well known that the solutions may have singularities around nonsmooth points on the boundary, namely, the corners in 2D, the vertices and edges in 3D. In fact, the solution may be only in H1+s, 0 < s < 1, even if the give data is quite smooth. On the other hand, the lack of the regularity of the solution in usual Sobolev spaces Hm will destroy the convergence rate for the numerical solution in the finite element method, which is expected for the smooth solution. In this talk, I will introduce the proper weighted Sobolev space Km a in which the corner singularities can be treated very well. To fix ideas, we use the Poissons equation with Dirichlet boundary condition as the model problem. Classical regularity arguments for elliptic equations in usual Sobolev spaces for smooth domains will be restated in terms of weighted Sobolev spaces for Lipschitz domains. Based on this theoretical framework, special finite elements can be designed to overcome the difficulty in convergence rate from the singularity. Our efforts on recovering the optimal convergence rate for the numerical solution have proved to be successful. Numerical results for different cornerlike singularities will be shown. 
12/4/06  Jesse Barlow
Penn State University 
TITLE: Structured Total Least Squares and Image Processing
Abstract: Color image restoration is considered. As the point spread function and the observed image are contaminated by noise, total least squares (TLS) methods are an appropriate approach to restoring the original image. The way in which the blurring matrices represent the errors leads toward a structured total least squares (STLS) approach to compensate for the contamination of the point spread function. The illconditioning of said blurring matrices requires Tikhanov regualization to stabilize the solution. Since the STLS method requires the solution of a nonlinear least squares problem based upon imposing Neumann boundary conditions, solving the large linear systems resulting from a GaussNewton iteration can be done very efficiently using a conjugate gradient method preconditioned by a Discrete Cosine Transform (DCT) based preconditioner. 
12/11/06  Anna Mazzucato
Penn State University 
TITLE: Inverse Problems in Elasticity
Abstract: I will discuss the inverse problem of identifying the elastic parameters in the interior of an anisotropic inhomogeneus body from dynamic displacementtraction measurements made at the surface. This problem has applications to medical imaging and exploration seismology. I will consider in particular whether the surface measurements allow to uniquely determine the parameters and discuss some results for transversely isotropic media. This is joint work with L. Rachele (RPI). 


Sponsored in part by CCMA and by individual faculty grants  
