August 2005
Embedding properties for Weighted-Sobolev spaces in unbounded domains
Friday, 8/12/2005 3:00 PM-4:00 PM
Stratton Hall, 203
Colloquium/PDE Seminar
Speaker: Hirokazu Ohya
Department of Mathematics, Waseda University, Japan
I am concerned with the embedding properties of certain functional spaces in unbounded domains. In this talk we discuss about Weighted-Sobolev (Banach) spaces with unbounded coefficients, especially for exponentially growing ones. In such cases, Escobedo and Kavian have shown the properties of continuous and compact embedding of Weighted-Sobolev (Hilbert) spaces with simple calculations. By using similar arguments given in above, we can derive (or relax) the sufficient conditions on these coefficients where the embedding properties work well. The above properties are used in showing the existence of solutions for some quasilinear elliptic problems in unbounded domains. I also talk about some mechanism which the embedding of Weighted-Sobolev spaces is compact or not.
Sponsored by: Mathematical Sciences Department
For more information, e-mail ma-chair@wpi.edu or call 508-831-5241.
September 2005
The Aftermath of Calculus "Reform": What Changed, What Didn't, What to Do Now
Thursday, 9/15/2005 11:00 AM-12:00 PM
Washburn Shops & Stoddard Laboratories, 229
Speaker: Thomas Tucker (Colgate University)
Abstract: It is almost 20 years since the Tulane Conference, followed by a six-year NSF program, called for a change in calculus instruction. Some things stuck: encouraging students to work together, broadening conceptual understanding with graphical, numerical, and verbal viewpoints. Some things did not: technology, leaner and cleaner textbooks. A veteran of the "reform" wars (chair of CRAFTY 1987-1990 and co-author of the Hughes-Hallett textbook) offers some thoughts on a sensible approach to teaching calculus.
For more information, e-mail ma-chair@wpi.edu.
Distinguishability of Maps
Thursday, 9/15/2005 4:00 PM-5:00 PM
Stratton Hall, 203
Speaker: Thomas Tucker (Colgate University)
Abstract: A faithful action of a group A on a set X has distinguishing number D(A,X)=k if the elements of X can be colored with k colors so that the action is not color-preserving and no smaller k suffices. Alberston and Collins first introduced the idea about seven years ago in the context of graphs, although it goes back to an old "recreational math'' problem about coloring keys on a keyring. Here it is applied to maps (i.e., graphs embedded in surfaces) and the results are striking.
The technical difficulties are significant and require many new ideas about map automorphisms.
For more information, e-mail ma-chair@wpi.edu.
Bayesian Predictive Inference for a Binary Random Variable: Risk-adjusted Assessments of Medical Outcomes
Friday, 9/23/2005 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Michael Racz, Ph.D. (NYS Department of Health/School of Public Health)
Abstract: This talk will examine models and methodologies that are commonly used for provider
profiling, and evaluation of the quality of health care. The conventional approach to computing these
provider assessments is to use a likelihood-based frequentist methodology. The use of locally uniform
prior distributions allows direct comparison between the likelihood-based and Bayesian methods. I
compare the frequentist and Bayesian results for each of three models. The models are applied to the
data used by the New York State Department of Health (DOH) for its annually released reports that
profile hospitals permitted to perform coronary artery bypass graft (CABG) surgery. With the
advances of Markov Chain Monte Carlo methods, Bayesian methods are easily implemented and are
preferable to standard frequentist methods for models with a binary dependent variable since the latter
always rely on asymptotic approximations. One of the three types of models presented assumes
exchangeability amongst hospitals. The results from fitting these exchangeable models, either frequentist
or Bayesian, exhibit the phenomenon of shrinkage. In addition to comparisons between the frequentist
and Bayesian shrinkage methods, results are compared from the non-shrinkage and shrinkage methods
using data from the DOH CABG reports.
For more information, e-mail ma-chair@wpi.edu.
October 2005
Harold J. Gay Lecture Series
Tuesday, 10/4/2005 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Louis Nirenberg (NYU)
Title: "Distance to the boundary and Hamilton-Jacobi equations"
Abstract: We study the set of points where the distance function to the boundary is not smooth. Its dimension is estimated. A similar result is derived for the singular set of solutions of some Hamilton-Jacobi equations.
Refreshments available one half hour before lecture time.
For more information, e-mail ma-chair@wpi.edu.
Harold J. Gay Lecture Series
Friday, 10/7/2005 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Louis Nirenberg (NYU)
Title: Estimates for laminar materials
Abstract: Some problems on laminar materials lead to elliptic systems, with coefficients that are smooth in subregions but may jump from region to region. It is of interest to get estimates on the solution and its derivatives, in each subregion, which are independent of the narrowness of the regions. Some such estimates are presented.
Refreshments available one half hour before the lecture.
For more information, e-mail ma-chair@wpi.edu.
Automatic Differentiation and its Rold in Large-Scale Optimization
Thursday, 10/13/2005 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Paul Hovland (Mathematics and Computer Science Division, Argonne National Laboratory)
Abstract: Automatic differentiation is a technique for computing the derivatives of a function defined by a computer subprogram. We provide an introduction to automatic differentiation tools and theory and describe the role of current and next generation automatic differentiation tools in large-scale optimization. We focus on the computational costs of automatic differentiation and the trend toward tools that are more robust, easier to use, and more powerful. We conclude with a short description of our research agenda, with an emphasis on the motivational role played by current and future applications.
For more information, e-mail ma-chair@wpi.edu.
November 2005
BDDC Domain Decomposition Algorithms: Methods with Three Levels and for Flow in Porous Media
Friday, 11/11/2005 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Xuemin Tu (Courant Institute, NYU)
Abstract: Two inexact coarse solvers for Balancing Domain Decomposition by Constraints (BDDC) algorithms are introduced and analyzed. These solvers help remove a bottleneck for the two-level BDDC algorithms related to the cost of the coarse problem when the number of subdomains is large. At the same time, a good convergence rate is maintained.
BDDC algorithms are also developed for the linear systems arising from flow in porous media discretized with mixed and hybrid finite elements.
Our methods are proven to be scalable and the condition numbers of the operators with our BDDC preconditioners grow only polylogarithmically with the size of the subdomain problems.
For more information, e-mail ma-chair@wpi.edu.
Methods of Motion Analysis in Microscopic and Echocardiographic Images
Friday, 11/18/2005 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Ed Marcus, Ph.D
Department of Cardiology
Children's Hospital Boston
Harvard Medical School
Abstract: Moving structures of the body present a real challenge to recording and analyzing image data. We will look at two examples in biological imaging where the understanding of movement provides fresh insight into underlying function and physiology. Case studies include 1) microscopic images of moving cells and 2) echocardiogram images of beating hearts. Special techniques developed to process these moving image structures will be described, and the resulting clinical implications of the motion analysis will be open for discussion.
For more information, e-mail ma-chair@wpi.edu.
December 2005
Mathematics as a Path Integral to Physics
Friday, 12/2/2005 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Carolann Koleci (Physics Department, WPI)
Abstract: Most of the world around us can be described by physics. On the large scale we have planetary motion, while on the small scale we have quantum mechanics.
Fundamental to our knowledge construction of the physical sciences and engineering sciences is mathematics. We can learn about fundamental concepts in various contexts. For example, calculus provides us with fundamental knowledge concerning the interrelation of displacement, velocity, and acceleration. This talk will feature a physicist's perspective on the scope of mathematics taught in the context of introductory physics. All roads may lead to Rome, but mathematics is a path integral to the rudimentary teachings of physics, and physics is a path integral to the rudimentary teachings of mathematics.
This presentation is part of the "Calculus Dialog" series, supported by the WPI Educational Development Council.
Students and faculty are welcome. Refreshments in SH107, 10:30 - 11:00AM
For more information, e-mail ma-chair@wpi.edu.
Distinguished Statistics Lecture--"Sampling in a Data Rich World"
Friday, 12/9/2005 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Dr. Fritz Scheuren (President, American Statistical Association)
Abstract: The paradigm of sampling is shifting in business surveys in response to large operational data sets and cheap computing, combined with expensive (and necessarily) small samples.
In this environment the role of randomization at inference inevitably has been increasingly overshadowed by the role of models and diagnostics that robustness and representativeness have been achieved. Some history will be given of this movement and some new and old tools discussed in ensemble, using a case study example from practice.
For more information, e-mail ma-chair@wpi.edu.
Level set method in moving boundary problems
Friday, 12/16/2005 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Roger Y. Lui (Mathematical Sciences, WPI) and Ravi Vuta (Mechanical Engineering, WPI)
Abstract: Level set method is a powerful numerical method introduced by Osher and Sethian (J. Computational Physics (1988), vol. 79, 12-49) to compute motion of a front whose speed depends on the local curvature. Since then, the method has been improved and applied in many areas of applied mathematics and physics. In this talk, I will give an introduction to the level set method by using it to solve a moving boundary value problem which consists of a porous medium type equation satisfied in a closed bounded domain in the plane that moves with time. The boundary of the domain is given by the zero level curve of a distance function which satisfies some kind of Hamilton-Jacobi equation. We allow the normal speed at the boundary of the domain to depend not only on the curvature of the domain but also on its perimeter and magnitude of the gradient of the solution of the pde. We then run some simulations and see what happen. This moving boundary problem arises from a model in cell motility. Second part of the talk will be delivered by Ravi Vuta.
For more information, e-mail ma-chair@wpi.edu.
January 2006
Flux-implicit weighted essentially non-oscillatory methods
Friday, 1/20/2006 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Sigal Gottlieb (University of Massachusetts - Dartmouth)
Abstract: The weighted essentially non-oscillatory method (WENO) is an
excellent spatial discretization for hyperbolic partial differential
equations with discontinuous solutions. However, the time-step restriction
associated with explicit methods may pose severe limitations on their use in
applications requiring large scale computations. An efficient implicit WENO
method is necessary. In this talk, I will discuss a prototype flux-implicit
WENO (iWENO) method. Numerical tests on classical scalar equations show that
this is a viable and stable method, which requires appropriate time stepping
methods. I will also discuss the extension of this method to a multi-domain
setting.
For more information, e-mail ma-chair@wpi.edu.
Anisotropic Elasto-plastic Model for Large Metal Forming Deformation
Friday, 1/27/2006 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Sanda Cleja-Tigoiu (Department of Mechanical Engineering, Texas A&M University and School of Mathematics, University of Bucharest)
Abstract: A rate-type model concerning addressing the influence of the microstructure on the macroscopic behavior of crystalline material is presented. Material symmetry concepts as well as the presence of the plastic spin are considered. In particular, two types of anisotropy are considered: the preexisting initial anisotropy (say orthotropy) and anisotropy induced by the plastic deformations. This model pertains to small elastic strains but large elastic rotations and plastic deformations - feature commonly found in metal forming deformation processes. Cases of plane strain and plain stress state behavior are also analyzed and locally homogeneous processes are simulated. Computational results underline the role played in the theory by different material constants, as well as the explicit role of the plastic spin. A discussion of the results in the context of recent experimental results is presented.
For more information, e-mail ma-chair@wpi.edu.
February 2006
A Stochastic Immersed Boundary Method Incorporating Thermal Fluctuations : Toward Modeling Cellular Micromechanics
Friday, 2/3/2006 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Paul Atzberger (RPI)
Abstract: The mechanics of many cellular systems involve elastic structures which interact with a fluid, for example the outer cell membrane deforms during protrusions generated during motility and cell organelles such as the Golgi Apparatus and Mitochondria involve membranes which deform and bud vesicular and tubular structures during biological processes. Modeling, analyzing, and simulating the mechanics of such systems presents many mathematical challenges.
The immersed boundary method is one modeling approach for such systems, and has been applied to many macroscopic biological problems, such as blood flow in the heart and lift generation in insect flight.
At the length scales of cells and cell organelles, thermal fluctuations also become significant and must be taken into account. In this talk we discuss an extension of the immersed boundary method framework which incorporates thermal fluctuations through appropriate stochastic forcing terms in the fluid equations. This gives a system of stiff SPDE's for which standard numerical approaches perform poorly. We discuss a novel stochastic numerical method which exploits stochastic calculus to handle stiff features of the equations. We further show how this numerical method can be applied in practice to model the basic microscopic mechanics of polymers, polymer knots, membrane sheets, and vesicles. We also discuss preliminary work on modeling the dynamics of cell organelle structures.
For more information, e-mail ma-chair@wpi.edu.
Applications of Potential Theory to Numerical Linear Algebra
Friday, 2/10/2006 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: John Rossi (Virginia Tech)
Abstract: It is shown that convergence rates of iterative methods based on Krylov subspace
techniques can be quantified via elementary results in 2 dimensional potential theory involving Green's functions and capacity. We will demonstrate our results by looking at three different problems involving a very large matrix A:
- Solving Ax=b
- Finding the eigenspaces of A
- Computing f(A) where f is analytic in a domain containing the eigenvalues of A.
For more information, e-mail ma-chair@wpi.edu.
An Application of Stochastic Analysis
Tuesday, 2/14/2006 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Luis Roman (WPI)
Abstract: In mathematical studies of the transport of pollutants in groundwater and of oil recovery processes the permeability tensor is assumed to be random and models the properties of the rocks, which are not know with certainty. The governing system of equations in the laboratory scale is derived using Darcy's Law combined with the conservation laws for the two phases (oil and water), and can be expressed in the familiar form of "pressure" and "saturation" coupled equations. Typically, this system is solved through decoupling procedures, which leads to an elliptic stochastic partial differential equation for the pressure. The use of Monte Carlo methods to approximate the expected value, higher moments, or other functionals of the pressure require solving similar system of equations as many times as samples of the permeability are taken, thus it becomes expensive and impractical. In this talk, we will explain how Stochastic Analysis can be used to solve this problem and obtain an efficient numerical method to approximate the pressure. This method reduces the simulation of the approximation to solving a linear system of equations only once and simulation of a finite number of independent normally distributed random variables. We will also discuss the main differences of this approach to other related work in the field and provide open problems for future research.
For more information, e-mail ma-chair@wpi.edu.
Forward-backward stochastic differential equations and affine models in finance
Tuesday, 2/21/2006 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Cody Hyndman (University of Calgary)
Abstract: Backward stochastic differential equations have been applied to pricing and hedging various contingent claims in finance. In the first part of this talk a brief introduction to some basic contingent claims and to backward stochastic differential equations shall be provided.
In the remainder of the talk we shall consider affine term-structure models for the bond price where the risk-neutral dynamics of the factors process are modeled by a square-root affine diffusion. By linking the existence and uniqueness of certain forward-backward stochastic differential equations with the solvability of Riccati ordinary differential equations we are able to characterize the bond price as an exponential affine function of the factors in a new way.
We also show how our approach can be generalized to other contingent claims such as futures and forward contracts. We consider a class of affine factor models driving the price of a risky asset paying stochastic dividends in an economy with stochastic interest rates. We are able to characterize the forward price and futures price as exponential affine functions of the factors process using the forward-backward stochastic differential equation approach.
For more information, e-mail ma-chair@wpi.edu.
March 2006
Thin film equations for fluid motion driven by surfactants
Friday, 3/17/2006 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Michael Shearer (North Carolina State University)
Abstract: The study of thin liquid films is motivated by two main areas of application, namely flow of mucus in the lung, and coating flows in industrial processes such as microchip manufacture. In the lubrication approximation, the motion of a thin liquid film is described by a single fourth-order partial differential equation that models the evolution of the height of the film. When the fluid is driven by a Marangoni force generated by a distribution of insoluble surfactant, the thin film equation is coupled to an equation for the concentration of surfactant. In this talk, I show the basic structure of this system, and begin an analysis of wave-like solutions in the specific context of a thin film flowing down an inclined plane. Numerical simulations reveal an array of traveling waves, which
persists when capillarity and surface diffusion are neglected. The analysis of the limiting system has some surprises, and in this talk, I show how far we have come in understanding the numerical results analytically, and the analytical results numerically.
For more information, e-mail ma-chair@wpi.edu.
H.J. Gay Lecture Series: Degenerate Symmetric Matrices
Friday, 3/31/2006 2:00 PM-3:00 PM
Higgins Laboratories, 218
Speaker: Peter D. Lax (Courant Institute of Mathematical Sciences, NYU)
Abstract: A symmetric matrix is called degenerate by physicists if it has a multiple eigenvalue.Wigner and von Neumann have shown long ago that the degenerate matrices form a variety of codimension two in the space of all symmetric matrices.This explains the phenomenon of "avoidance of crossing".On the other hand the degenerate matrices are characterised by the single equation discr(S)=0, where discr(S) is the discriminant of S.In this talk we investigate the nature of the discriminant, especially its representation as a sum of squares.
In the second part it will be shown that some pencils of real symmetric matrices always contain a degenerate one.
For more information, e-mail ma-chair@wpi.edu.
April 2006
Introduction to the Mathematical Theory of Intertype Interaction
Friday, 4/7/2006 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Yury Shestopalov (Karlstad University, Sweden)
Abstract: We consider some mathematical aspects of the theory of interaction of oscillations and waves in electromagnetics and acoustics. The problem is considered in terms of the analysis of the spectral and various critical points of multi-parameter operator-valued functions (OVFs), in particular, integral and infinite-matrix (summation) OVFs, when some of the operator parameters are varied. Interaction of oscillations in a cylindrical slotted resonator whose cross section is formed by two rectangular domains is taken as an example. We apply the reduction of the initial boundary eigenvalue problem for the Helmholtz equation to a Fredholm integral equation of the first kind with logarithmic singularity and then to infinite systems with respect to the unknown Fourier coefficients of the solution. We demonstrate that under certain conditions the electromagnetic field distributions become unstable with respect to small variations of certain parameters of the structure (geometric, permittivity etc.) and the interaction of oscillations takes place. We show that such an unstable behavior reveals a specific interaction phenomenon and occurs when the frequency of the excited oscillation is varied in vicinity of the resonant frequencies (eigenfrequencies) of one of the partial domains.
For more information, e-mail ma-chair@wpi.edu.
Solitary Wave Dynamics In Optical and Soft-Condensed Matter
Friday, 4/28/2006 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Panayotis Kevrekidis (Department of Mathematics & Statistics, University of Massachusetts-Amherst)
Abstract: In the past few years, there has been a tremendous increase
in the interest on nonlinear wave and coherent structures in optical
applications (such as waveguide arrays and photorefractive
materials) and soft-condensed matter ones (such as Bose-Einstein
condensates). In these areas, modelling through nonlinear partial
differential equations and differential-difference equations has
proved extremely valuable in predicting the type of waveforms that
spontaneously emerge as well as the ones that can be engineered and
are supported by nonlinearity. I will try to give a series of relevant
examples where theoretical results and numerical computations have
been directly or indirectly confirmed by experiments.
Such results will include the development of instabilities (such as
the modulational instability), the formation of waves (such as
continuum and "discrete'' vortices) and the generation of more exotic
structures (such as ring patterns) among others.
Jointly sponsored by The Department of Chemical Engineering, WPI.
For more information, e-mail ma-chair@wpi.edu.
August 2005
Embedding properties for Weighted-Sobolev spaces in unbounded domains
Friday, 8/12/2005 3:00 PM-4:00 PM
Stratton Hall, 203
Colloquium/PDE Seminar
Speaker: Hirokazu Ohya
Department of Mathematics, Waseda University, Japan
I am concerned with the embedding properties of certain functional spaces in unbounded domains. In this talk we discuss about Weighted-Sobolev (Banach) spaces with unbounded coefficients, especially for exponentially growing ones. In such cases, Escobedo and Kavian have shown the properties of continuous and compact embedding of Weighted-Sobolev (Hilbert) spaces with simple calculations. By using similar arguments given in above, we can derive (or relax) the sufficient conditions on these coefficients where the embedding properties work well. The above properties are used in showing the existence of solutions for some quasilinear elliptic problems in unbounded domains. I also talk about some mechanism which the embedding of Weighted-Sobolev spaces is compact or not.
Sponsored by: Mathematical Sciences Department
For more information, e-mail ma-chair@wpi.edu or call 508-831-5241.
September 2005
The Aftermath of Calculus "Reform": What Changed, What Didn't, What to Do Now
Thursday, 9/15/2005 11:00 AM-12:00 PM
Washburn Shops & Stoddard Laboratories, 229
Speaker: Thomas Tucker (Colgate University)
Abstract: It is almost 20 years since the Tulane Conference, followed by a six-year NSF program, called for a change in calculus instruction. Some things stuck: encouraging students to work together, broadening conceptual understanding with graphical, numerical, and verbal viewpoints. Some things did not: technology, leaner and cleaner textbooks. A veteran of the "reform" wars (chair of CRAFTY 1987-1990 and co-author of the Hughes-Hallett textbook) offers some thoughts on a sensible approach to teaching calculus.
For more information, e-mail ma-chair@wpi.edu.
Distinguishability of Maps
Thursday, 9/15/2005 4:00 PM-5:00 PM
Stratton Hall, 203
Speaker: Thomas Tucker (Colgate University)
Abstract: A faithful action of a group A on a set X has distinguishing number D(A,X)=k if the elements of X can be colored with k colors so that the action is not color-preserving and no smaller k suffices. Alberston and Collins first introduced the idea about seven years ago in the context of graphs, although it goes back to an old "recreational math'' problem about coloring keys on a keyring. Here it is applied to maps (i.e., graphs embedded in surfaces) and the results are striking.
The technical difficulties are significant and require many new ideas about map automorphisms.
For more information, e-mail ma-chair@wpi.edu.
Bayesian Predictive Inference for a Binary Random Variable: Risk-adjusted Assessments of Medical Outcomes
Friday, 9/23/2005 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Michael Racz, Ph.D. (NYS Department of Health/School of Public Health)
Abstract: This talk will examine models and methodologies that are commonly used for provider
profiling, and evaluation of the quality of health care. The conventional approach to computing these
provider assessments is to use a likelihood-based frequentist methodology. The use of locally uniform
prior distributions allows direct comparison between the likelihood-based and Bayesian methods. I
compare the frequentist and Bayesian results for each of three models. The models are applied to the
data used by the New York State Department of Health (DOH) for its annually released reports that
profile hospitals permitted to perform coronary artery bypass graft (CABG) surgery. With the
advances of Markov Chain Monte Carlo methods, Bayesian methods are easily implemented and are
preferable to standard frequentist methods for models with a binary dependent variable since the latter
always rely on asymptotic approximations. One of the three types of models presented assumes
exchangeability amongst hospitals. The results from fitting these exchangeable models, either frequentist
or Bayesian, exhibit the phenomenon of shrinkage. In addition to comparisons between the frequentist
and Bayesian shrinkage methods, results are compared from the non-shrinkage and shrinkage methods
using data from the DOH CABG reports.
For more information, e-mail ma-chair@wpi.edu.
October 2005
Harold J. Gay Lecture Series
Tuesday, 10/4/2005 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Louis Nirenberg (NYU)
Title: "Distance to the boundary and Hamilton-Jacobi equations"
Abstract: We study the set of points where the distance function to the boundary is not smooth. Its dimension is estimated. A similar result is derived for the singular set of solutions of some Hamilton-Jacobi equations.
Refreshments available one half hour before lecture time.
For more information, e-mail ma-chair@wpi.edu.
Harold J. Gay Lecture Series
Friday, 10/7/2005 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Louis Nirenberg (NYU)
Title: Estimates for laminar materials
Abstract: Some problems on laminar materials lead to elliptic systems, with coefficients that are smooth in subregions but may jump from region to region. It is of interest to get estimates on the solution and its derivatives, in each subregion, which are independent of the narrowness of the regions. Some such estimates are presented.
Refreshments available one half hour before the lecture.
For more information, e-mail ma-chair@wpi.edu.
Automatic Differentiation and its Rold in Large-Scale Optimization
Thursday, 10/13/2005 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Paul Hovland (Mathematics and Computer Science Division, Argonne National Laboratory)
Abstract: Automatic differentiation is a technique for computing the derivatives of a function defined by a computer subprogram. We provide an introduction to automatic differentiation tools and theory and describe the role of current and next generation automatic differentiation tools in large-scale optimization. We focus on the computational costs of automatic differentiation and the trend toward tools that are more robust, easier to use, and more powerful. We conclude with a short description of our research agenda, with an emphasis on the motivational role played by current and future applications.
For more information, e-mail ma-chair@wpi.edu.
November 2005
BDDC Domain Decomposition Algorithms: Methods with Three Levels and for Flow in Porous Media
Friday, 11/11/2005 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Xuemin Tu (Courant Institute, NYU)
Abstract: Two inexact coarse solvers for Balancing Domain Decomposition by Constraints (BDDC) algorithms are introduced and analyzed. These solvers help remove a bottleneck for the two-level BDDC algorithms related to the cost of the coarse problem when the number of subdomains is large. At the same time, a good convergence rate is maintained.
BDDC algorithms are also developed for the linear systems arising from flow in porous media discretized with mixed and hybrid finite elements.
Our methods are proven to be scalable and the condition numbers of the operators with our BDDC preconditioners grow only polylogarithmically with the size of the subdomain problems.
For more information, e-mail ma-chair@wpi.edu.
Methods of Motion Analysis in Microscopic and Echocardiographic Images
Friday, 11/18/2005 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Ed Marcus, Ph.D
Department of Cardiology
Children's Hospital Boston
Harvard Medical School
Abstract: Moving structures of the body present a real challenge to recording and analyzing image data. We will look at two examples in biological imaging where the understanding of movement provides fresh insight into underlying function and physiology. Case studies include 1) microscopic images of moving cells and 2) echocardiogram images of beating hearts. Special techniques developed to process these moving image structures will be described, and the resulting clinical implications of the motion analysis will be open for discussion.
For more information, e-mail ma-chair@wpi.edu.
December 2005
Mathematics as a Path Integral to Physics
Friday, 12/2/2005 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Carolann Koleci (Physics Department, WPI)
Abstract: Most of the world around us can be described by physics. On the large scale we have planetary motion, while on the small scale we have quantum mechanics.
Fundamental to our knowledge construction of the physical sciences and engineering sciences is mathematics. We can learn about fundamental concepts in various contexts. For example, calculus provides us with fundamental knowledge concerning the interrelation of displacement, velocity, and acceleration. This talk will feature a physicist's perspective on the scope of mathematics taught in the context of introductory physics. All roads may lead to Rome, but mathematics is a path integral to the rudimentary teachings of physics, and physics is a path integral to the rudimentary teachings of mathematics.
This presentation is part of the "Calculus Dialog" series, supported by the WPI Educational Development Council.
Students and faculty are welcome. Refreshments in SH107, 10:30 - 11:00AM
For more information, e-mail ma-chair@wpi.edu.
Distinguished Statistics Lecture--"Sampling in a Data Rich World"
Friday, 12/9/2005 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Dr. Fritz Scheuren (President, American Statistical Association)
Abstract: The paradigm of sampling is shifting in business surveys in response to large operational data sets and cheap computing, combined with expensive (and necessarily) small samples.
In this environment the role of randomization at inference inevitably has been increasingly overshadowed by the role of models and diagnostics that robustness and representativeness have been achieved. Some history will be given of this movement and some new and old tools discussed in ensemble, using a case study example from practice.
For more information, e-mail ma-chair@wpi.edu.
Level set method in moving boundary problems
Friday, 12/16/2005 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Roger Y. Lui (Mathematical Sciences, WPI) and Ravi Vuta (Mechanical Engineering, WPI)
Abstract: Level set method is a powerful numerical method introduced by Osher and Sethian (J. Computational Physics (1988), vol. 79, 12-49) to compute motion of a front whose speed depends on the local curvature. Since then, the method has been improved and applied in many areas of applied mathematics and physics. In this talk, I will give an introduction to the level set method by using it to solve a moving boundary value problem which consists of a porous medium type equation satisfied in a closed bounded domain in the plane that moves with time. The boundary of the domain is given by the zero level curve of a distance function which satisfies some kind of Hamilton-Jacobi equation. We allow the normal speed at the boundary of the domain to depend not only on the curvature of the domain but also on its perimeter and magnitude of the gradient of the solution of the pde. We then run some simulations and see what happen. This moving boundary problem arises from a model in cell motility. Second part of the talk will be delivered by Ravi Vuta.
For more information, e-mail ma-chair@wpi.edu.
January 2006
Flux-implicit weighted essentially non-oscillatory methods
Friday, 1/20/2006 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Sigal Gottlieb (University of Massachusetts - Dartmouth)
Abstract: The weighted essentially non-oscillatory method (WENO) is an
excellent spatial discretization for hyperbolic partial differential
equations with discontinuous solutions. However, the time-step restriction
associated with explicit methods may pose severe limitations on their use in
applications requiring large scale computations. An efficient implicit WENO
method is necessary. In this talk, I will discuss a prototype flux-implicit
WENO (iWENO) method. Numerical tests on classical scalar equations show that
this is a viable and stable method, which requires appropriate time stepping
methods. I will also discuss the extension of this method to a multi-domain
setting.
For more information, e-mail ma-chair@wpi.edu.
Anisotropic Elasto-plastic Model for Large Metal Forming Deformation
Friday, 1/27/2006 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Sanda Cleja-Tigoiu (Department of Mechanical Engineering, Texas A&M University and School of Mathematics, University of Bucharest)
Abstract: A rate-type model concerning addressing the influence of the microstructure on the macroscopic behavior of crystalline material is presented. Material symmetry concepts as well as the presence of the plastic spin are considered. In particular, two types of anisotropy are considered: the preexisting initial anisotropy (say orthotropy) and anisotropy induced by the plastic deformations. This model pertains to small elastic strains but large elastic rotations and plastic deformations - feature commonly found in metal forming deformation processes. Cases of plane strain and plain stress state behavior are also analyzed and locally homogeneous processes are simulated. Computational results underline the role played in the theory by different material constants, as well as the explicit role of the plastic spin. A discussion of the results in the context of recent experimental results is presented.
For more information, e-mail ma-chair@wpi.edu.
February 2006
A Stochastic Immersed Boundary Method Incorporating Thermal Fluctuations : Toward Modeling Cellular Micromechanics
Friday, 2/3/2006 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Paul Atzberger (RPI)
Abstract: The mechanics of many cellular systems involve elastic structures which interact with a fluid, for example the outer cell membrane deforms during protrusions generated during motility and cell organelles such as the Golgi Apparatus and Mitochondria involve membranes which deform and bud vesicular and tubular structures during biological processes. Modeling, analyzing, and simulating the mechanics of such systems presents many mathematical challenges.
The immersed boundary method is one modeling approach for such systems, and has been applied to many macroscopic biological problems, such as blood flow in the heart and lift generation in insect flight.
At the length scales of cells and cell organelles, thermal fluctuations also become significant and must be taken into account. In this talk we discuss an extension of the immersed boundary method framework which incorporates thermal fluctuations through appropriate stochastic forcing terms in the fluid equations. This gives a system of stiff SPDE's for which standard numerical approaches perform poorly. We discuss a novel stochastic numerical method which exploits stochastic calculus to handle stiff features of the equations. We further show how this numerical method can be applied in practice to model the basic microscopic mechanics of polymers, polymer knots, membrane sheets, and vesicles. We also discuss preliminary work on modeling the dynamics of cell organelle structures.
For more information, e-mail ma-chair@wpi.edu.
Applications of Potential Theory to Numerical Linear Algebra
Friday, 2/10/2006 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: John Rossi (Virginia Tech)
Abstract: It is shown that convergence rates of iterative methods based on Krylov subspace
techniques can be quantified via elementary results in 2 dimensional potential theory involving Green's functions and capacity. We will demonstrate our results by looking at three different problems involving a very large matrix A:
- Solving Ax=b
- Finding the eigenspaces of A
- Computing f(A) where f is analytic in a domain containing the eigenvalues of A.
For more information, e-mail ma-chair@wpi.edu.
An Application of Stochastic Analysis
Tuesday, 2/14/2006 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Luis Roman (WPI)
Abstract: In mathematical studies of the transport of pollutants in groundwater and of oil recovery processes the permeability tensor is assumed to be random and models the properties of the rocks, which are not know with certainty. The governing system of equations in the laboratory scale is derived using Darcy's Law combined with the conservation laws for the two phases (oil and water), and can be expressed in the familiar form of "pressure" and "saturation" coupled equations. Typically, this system is solved through decoupling procedures, which leads to an elliptic stochastic partial differential equation for the pressure. The use of Monte Carlo methods to approximate the expected value, higher moments, or other functionals of the pressure require solving similar system of equations as many times as samples of the permeability are taken, thus it becomes expensive and impractical. In this talk, we will explain how Stochastic Analysis can be used to solve this problem and obtain an efficient numerical method to approximate the pressure. This method reduces the simulation of the approximation to solving a linear system of equations only once and simulation of a finite number of independent normally distributed random variables. We will also discuss the main differences of this approach to other related work in the field and provide open problems for future research.
For more information, e-mail ma-chair@wpi.edu.
Forward-backward stochastic differential equations and affine models in finance
Tuesday, 2/21/2006 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Cody Hyndman (University of Calgary)
Abstract: Backward stochastic differential equations have been applied to pricing and hedging various contingent claims in finance. In the first part of this talk a brief introduction to some basic contingent claims and to backward stochastic differential equations shall be provided.
In the remainder of the talk we shall consider affine term-structure models for the bond price where the risk-neutral dynamics of the factors process are modeled by a square-root affine diffusion. By linking the existence and uniqueness of certain forward-backward stochastic differential equations with the solvability of Riccati ordinary differential equations we are able to characterize the bond price as an exponential affine function of the factors in a new way.
We also show how our approach can be generalized to other contingent claims such as futures and forward contracts. We consider a class of affine factor models driving the price of a risky asset paying stochastic dividends in an economy with stochastic interest rates. We are able to characterize the forward price and futures price as exponential affine functions of the factors process using the forward-backward stochastic differential equation approach.
For more information, e-mail ma-chair@wpi.edu.
March 2006
Thin film equations for fluid motion driven by surfactants
Friday, 3/17/2006 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Michael Shearer (North Carolina State University)
Abstract: The study of thin liquid films is motivated by two main areas of application, namely flow of mucus in the lung, and coating flows in industrial processes such as microchip manufacture. In the lubrication approximation, the motion of a thin liquid film is described by a single fourth-order partial differential equation that models the evolution of the height of the film. When the fluid is driven by a Marangoni force generated by a distribution of insoluble surfactant, the thin film equation is coupled to an equation for the concentration of surfactant. In this talk, I show the basic structure of this system, and begin an analysis of wave-like solutions in the specific context of a thin film flowing down an inclined plane. Numerical simulations reveal an array of traveling waves, which
persists when capillarity and surface diffusion are neglected. The analysis of the limiting system has some surprises, and in this talk, I show how far we have come in understanding the numerical results analytically, and the analytical results numerically.
For more information, e-mail ma-chair@wpi.edu.
H.J. Gay Lecture Series: Degenerate Symmetric Matrices
Friday, 3/31/2006 2:00 PM-3:00 PM
Higgins Laboratories, 218
Speaker: Peter D. Lax (Courant Institute of Mathematical Sciences, NYU)
Abstract: A symmetric matrix is called degenerate by physicists if it has a multiple eigenvalue.Wigner and von Neumann have shown long ago that the degenerate matrices form a variety of codimension two in the space of all symmetric matrices.This explains the phenomenon of "avoidance of crossing".On the other hand the degenerate matrices are characterised by the single equation discr(S)=0, where discr(S) is the discriminant of S.In this talk we investigate the nature of the discriminant, especially its representation as a sum of squares.
In the second part it will be shown that some pencils of real symmetric matrices always contain a degenerate one.
For more information, e-mail ma-chair@wpi.edu.
April 2006
Introduction to the Mathematical Theory of Intertype Interaction
Friday, 4/7/2006 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Yury Shestopalov (Karlstad University, Sweden)
Abstract: We consider some mathematical aspects of the theory of interaction of oscillations and waves in electromagnetics and acoustics. The problem is considered in terms of the analysis of the spectral and various critical points of multi-parameter operator-valued functions (OVFs), in particular, integral and infinite-matrix (summation) OVFs, when some of the operator parameters are varied. Interaction of oscillations in a cylindrical slotted resonator whose cross section is formed by two rectangular domains is taken as an example. We apply the reduction of the initial boundary eigenvalue problem for the Helmholtz equation to a Fredholm integral equation of the first kind with logarithmic singularity and then to infinite systems with respect to the unknown Fourier coefficients of the solution. We demonstrate that under certain conditions the electromagnetic field distributions become unstable with respect to small variations of certain parameters of the structure (geometric, permittivity etc.) and the interaction of oscillations takes place. We show that such an unstable behavior reveals a specific interaction phenomenon and occurs when the frequency of the excited oscillation is varied in vicinity of the resonant frequencies (eigenfrequencies) of one of the partial domains.
For more information, e-mail ma-chair@wpi.edu.
Solitary Wave Dynamics In Optical and Soft-Condensed Matter
Friday, 4/28/2006 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Panayotis Kevrekidis (Department of Mathematics & Statistics, University of Massachusetts-Amherst)
Abstract: In the past few years, there has been a tremendous increase
in the interest on nonlinear wave and coherent structures in optical
applications (such as waveguide arrays and photorefractive
materials) and soft-condensed matter ones (such as Bose-Einstein
condensates). In these areas, modelling through nonlinear partial
differential equations and differential-difference equations has
proved extremely valuable in predicting the type of waveforms that
spontaneously emerge as well as the ones that can be engineered and
are supported by nonlinearity. I will try to give a series of relevant
examples where theoretical results and numerical computations have
been directly or indirectly confirmed by experiments.
Such results will include the development of instabilities (such as
the modulational instability), the formation of waves (such as
continuum and "discrete'' vortices) and the generation of more exotic
structures (such as ring patterns) among others.
Jointly sponsored by The Department of Chemical Engineering, WPI.
For more information, e-mail ma-chair@wpi.edu.