September 2009
Katharine Ott (University of Kentucky) - The mixed boundary value problem for the Laplacian in Lipschitz domains
Friday, 9/25/2009 11:00 AM-12:00 PM
Stratton Hall, 203
TITLE: The mixed boundary value problem for the Laplacian in Lipschitz domains
ABSTRACT: Two classical boundary value problems for the Laplacian are the Dirichlet and Neumann problems. A third type of boundary value problem is the mixed problem, or Zaremba's problem, where we specify Dirichlet data on a portion of the boundary and Neumann data on the remaining portion. In this talk I will discuss existence, uniqueness, and regularity of solutions to the mixed boundary value problem for the Laplacian in Lipschitz domains.
For more information, e-mail ma-chair@wpi.edu.
October 2009
Konstantin Lurie (WPI)-On Dynamic Materials with Colliding Characteristics
Thursday, 10/1/2009 4:00 PM-5:00 PM
Stratton Hall, 202
ABSTRACT: In this talk I'll discuss a concept of dynamic materials in the presence of collision of characteristics. Generally, the problem allows for various formulations depending on the physical models and assumptions. A special extension related to a traffic flow described by the continuity equation in 1D without a takever will be examined in detail. Collision of characteristics in this model generates the formation of clots (delta singularities) in the solution for the mass density. The procedure resembles the "sticky particles" method used in pressureless gas dynamics, though there are some conceptual differences between the two approaches. This extension applies to optimization of mass transport in one line traffic.
This example shows that the concept of dynamic materials that is linear when the characteristics do not collide may become substantially nonlinear in the presence of such collisions.
For more information, e-mail ma-chair@wpi.edu.
Constantin Bacuta (University of Delaware)-Regularity and Discretization of Stokes type Systems
Thursday, 10/8/2009 4:00 PM-5:00 PM
Stratton Hall, 202
ABSTRACT: We present a regularity result for the Stokes system on polygonal domains based on regularity for the biharmonic problem. Next, we provide an abstract framework for discretizing saddle point systems of Stokes type. We introduce new algorithms which combine the inexact Uzawa method at the continuous level with standard multilevel techniques. The discrete inf-sup or LBB stability condition might not be satisfied and a posteriori error estimates are required only for solutions of symmetric and positive definite problems. In the end we provide an application to solving first order systems and numerical results supporting the new approach.
For more information, e-mail ma-chair@wpi.edu.
Luke Rogers (University of Connecticut)-The use of self-similarity in analysis on fractal sets, and its limitations
Thursday, 10/29/2009 4:00 PM-5:00 PM
Stratton Hall, 202
ABSTRACT: Most of my work over the past few years has been in analysis on post critically finite self-similar fractal sets. What makes these sets special is that one can sometimes define a Laplacian operator on them, and we can use this operator as the basic differential operator in an analytic theory. This opens up the possibility of studying physically interesting differential equations on these fractals.
A significant difficulty in this theory is that smoothness on these fractal sets is very different than it is on Euclidean spaces, so it is typically non trivial to get a handle on what smooth functions look like and to construct smooth functions with useful properties. This is where the self-similarity of the fractal can be an invaluable tool. I will describe a number of results in which self-similarity is exploited to construct or describe functions with useful properties, some of which are from papers I wrote with Bob Strichartz and various other coauthors. If time permits, I will speculate a little on the challenges that remain, particularly in producing a theory for more fractals that are more realistic models of physical objects.
For more information, e-mail ma-chair@wpi.edu.
November 2009
Abdelmalek Abdesselam (University of Virginia)-Introduction to the Renormalization Group
Friday, 11/6/2009 11:00 AM-12:00 PM
Stratton Hall, 203
ABSTRACT: Ever since its introduction by Kenneth G. Wilson in the seventies, the renormalization group has been the main conceptual tool used by physicists in order to make meaningful calculations with functional integrals. The latter are, largely conjectural, infinite-dimensional probability measures over spaces of functions which one can try to construct rigorously using a scaling limit of similar measures where the continuum is discretized by finer and finer grids.
The renormalization group is a dynamical system corresponding to averaging over the short distance fluctuations of the random function and zooming out by a fixed scale ratio. Fixed points of this dynamical system correspond to the possible scaling limits one can achieve. The renormalization group provides a far reaching generalization of the familiar central limit theorem, in a situation where the random variables are dependent, in a way which is subordinated to the geometry of the space labeling these variables. In this nontechnical presentation, we will provide an introduction to the basic ideas of the renormalization group on the example of the usual central limit theorem, and the next simplest example: the so-called hierarchical model.
For more information, e-mail ma-chair@wpi.edu.
Homer Walker (WPI)-Anderson Acceleration for Fixed-Point Iteration, with PDE Applications
Thursday, 11/12/2009 4:00 PM-5:00 PM
Stratton Hall, 202
ABSTRACT: Fixed-point iterations occur naturally and are commonly used in a broad variety of computational science and engineering applications.
In practice, fixed-point iterates often converge undesirably slowly, if at all, and procedures for accelerating the convergence are desirable. This talk will focus on a particular acceleration method that originated in work of Anderson (1965). This method has enjoyed considerable success in electronic-structure computations but seems to have been untried or underexploited in many other important applications. Moreover, while other acceleration methods have been extensively studied by mathematicians and numerical analysts, Anderson acceleration has received relatively little attention from them, despite there being many significant unanswered mathematical questions. In this talk, I will outline Anderson acceleration, discuss some of its theoretical properties, and demonstrate its performance in several applications, with a particular focus on PDE applications.
For more information, e-mail ma-chair@wpi.edu.
Matthew Wright (Missouri State University) - Applications of Transmission Boundary Value Problems
Thursday, 11/19/2009 4:00 PM-5:00 PM
Stratton Hall, 202
ABSTRACT: A transmission boundary value problem seeks to find solutions of a partial differential equation in both the interior and exterior of a fixed boundary that interact in a precise fashion along the boundary. Applications include studying heat conduction across an interface or the interactions between fluids of different viscosities. I will talk about the connections between transmission problems and other boundary value problems, and explain how the method of layer potentials can be used to construct solutions in non-smooth domains. Although quite complicated, I will demonstrate that transmission problems are, in a sense, more fundamental than standard Dirichlet or Neumann problems.
I will give examples involving various partial differential equations including Laplace’s equation, the Stokes system, the Lamé system, and polyharmonic equations.
For more information, e-mail ma-chair@wpi.edu.
December 2009
Mohamed Sulman (WPI) - Higher Dimensional Adaptive Grid Methods Based on Solving the Monge-Kantorovich Problem
Thursday, 12/3/2009 4:00 PM-5:00 PM
Stratton Hall, 202
ABSTRACT: The use of adaptive grid methods is fundamental to the numerical solution of differential equations that involve large solution variations. These methods seek to provide efficient numerical solutions with high accuracy. In this talk, I will describe a newly developed approach for adaptive gird generation in higher dimensions based on solving the optimal mass transportation problem also known as the L^2 Monge–Kantorovich problem (MKP). The coordinate transformation used to define the grid adaptation is determined using two different methods. In the first method, the transformation is computed by solving a parabolic Monge-Amp`ere equation for a steady state solution. For the second method, the grid movement is determined from the velocity field obtained by solving the fluid dynamics formulation of the L^2 MKP. The experimental results show that the MKP methods show promise as effective and reliable methods for higher dimensional adaptive grid generation.
For more information, e-mail ma-chair@wpi.edu.
Irina Mitrea (WPI)-Fredholm Theory for Higher Order Elliptic Boundary Value Problems
Thursday, 12/10/2009 4:00 PM-5:00 PM
Stratton Hall, 202
ABSTRACT: In this talk I will briefly review the Fredholm theory approach for second order elliptic boundary value problems in $C^1$ domains and I will discuss a number of obstacles, which are due to the intricacies of the higher order setting, that have been recently overcome for the employment of this method for boundary value problems for higher order elliptic differential operators. This can be done in a very general class of domains which is in the nature of best possible from the point of view of geometric measure theory.
For more information, e-mail ma-chair@wpi.edu.
Mark Lyon (University of New Hampshire) - Efficient solution of PDEs on complex domains: Fourier Continuation - Alternating Direction (FC-AD) methods
Thursday, 12/17/2009 4:00 PM-5:00 PM
Stratton Hall, 202
ABSTRACT: A new methodology for the numerical solution of Partial Differential Equations (PDEs) in general spatial domains will be presented. Our algorithms are based on the use of a certain "Fourier Continuation" (FC) method for the resolution of the Gibbs phenomenon in conjunction with well-known alternating direction (AD) strategies. This FC-AD methodology will be demonstrated through a variety of examples including the Heat, Laplace, and Wave Equations and both Dirichlet and Neumann problems. The high-order algorithms possess the desirable property of unconditional stability for general domains in computational time that grows in an essentially linear manner with the number of unknowns. The significant improvements that these new algorithms can provide over the computing times required by alternative general-domain solvers will be discussed. In particular, the debilitating "pollution error," which arises as finite-difference and finite-element solvers are applied to the solution of wave propagation problems, is absent from our calculations due to the Fourier basis used in the FC-AD calculations and will be discussed.
For more information, e-mail ma-chair@wpi.edu.
March 2010
Silvia Jimenez (Louisiana State University)-Correctors and Field Fluctuations for the $p_{\epsilon}(x)$-Laplacian
Thursday, 3/18/2010 4:00 PM-5:00 PM
Stratton Hall, 203
ABSTRACT: Properties of local fields inside mixtures of two nonlinear power-law materials are studied. This simple constitutive model is frequently used to describe several phenomena ranging from plasticity to nonlinear dielectric media. We provide the corrector theory for the strong approximation of fields inside composites made from power-law materials. These results are applied to deliver new multiscale tools for bounding the local singularity strength inside microstructured media in terms of the macroscopic applied fields.
For more information, e-mail ma-chair@wpi.edu.
April 2010
Burt Tilley (WPI) - Title TBA
Thursday, 4/8/2010 4:00 PM-5:00 PM
Stratton Hall, 203
ABSTRACT: TBA
For more information, e-mail ma-chair@wpi.edu.
Mayer Humi (WPI) - Title TBA
Thursday, 4/15/2010 4:00 PM-5:00 PM
Stratton Hall, 203
ABSTRACT: TBA
For more information, e-mail ma-chair@wpi.edu.
Darko Volkov (WPI) - Topic TBA
Thursday, 4/22/2010 4:00 PM-5:00 PM
Stratton Hall, 203
ABSTRACT: TBA
For more information, e-mail ma-chair@wpi.edu.
Umberto Mosco (WPI) - Title TBA
Thursday, 4/29/2010 4:00 PM-5:00 PM
Stratton Hall, 203
ABSTRACT: TBA
For more information, e-mail ma-chair@wpi.edu.
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September 2009
Katharine Ott (University of Kentucky) - The mixed boundary value problem for the Laplacian in Lipschitz domains
Friday, 9/25/2009 11:00 AM-12:00 PM
Stratton Hall, 203
TITLE: The mixed boundary value problem for the Laplacian in Lipschitz domains
ABSTRACT: Two classical boundary value problems for the Laplacian are the Dirichlet and Neumann problems. A third type of boundary value problem is the mixed problem, or Zaremba's problem, where we specify Dirichlet data on a portion of the boundary and Neumann data on the remaining portion. In this talk I will discuss existence, uniqueness, and regularity of solutions to the mixed boundary value problem for the Laplacian in Lipschitz domains.
For more information, e-mail ma-chair@wpi.edu.
October 2009
Konstantin Lurie (WPI)-On Dynamic Materials with Colliding Characteristics
Thursday, 10/1/2009 4:00 PM-5:00 PM
Stratton Hall, 202
ABSTRACT: In this talk I'll discuss a concept of dynamic materials in the presence of collision of characteristics. Generally, the problem allows for various formulations depending on the physical models and assumptions. A special extension related to a traffic flow described by the continuity equation in 1D without a takever will be examined in detail. Collision of characteristics in this model generates the formation of clots (delta singularities) in the solution for the mass density. The procedure resembles the "sticky particles" method used in pressureless gas dynamics, though there are some conceptual differences between the two approaches. This extension applies to optimization of mass transport in one line traffic.
This example shows that the concept of dynamic materials that is linear when the characteristics do not collide may become substantially nonlinear in the presence of such collisions.
For more information, e-mail ma-chair@wpi.edu.
Constantin Bacuta (University of Delaware)-Regularity and Discretization of Stokes type Systems
Thursday, 10/8/2009 4:00 PM-5:00 PM
Stratton Hall, 202
ABSTRACT: We present a regularity result for the Stokes system on polygonal domains based on regularity for the biharmonic problem. Next, we provide an abstract framework for discretizing saddle point systems of Stokes type. We introduce new algorithms which combine the inexact Uzawa method at the continuous level with standard multilevel techniques. The discrete inf-sup or LBB stability condition might not be satisfied and a posteriori error estimates are required only for solutions of symmetric and positive definite problems. In the end we provide an application to solving first order systems and numerical results supporting the new approach.
For more information, e-mail ma-chair@wpi.edu.
Luke Rogers (University of Connecticut)-The use of self-similarity in analysis on fractal sets, and its limitations
Thursday, 10/29/2009 4:00 PM-5:00 PM
Stratton Hall, 202
ABSTRACT: Most of my work over the past few years has been in analysis on post critically finite self-similar fractal sets. What makes these sets special is that one can sometimes define a Laplacian operator on them, and we can use this operator as the basic differential operator in an analytic theory. This opens up the possibility of studying physically interesting differential equations on these fractals.
A significant difficulty in this theory is that smoothness on these fractal sets is very different than it is on Euclidean spaces, so it is typically non trivial to get a handle on what smooth functions look like and to construct smooth functions with useful properties. This is where the self-similarity of the fractal can be an invaluable tool. I will describe a number of results in which self-similarity is exploited to construct or describe functions with useful properties, some of which are from papers I wrote with Bob Strichartz and various other coauthors. If time permits, I will speculate a little on the challenges that remain, particularly in producing a theory for more fractals that are more realistic models of physical objects.
For more information, e-mail ma-chair@wpi.edu.
November 2009
Abdelmalek Abdesselam (University of Virginia)-Introduction to the Renormalization Group
Friday, 11/6/2009 11:00 AM-12:00 PM
Stratton Hall, 203
ABSTRACT: Ever since its introduction by Kenneth G. Wilson in the seventies, the renormalization group has been the main conceptual tool used by physicists in order to make meaningful calculations with functional integrals. The latter are, largely conjectural, infinite-dimensional probability measures over spaces of functions which one can try to construct rigorously using a scaling limit of similar measures where the continuum is discretized by finer and finer grids.
The renormalization group is a dynamical system corresponding to averaging over the short distance fluctuations of the random function and zooming out by a fixed scale ratio. Fixed points of this dynamical system correspond to the possible scaling limits one can achieve. The renormalization group provides a far reaching generalization of the familiar central limit theorem, in a situation where the random variables are dependent, in a way which is subordinated to the geometry of the space labeling these variables. In this nontechnical presentation, we will provide an introduction to the basic ideas of the renormalization group on the example of the usual central limit theorem, and the next simplest example: the so-called hierarchical model.
For more information, e-mail ma-chair@wpi.edu.
Homer Walker (WPI)-Anderson Acceleration for Fixed-Point Iteration, with PDE Applications
Thursday, 11/12/2009 4:00 PM-5:00 PM
Stratton Hall, 202
ABSTRACT: Fixed-point iterations occur naturally and are commonly used in a broad variety of computational science and engineering applications.
In practice, fixed-point iterates often converge undesirably slowly, if at all, and procedures for accelerating the convergence are desirable. This talk will focus on a particular acceleration method that originated in work of Anderson (1965). This method has enjoyed considerable success in electronic-structure computations but seems to have been untried or underexploited in many other important applications. Moreover, while other acceleration methods have been extensively studied by mathematicians and numerical analysts, Anderson acceleration has received relatively little attention from them, despite there being many significant unanswered mathematical questions. In this talk, I will outline Anderson acceleration, discuss some of its theoretical properties, and demonstrate its performance in several applications, with a particular focus on PDE applications.
For more information, e-mail ma-chair@wpi.edu.
Matthew Wright (Missouri State University) - Applications of Transmission Boundary Value Problems
Thursday, 11/19/2009 4:00 PM-5:00 PM
Stratton Hall, 202
ABSTRACT: A transmission boundary value problem seeks to find solutions of a partial differential equation in both the interior and exterior of a fixed boundary that interact in a precise fashion along the boundary. Applications include studying heat conduction across an interface or the interactions between fluids of different viscosities. I will talk about the connections between transmission problems and other boundary value problems, and explain how the method of layer potentials can be used to construct solutions in non-smooth domains. Although quite complicated, I will demonstrate that transmission problems are, in a sense, more fundamental than standard Dirichlet or Neumann problems.
I will give examples involving various partial differential equations including Laplace’s equation, the Stokes system, the Lamé system, and polyharmonic equations.
For more information, e-mail ma-chair@wpi.edu.
December 2009
Mohamed Sulman (WPI) - Higher Dimensional Adaptive Grid Methods Based on Solving the Monge-Kantorovich Problem
Thursday, 12/3/2009 4:00 PM-5:00 PM
Stratton Hall, 202
ABSTRACT: The use of adaptive grid methods is fundamental to the numerical solution of differential equations that involve large solution variations. These methods seek to provide efficient numerical solutions with high accuracy. In this talk, I will describe a newly developed approach for adaptive gird generation in higher dimensions based on solving the optimal mass transportation problem also known as the L^2 Monge–Kantorovich problem (MKP). The coordinate transformation used to define the grid adaptation is determined using two different methods. In the first method, the transformation is computed by solving a parabolic Monge-Amp`ere equation for a steady state solution. For the second method, the grid movement is determined from the velocity field obtained by solving the fluid dynamics formulation of the L^2 MKP. The experimental results show that the MKP methods show promise as effective and reliable methods for higher dimensional adaptive grid generation.
For more information, e-mail ma-chair@wpi.edu.
Irina Mitrea (WPI)-Fredholm Theory for Higher Order Elliptic Boundary Value Problems
Thursday, 12/10/2009 4:00 PM-5:00 PM
Stratton Hall, 202
ABSTRACT: In this talk I will briefly review the Fredholm theory approach for second order elliptic boundary value problems in $C^1$ domains and I will discuss a number of obstacles, which are due to the intricacies of the higher order setting, that have been recently overcome for the employment of this method for boundary value problems for higher order elliptic differential operators. This can be done in a very general class of domains which is in the nature of best possible from the point of view of geometric measure theory.
For more information, e-mail ma-chair@wpi.edu.
Mark Lyon (University of New Hampshire) - Efficient solution of PDEs on complex domains: Fourier Continuation - Alternating Direction (FC-AD) methods
Thursday, 12/17/2009 4:00 PM-5:00 PM
Stratton Hall, 202
ABSTRACT: A new methodology for the numerical solution of Partial Differential Equations (PDEs) in general spatial domains will be presented. Our algorithms are based on the use of a certain "Fourier Continuation" (FC) method for the resolution of the Gibbs phenomenon in conjunction with well-known alternating direction (AD) strategies. This FC-AD methodology will be demonstrated through a variety of examples including the Heat, Laplace, and Wave Equations and both Dirichlet and Neumann problems. The high-order algorithms possess the desirable property of unconditional stability for general domains in computational time that grows in an essentially linear manner with the number of unknowns. The significant improvements that these new algorithms can provide over the computing times required by alternative general-domain solvers will be discussed. In particular, the debilitating "pollution error," which arises as finite-difference and finite-element solvers are applied to the solution of wave propagation problems, is absent from our calculations due to the Fourier basis used in the FC-AD calculations and will be discussed.
For more information, e-mail ma-chair@wpi.edu.
March 2010
Silvia Jimenez (Louisiana State University)-Correctors and Field Fluctuations for the $p_{\epsilon}(x)$-Laplacian
Thursday, 3/18/2010 4:00 PM-5:00 PM
Stratton Hall, 203
ABSTRACT: Properties of local fields inside mixtures of two nonlinear power-law materials are studied. This simple constitutive model is frequently used to describe several phenomena ranging from plasticity to nonlinear dielectric media. We provide the corrector theory for the strong approximation of fields inside composites made from power-law materials. These results are applied to deliver new multiscale tools for bounding the local singularity strength inside microstructured media in terms of the macroscopic applied fields.
For more information, e-mail ma-chair@wpi.edu.
April 2010
Burt Tilley (WPI) - Title TBA
Thursday, 4/8/2010 4:00 PM-5:00 PM
Stratton Hall, 203
ABSTRACT: TBA
For more information, e-mail ma-chair@wpi.edu.
Mayer Humi (WPI) - Title TBA
Thursday, 4/15/2010 4:00 PM-5:00 PM
Stratton Hall, 203
ABSTRACT: TBA
For more information, e-mail ma-chair@wpi.edu.
Darko Volkov (WPI) - Topic TBA
Thursday, 4/22/2010 4:00 PM-5:00 PM
Stratton Hall, 203
ABSTRACT: TBA
For more information, e-mail ma-chair@wpi.edu.
Umberto Mosco (WPI) - Title TBA
Thursday, 4/29/2010 4:00 PM-5:00 PM
Stratton Hall, 203
ABSTRACT: TBA
For more information, e-mail ma-chair@wpi.edu.
Powered by the Social Web - Bringing people together through Events, Places, & Common Interests