September 2006
In vivo MRI-Based 3D Models for Computer-Aided Human Right Ventricle
Friday, 9/8/2006 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Dalin Tang (Mathematical Sciences Department, WPI)
ABSTRACT: Right ventricular dysfunction is one of the more common causes
of heart failure in patients with congenital heart defects. Use of
computer-assisted procedures is becoming more popular in clinical decision
making process and computer-aided surgeries. A 3D in vivo MRI-based
RV/LV combination model with fluid-structure interactions (FSI), RV-LV
interaction, and RV-patch interaction was introduced to perform mechanical
analysis for human right ventricle with potential clinical applications.
Patient-specific RV-LV morphologies were acquired by cardiac MRI.
Material properties of ventricles and pressure conditions were chosen to
match patient-specific data. Two major measures of RV cardiac functions
(stroke volume (SV) and ejection fraction (EF)) which are verifiable by
MRI data were chosen as the end points for measurement of RV cardiac
function improvement after surgery. Our preliminary results indicated
that smaller patches and patches with material properties better matching
RV tissue properties lead to better RV function recoveries. More patient
studies are needed to establish our computational modeling and
optimization procedures so that they can be used to replace empirical and
often risky clinical experimentation to examine the efficiency and
suitability of various reconstructive procedures in diseased hearts.
Joint research with: Chun Yang, Tal Geva, and Pedro J. del Nido.
For more information, e-mail ma-chair@wpi.edu.
Numerical implementation of a variational model of brittle fracture
Friday, 9/29/2006 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Blaise Bourdin (Louisiana State University)
ABSTRACT: Fracture mechanics is a very active area of research, with vital
applications. In recent years, the unexpected collapse of terminal 2F at
Charles de Gaulle airport in France or the Columbia space shuttle
disintegration upon re-entry illustrate the importance of a better
understanding and numerical simulation of the mechanisms of fracture.
In the area of brittle fracture, the most widely accepted theories are
based on Griffith's criterion and limited to the propagation of an
isolated, pre-existing crack along a given path. Extending Griffith's
theory into a global minimization principle, while preserving its
essence, the concept of energy restitution in between surface and bulk
terms, G. Francfort and J.-J. Marigo proposed a new formulation for the
brittle fracture problem. The basis of their model is the minimization
of a total energy with respect to any admissible displacement and crack
field. The main advantage of this approach is to be capable of
predicting the initiation of new cracks, computing their path, and
accounting the interactions betwenn several cracks, in two and three
space dimensions. Of course, this has a price both theoretically and
numerically. In particular, in order to acheive global minimization with
respect to any crack set, one has to devise special numerical methods.
After briefly reviewing the issues of brittle fracture mechanics, I
will present the Francfort-Marigo model. I will rapidly describe some
elements of its analysis, and present a numerical approximation based on
the properties of Gamma-convergence. I will derive necessary optimality
condiions with respect to the global time evolution, and show how to use
them in a minimization algorithm. Then, I will present some extensions
of the original model, accounting for body forces (under some
restrictions) or thermal loads, and describe how to adapt the numerical
implementation. I will illustrate my talk with several large scale two
and three dimensional experiments.
For more information, e-mail ma-chair@wpi.edu.
October 2006
Computational Mechanical Analysis for Aortic Dissection
Friday, 10/13/2006 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Feng Gao (School of Information Science Japan Advanced
Institute of Science and Technology)
ABSTRACT: The aorta is the largest artery that delivers blood from the heart to the
rest of the body, and many cardiovascular diseases often occur on the
aortic arch. Aortic dissection is a pathological state in which a tear
develops in the intima layer of the aorta. Blood enters at the site of
the tear, separates the layers of the aorta, and spreads the dissection.
Identifying the factors causing aortic dissection may provide useful data
for inclusion in the design of prevention systems.
The aorta is a large-caliber vessel that consists of three distinct
layers: a tunica intima, a tunica media, and a tunica adventitia. A three
layered aortic arch model was constructed based on the aorta structure. A
loosely coupled method was used to study the complex mechanical
interaction under blood flow in a three-layered aortic arch model. The
complex mechanical interaction between blood flow and wall dynamics in the
layered aortic arch model was simulated. This study investigated the flow
field in the aortic arch, showed the wall displacement and aorta
deformation, determined wall stresses distribution on aortic arch and
across aortic wall. The stress distributions were continuous in the
one-layered model, while for the three-layered model, the stresses are
much higher in the media than that in the intima and adventitia.
Arterial stiffness is an important determinant of cardiovascular risk, and
patients with aortic aneurysm are prone to have aortic dissection. We
therefore also evaluated the influence of wall stiffness and aortic arch
aneurysm on wall stress distribution in the layered aortic arch model. The
result showed increasing the medial stiffness caused an increase in wall
stress in media and arch aneurysm caused high wall stress in aortic wall.
The numerical layered aortic model may prove useful for studying
biomechanical analysis and the pathogeneses of aortic dissection.
For more information, e-mail ma-chair@wpi.edu.
Towards an Understanding of Nonlinear Electrochemical Transport
Friday, 10/27/2006 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Kevin Chu (Princeton University)
ABSTRACT: Charge transport plays a crucial role in the response of many colloidal,
micro-fluidic, micro-electrochemical and biological systems subjected to
applied electric fields. As a result, studying the electrochemical transport
properties of these systems is an important first step towards understanding
their overall response to applied electric fields. To avoid the nonlinear
coupling between the electric field and ion concentration fields, analysis of
charge transport problems has traditionally relied on simple circuit models
which are only valid in the weak field regime. Unfortunately, to understand
the behavior of systems in strong applied fields (which are easy achieved at
the micro- and nano-scale), we are forced to face the full nonlinear charge
transport equations. In this talk, we will discuss recent advances in our
theoretical understanding of the nonlinear charge transport equations in the
thin double-layer limit. Our main results include a general formulation of
of "surface conservation laws" for diffuse boundary layers and
characterization of the structure of the electric and ion concentration fields
around spherical, metallic (i.e. highly-polarizable) colloid particles.
Along the way, we will discuss various practical issues associated with
numerical solution of the nonlinear governing equations.
For more information, e-mail ma-chair@wpi.edu.
November 2006
Integrative Analysis of Genomic Data
Friday, 11/3/2006 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Peter J. Park, Ph.D. (Childrens Hospital, Harvard Medical
School, Assitant Professor Harvard-Partners Center for Genetics and
Genomics, Associate Director of Bioinformatics)
ABSTRACT: Several new technologies have enabled generation of biological
data in an unprecedented scale. Examples of such data are genome
sequences, gene expression data, protein interactions maps, and RNA
interference screens. A central question confronted by scientists in
many areas of biology and medicine now is how to derive biological
insights by computational analysis of large data sets. I will describe
some of these data sets and give examples of how integration of multiple
data sources can lead to a new biological understanding. No biology
background will be assumed.
Information on the speaker: Peter studied applied mathematics at Harvard
and Caltech. His doctoral thesis in 1999 was on numerical analysis of
partial differential equations with Prof. Thomas Hou. Seeking a more
interactive and rapidly evolving field, he did his postdoctoral fellowship
in the Biostatistics Department at the Harvard School of Public Health,
where he was introduced to molecular biology and genomics. He then moved
to Children's Hospital Boston in 2001 as an instructor at the Medical
School and has been there since, becoming an assistant professor in 2006.
In addition to research, he teaches a statistics course at the School of
Public Health and serves as a guest lecturer in a number of courses. He is
Associate Director of Bioinformatics at the Harvard-Partners Center for
Genetics and Genomics and a member of the Affiliated Faculty at the
Harvard-MIT Division of Health Sciences and Technology.
He frequently serves as a referee for journals, including Science, PNAS,
Bioinformatics, Nucleic Acids Research, Genome Biology, PLoS Computational
Biology, PLoS Biology, PLoS Medicine, BMC Bioinformatics, BMC Genomics,
BMC Biology, Genomics, Journal of the Royal Statistical Society,
Biometrics, Bulletin of Mathematical Biology, BioTechniques, Pacific
Symposium for Biocomputing, Statistics in Medicine, Lifetime Data
Analysis, Physiological Genomics, FEBS Letters, Expert Review of Molecular
Diagnostics, In Silico Biology, International Journal of Cancer, American
Medical Informatics Association Symposium.
For more information, e-mail ma-chair@wpi.edu.
The Mathematics of Automobile Cams and Valve Motion
Friday, 11/10/2006 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Ron Mosier (Applied Mathematician, DaimlerChrysler, AG (Retired))
ABSTRACT: The cam is the most mathematical part of the automobile. The shape of the cam causes the automobile's valves to move with a motion prescribed by the cam designer, whose task can be made easier by the knowledge and use of differential geometry, linear algebra, complex analysis, spline theory, and, of course, the calculus. Still, for whatever reason, most books and manuals for engineers still teach cam design using myriad numbers of special cases, estimates, and seldom any mathematics of a level higher than trigonometry. With only one exception, splines are never mentioned.
This talk will be in two parts. The first part will illustrate the use of fundamental differential geometry in cam analysis. Then we will show how simple linear algebra and working in the complex plane can make cam shape determination very easy. The second part will show how Chrysler group designs valve motion using shape preserving splines. Until now, Chrysler has kept the design methods of valve motion secret. This talk will be the first public description.
The first part of the talk will be quite easy and accessible to most undergraduates and all graduate students. The second part will be somewhat more difficult. The faculty will easily follow all of the talk and teachers will be able to find numerous easy, but practical, examples to illustrate topics in the their classrooms.
It is hoped, but not promised, that handouts will be available.
For more information, e-mail ma-chair@wpi.edu.
Harold J. Gay Lecture Series--Fractal Roughness: Beautiful, damn hard, and surprisingly useful
Friday, 11/10/2006 3:00 PM-4:00 PM
Olin Hall, 107
SPEAKER: Benoit Mandelbrot (Yale and Pacific Northwest National Laboratory)
ABSTRACT: Some fractals imitate mountains, clouds, stock markets, and many other aspects of nature and culture. Others yield wild and wonderful new patterns that a child can draw but great masters struggle or fail to understand. All are shapes that look the same from any distance, far away or close by. Since time immemorial, some have been used by great artists. A hundred years ago, mathematicians called them monsters and an excuse to split from physics. Nowespecially since my 1975 term fractalthey help heal this split. Fractal geometry helps mathematics and the sciences to cross a long avoided boundary between the smooth and the rough. Partial differential equations must allow very rough solutions. Man's basic sensation of roughness can now be measured intrinsically by fractal dimension, first step to being mastered. An introduction to fractal geometry with updates on some current developments including finance.
For more information, e-mail ma-chair@wpi.edu.
Asymptotic analysis of utility-based hedging strategies for small number of contingent claims
Friday, 11/17/2006 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Mihai Sirbu
ABSTRACT: In the framework of incomplete financial models the role of hedging
strategy is to provide the optimal trade-off between risk (error of
replication) and return. We study the linear approximation of
utility-based hedging strategies for small number of contingent
claims. We show that this approximation is actually a mean-variance
hedging strategy under an appropriate choice of a numeraire and a
risk-neutral probability. The presentation is based on joint work with Dmitry Kramkov.
For more information, e-mail ma-chair@wpi.edu.
December 2006
Harold J. Gay Lecture Series--Multidimensional Conservation Laws - More Questions than Answers
Friday, 12/1/2006 11:00 AM-12:00 PM
Bartlett Center
SPEAKER: Barbara Keyfitz (Fields Institute and University of Houston)
ABSTRACT: The analysis of quasilinear hyperbolic partial differential
equations presents a number of challenges. Although equations of this
type are important in a number of applications, ranging from high-speed
aerodynamics, through magnetohydrodynamics, to multiphase flows important
in industrial technology, there is little theory against which even to check
the reliability of numerical simulations.
Development of a theory for conservation laws in a single space variable has
led to remarkable advances in analysis, including the theory of compensated
compactness and the study of novel function spaces. Recently, a number of
groups have begun to approach multidimensional systems via self-similar
solutions.
In this talk, I will give some history of the development of conservation
law theory, including an indication of why the applications are important.
I will describe some of the recent results on self-similar solutions, and
the interesting results in analysis that they involve. Finally, I will
outline some of the paradoxical questions that remain.
For more information, e-mail ma-chair@wpi.edu.
Harold J. Gay Lecture Series--From Collisionless Shocks to Integrable Systems
Wednesday, 12/6/2006 11:00 AM-12:00 PM
Salisbury Laboratories, 104
SPEAKER: Cathleen Morawetz
ABSTRACT: Collisionless shocks have been studied twice. First in the 1950s
they were proposed as a mechanism for heating up a controlled nuclear fusion
machine for creating energy. But their mathematical structure was an open
question. Now such a shock has been observed by Voyager 2 in its travels
through space. The lecture will first describe how collisionless shocks
occur in the solar system .Then we will examine in a simple model what we
mean mathematically by a collisionless shock and why its structure is a
puzzle. Finally we look at how these investigations led to the study of
completely integrable systems of partial differential equations.
For more information, e-mail ma-chair@wpi.edu.
Thin Films of Martensitic Materials
Friday, 12/8/2006 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Marian Bocea (North Dakota State University)
ABSTRACT: The shape memory effect is an important manifestation of the martensitic phase transformation, a phenomenon observed in various metallic alloys, ceramics, and biological systems. In view of their technological applications as microactuators, thin films of martensitic materials utilizing the shape memory effect have been intensively studied in recent years. The key analytical issue in this context is to rigorously understand how effective thin film models stem from three-dimensional Nonlinear Elasticity via dimensional reduction. After giving an overview on the martensitic phase transformation and the shape memory effect, I will motivate the need for dimensional reduction, and I will revisit the derivation of Bhattacharya and James' theory of martensitic thin films in the absence of interfacial energy contributions to the total bulk energy.
For more information, e-mail ma-chair@wpi.edu.
The depolymerization which drives nematode sperm motility
Friday, 12/15/2006 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Charles Wolgemuth (University of Connecticut Health Center)
ABSTRACT: In this talk I will discuss a model for how contractile force is generated in crawling nematode sperm cells and use this model to describe the biomechanics of the translocation of these cells. Nematode sperm are a good model system for studying cellular crawling as their primary function is motility. The cytoskeleton of nematode sperm is composed of Major Sperm Protein (MSP) rather than actin. MSP forms apolar filaments and there do not appear to be molecular motors that associate with it. Therefore, it is believed that the cytoskeletal meshwork of MSP is the sole force-producing agent driving motility. Recent experiments suggest that depolymerization of the MSP cytoskeleton can produce contractile force. I will describe a model that explains how depolymerization of the cytoskeleton (which can be modeled as an elastic meshwork immersed in a fluid) can produce a contractile force. This model agrees well with in vitro experiments on the contraction of MSP fiber bundles. Then, I will apply this model to the crawling of nematode sperm cells. We find that the model predicts that larger cells crawl faster, which has been observed experimentally. In addition, the model makes definite predictions about how the size, adhesion, and permeability of the membrane affect crawling. I will discuss these results and describe some experiments that we have been doing to test them. Finally, coupling of this model to a stick-slip mechanism for adhesion provides an explanation for the periodic contractions that are observed in other crawling cells.
For more information, e-mail ma-chair@wpi.edu.
January 2007
Quasi-static evolution for damage
Friday, 1/19/2007 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Adriana Garroni (University of Rome)
ABSTRACT: We consider a variational model for damage proposed by Francfort and
Marigo. This energy based model is nonconvex and in the minimization procedure
microstructures can be produced. A relaxed incremental problemthat accounts for
irreversibility can be defined and, by means of time discretization, a relaxed
quasi-static evolution can be obtained. We give an alternative model for damage
based on a threshold criterion. We prove that an 'energy based' solution is also
a 'threshold' solution. As a byproduct we also obtain that local minimizers for
the energy based model are actually global minimizers.
For more information, e-mail ma-chair@wpi.edu.
Pricing Equity Swaps in an Economy with Jumps
Friday, 1/26/2007 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Mia Hinnerich (Stockholm School of Economics)
ABSTRACT: Empirical evidence confirms that asset price processes exhibits jumps
and that asset returns are not Gaussian. Hence pricing formulas for
equity swaps that rely on the assumption of a Gaussian economy may
potentially be wrong. We provide a pricing model for equity swaps, when
all the asset price processes in the economy are allowed to jump. The
market is driven by a general marked point process as well as by a
standard multidimensional Wiener process. In order to obtain closed-form
solutions of the swap values, we assume that all parameters in the asset
price processes are deterministic, but possibly functions of time. We
derive swap values using martingale methods and the technique of
convexity corrections rather than using replicating portfolios. Our
results are an extension of the results of Liao & Wang (2003). The
martingale method is the key that enables the extension to
jump-diffusions. We find that the pricing formula for quanto equity
swaps does indeed change when we allow for jumps in the economy. We
provide a simple numerical illustration of how to quantify the pricing
errors that occurs if the true process is a jump-diffusion but the
Gaussian model is used instead. We find that the pricing errors can be
of economically significant size.
For more information, e-mail ma-chair@wpi.edu.
Parameter Estimation for Stochastic PDE's with Multiplicative Noise
Tuesday, 1/30/2007 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Igor Cialenco (University of Southern California)
ABSTRACT: We will investigate a parameter estimation
problem for some parabolic SPDEs driven by multiplicative noise.
Some Maximum Likelihood Estimators of the parameter,
based on finite-dimensional approximation of the solution will be
presented, and consistency and asymptotic normality of these
estimators will be discussed.
Finally we will show how this problem can be applied to fixed income
market (modeling forward rates).
For more information, e-mail ma-chair@wpi.edu.
February 2007
Optimal Life Insurance Purchase, Consumption, and Portfolio under an Uncertain Life
Friday, 2/2/2007 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Jinchun Ye (Base-2 Capital, Chicago)
ABSTRACT: A continuous-time model is developed for determining a wage earner's optimal strategies for dividing lifetime income between the purchase of life insurance, consumption, and risky investment. The wage earner, whose lifetime is uncertain, seeks to maximize the expectation of (1) the utility of consumption while still alive and working, (2) the utility of the bequest upon premature death, and (3) the utility of the size of the estate upon retirement (if he or she lives that long). This talk will focus on how to analyze the proposed model by a dynamic programming approach and a martingale approach and how to numerically solve the relevant Hamilton-Jacobi-Bellman equation by a Markov chain approximation approach. Numerical examples will be presented in an effort to explore economic implications.
For more information, e-mail ma-chair@wpi.edu.
No Arbitrage without Semimartingales
Tuesday, 2/6/2007 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Hasan Sayit (University of California-Santa Barbara)
ABSTRACT: Absence of arbitrage is a basic equilibrium condition in financial mathematics. However, the fundamental theorem of asset pricing (Delbaen and Schachermayer (1994)) asserts that any process that is not a semimartingale is not consistent with this condition. A typical example of such processes is Fractional Brownian Motion, a process that has drawn much attention in recent years in financial mathematics due to its long-range dependence property.
In this talk, we show that with suitable restrictions on allowable trading strategies, one has no arbitrage in settings where the traditional theory would admit arbitrage possibilities. In particular, price processes that are not semimartingales are possible in our setting, for example Fractional Brownian Motion.
For more information, e-mail ma-chair@wpi.edu.
Hyperbolic Systems with Dissipation
Friday, 2/9/2007 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Cleopatra Christoforou (Northwestern University)
ABSTRACT: I will begin with an introduction to the theory of hyperbolic conservation laws that arise in Continuum Physics, emphasizing the major challenges, construction methods and most recent developments. The second part of the talk will be devoted to global entropy weak solutions of systems of conservation laws with dissipation constructed via the vanishing viscosity method. I will present results of existence, uniqueness and stability for systems with (i) dissipative source and (ii) fading memory. Last, I will show that the vanishing viscosity method is an effective tool to establish the relaxation limit.
For more information, e-mail ma-chair@wpi.edu.
Nonlinear Mechanical Property of Tracheal Cartilage and its Collapse Behavior
Tuesday, 2/20/2007 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Zhongzhao Teng (University of Zaragoza, Spain)
ABSTRACT: Despite being the stiffest airway of the bronchial tree, trachea nonetheless undergoes significant deformation under physiological pressures, which affects the flow in the airway and biological functions of lung. An analysis of the interaction requires a fundamental knowledge of mechanical property of tracheal rings and further study on its collapse behavior. In this talk, a mathematical model for obtaining the mechanical property of tracheal cartilage will be presented as well as the further collapse simulation. Experimental results for determination of the material properties of trachea will be displayed. I will demonstrate (a) the current model can capture the nonlinear material property of the cartilage very well, (b) the linear model under-estimates the tracheal stability.
For more information, e-mail ma-chair@wpi.edu.
Mathematical Study of Brain Tumor Therapies and Initiation of Brain Tumor Stem Cells
Friday, 2/23/2007 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Jianjun Paul Tian (The Ohio State University)
ABSTRACT: Glioma is the most serious malignant brain tumor. In order to improve the efficacy of therapies, it is important to understand its progression with therapies and its genesis. In this talk, I will first present our effort in understanding of glioma progression with different therapies in terms of mathematical models. The first model is about virotherapy of glioma, which is a free boundary problem with five nonlinear partial differential equations. Virotherapy is a promising treatment for malignant solid tumors, and it is now in animal experimental stage. In order to treat human glioma by virotherapy, it is critical to understand all factors involved in the therapy. Our model finds an important factor of the therapy, burst size of virus, and the effect of immunosuppression drug cyclophosphamide in animal experiments. The model prediction has been verified by experimental results. The second model is about radiotherapy plus chemotherapy after surgical resection, which is a two-component free boundary problem. After surgery, the tumor progression depends on the degree of resection and radiation, and a particular drug. We use human data to estimate parameter values, and the model can predict the mean survival times of patients who undergo different protocols of treatments. The third part of the talk is about initiation of brain tumor stem cells. This work is ongoing, and our focus now is on two molecular signals SHH and GABA in subventricular zone niche of adult neural stem cells. Hopefully, we can reconcile the contradiction that GABA has inhibitory effect both on stem cell proliferation and on the velocity of cells moving within the niche and that of speed of migration, and we can test hypothesis that stem cells will become cancer stem cells once they leave niche signal control.
For more information, e-mail ma-chair@wpi.edu.
March 2007
Elliptic problems in perforated domains with nonlinear boundary conditions: homogenization by the unfolding method
Friday, 3/30/2007 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Doina Cioranescu (Universite Pierre et Marie Curie (Paris 6))
ABSTRACT: We introduce the unfolding operator for functions defined on periodically perforated domains as well as a boundary unfolding operator. As applications, we study first the homogenization of some elliptic problems with a Robin condition on the boundary of the holes, proving convergence and corrector results. Then, we extend these results to the case of nonlinear boundary conditions on the boundary of the perforation. The use of unfolding operators avoids the introduction of extensions for functions defined on oscillating perforated domains.
For more information, e-mail ma-chair@wpi.edu.
April 2007
The strange term in nonlinear elliptic PDE's via the unfolding method
Monday, 4/2/2007 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Alain Damlamian (Universite Pierre 12 Val de Marne)
ABSTRACT: The well-known "Strange term" arising at the limit when considering solutions of Dirichlet problems in domains with periodically distributed small holes (with homogeneous Dirichlet condition) is extended in this presentation to the non-linear monotone Leray-Lions framework via the unfolding method. The strange term involves a maximal monotone graph whose expression is obtained in terms of limits of unfolded operators.
For more information, e-mail ma-chair@wpi.edu.
Digital Feedback Regulation and State Estimation Techniques for Nonlinear Systems
Friday, 4/6/2007 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Nikolaos Kazantzis
Abstract: The present work proposes a new approach to the digital feedback regulator synthesis problem for nonlinear systems. The problem under consideration is formulated and addressed in the context of functional equations theory, and a rather general set of necessary and sufficient conditions for solvability is derived. Within the proposed regulator synthesis framework, the time-constants/dynamic modes of the closed-loop system dynamics (controlled system) are favorably assigned, and process regulation is attained in the presence of adverse disturbances that drive the system far from the reference equilibrium point. Furthermore, the problem of designing nonlinear digital state estimators for the accurate dynamic reconstruction of unmeasurable state variables is addressed within a similar framework of analysis. It is demonstrated that the proposed nonlinear state estimator can be designed such that the estimation error, or the mismatch between the state estimate and the actual state variable, decays to zero with an assignable rate/speed of convergence. Finally, the proposed nonlinear digital feedback regulator synthesis method, as well as the state estimator design method, are applied to a biological reactor that exhibits steady-state multiplicity. The control objective is to regulate the reactor at the middle unstable steady state that is considered to be the operationally favorable one, by manipulating the dilution rate. Simulation studies have been conducted to evaluate the performance of the proposed nonlinear feedback regulator and state estimator, as well as to illustrate the main design aspects. It is shown that the nonlinear feedback regulator and state estimator clearly outperform the standard linear ones, especially in the presence of adverse conditions, under which linear regulation at the unstable steady state is not always feasible and linear state estimation unsatisfactory.
For more information, e-mail ma-chair@wpi.edu.
Implicit and Split Numerical Solutions for a Reaction-Diffusion System Modeling Radiation Diffusion with a Fusion Heat Source
Friday, 4/13/2007 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Carol S. Woodward (Center for Applied Scientific Computing, Lawrence Livermore National Laboratory)
Abstract: Modeling radiation diffusion processes has traditionally been accomplished through inaccurate and nonscalable solution methods based on decoupling and linearizing the model equations. We present an algorithm for the fully implicit solution of radiation diffusion with material energy coupling and a fusion heat source. Our method fully converges the nonlinear solution, but uses lagged coefficient information when developing the Krylov basis and preconditioner.
Numerical results will be shown illustrating the benefits of fully implicit solution methods using Newton-Krylov techniques as well as the parallel scalability of the method. In addition, we present results exploring the relative costs of the most common operator splitting and implicit methods for coupled diffusion and reaction systems applied to our system. We will discuss the practical application of recently developed stability theory (Ropp & Shadid, 2005) on these systems and point out both advantages and disadvantages of split and implicit strategies.
This work was performed under the auspices of the U.S. Department of Energy by University of California Lawrence Livermore National Laboratory under Contract W-7405-Eng-48.
For more information, e-mail ma-chair@wpi.edu.
Characterization, Modeling and Management of Inferential Risk, Data Quality Risk and Operational Risk in Survey Procedures
Friday, 4/20/2007 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: John Eltinge (Bureau of Labor Statistics)
Abstract: This paper explores some conceptual and methodological issues that are important in the design, operation and analysis of large-scale government surveys. We view the design of survey procedures (including initial planning, sample design, data collection, inference and dissemination) as a mixture of optimization and risk management efforts in the presence of constraints and incomplete information. This in turn suggests several potentially rich areas for research in mathematical and applied statistics.
Five topics receive principal attention, beginning with some relatively well-defined technical issues and then expanding to several broader topics related to data quality and risk management. First, a review of the goals, constraints and risk profiles of survey practice suggests a spectrum of potential approaches to survey work, ranging from rigorously predetermined survey procedures at one extreme to highly exploratory analyses of previously collected data at the other extreme. Classical randomization-based procedures are arguably compatible with a mandate for predetermined methodology. Nonetheless, these procedures have limitations arising from efficiency issues, the presence of nonsampling error, and prospective inferential interest beyond the finite population that was sampled. These limitations lead to review of a second class of approaches to the analysis of survey data, based on models for survey variables, auxiliary variables and nonsampling error processes.
Third, we use the framework of risk management to explore six dimensions of survey data quality suggested in Brackstone (1999): accuracy (incorporating all of the components of error considered in standard models for total survey error), timeliness, relevance, interpretability, accessibility and coherence. Fourth, we expand our discussion of risk management by considering operational risk, i.e., the risk that one or more steps in a survey procedure may not be carried out as specified. Finally, we note that work with large-scale surveys will involve a mixture of statistical science and statistical technology, and we suggest that the literature on adoption and diffusion of technology can offer important insights into the distribution of expectations, utility functions and behaviors of large survey organizations, data analysts and other data users.
For more information, e-mail ma-chair@wpi.edu.
Nonlinear Elasticity: Liquid Crystal Elastomers and Gels
Friday, 4/27/2007 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Maria-Carme Calderer
Abstract: Since the celebrated existence theorems in nonlinear elasticity by John Ball in 1977, many applications have emerged in the study of complex material phenomena involving diffusiveless phase transformations. This talk examines recent applications of calculus of variations to study liquid crystal elastomers and gels. In the former, we explore the significantly different properties of the energy that result from the interaction of the anisotropy of liquid crystals with the nonlinear elastic properties of polymers. In particular, we analyze the physical and mathematical significance of the "soft elastomer modes." Indeed, the basic signature of liquid crystal elastomers is their capability of experiencing large deformations along principal stretch directions parallel to certain nematic liquid crystal preferencial axis. In studies of gels, we examine how the combination of the chemistry and network elasticity opens up to exciting new applications in the field of pharmacology.
For more information, e-mail ma-chair@wpi.edu.
May 2007
Quasistatic evolution problems in plasticity with softening
Friday, 5/18/2007 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Gianni Dal Maso (SISSA, Trieste, Italy)
ABSTRACT: In plasticity theory the term softening refers to the reduction of the yield stress as plastic deformation proceeds. We deal with this problem in the quasistatic case, in the framework of small strain associative elastoplasticity. The presence of a nonconvex term due to the softening phenomenon requires the extension of a variational framework proposed by Mielke to the case of a nonconvex energy functional. In this problem the use of global minimizers in the corresponding incremental problems is not justified from the mechanical point of view. We analize a different selection criterion for the solutions of the quasistatic evolution problem, based on a viscous approximation. In view of the nonconvexity of the problem, taking the limit as the artificial viscosity parameter tends to zero leads to a weak formulation of the problem in a space of Young measures. Moreover, since the growth exponent of the energy is one, we need a suitable notion of generalized Young measure in order to deal with concentration effects. Finally, the classical notion of total variation of a time-dependent function on a time interval has to be extended to time-dependent families of Young measures. This enables us to define, in this generalized context, a notion of dissipation, which plays a crucial role in Mielke's variational approach. Some examples show that smooth initial data may lead, after a critical time, to a Young measure solution with concentration phenomena.
These results have been obtained in collaboration with Antonio DeSimone, Maria Giovanna Mora and Massimiliano Morini.
For more information, e-mail ma-chair@wpi.edu.
September 2006
In vivo MRI-Based 3D Models for Computer-Aided Human Right Ventricle
Friday, 9/8/2006 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Dalin Tang (Mathematical Sciences Department, WPI)
ABSTRACT: Right ventricular dysfunction is one of the more common causes
of heart failure in patients with congenital heart defects. Use of
computer-assisted procedures is becoming more popular in clinical decision
making process and computer-aided surgeries. A 3D in vivo MRI-based
RV/LV combination model with fluid-structure interactions (FSI), RV-LV
interaction, and RV-patch interaction was introduced to perform mechanical
analysis for human right ventricle with potential clinical applications.
Patient-specific RV-LV morphologies were acquired by cardiac MRI.
Material properties of ventricles and pressure conditions were chosen to
match patient-specific data. Two major measures of RV cardiac functions
(stroke volume (SV) and ejection fraction (EF)) which are verifiable by
MRI data were chosen as the end points for measurement of RV cardiac
function improvement after surgery. Our preliminary results indicated
that smaller patches and patches with material properties better matching
RV tissue properties lead to better RV function recoveries. More patient
studies are needed to establish our computational modeling and
optimization procedures so that they can be used to replace empirical and
often risky clinical experimentation to examine the efficiency and
suitability of various reconstructive procedures in diseased hearts.
Joint research with: Chun Yang, Tal Geva, and Pedro J. del Nido.
For more information, e-mail ma-chair@wpi.edu.
Numerical implementation of a variational model of brittle fracture
Friday, 9/29/2006 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Blaise Bourdin (Louisiana State University)
ABSTRACT: Fracture mechanics is a very active area of research, with vital
applications. In recent years, the unexpected collapse of terminal 2F at
Charles de Gaulle airport in France or the Columbia space shuttle
disintegration upon re-entry illustrate the importance of a better
understanding and numerical simulation of the mechanisms of fracture.
In the area of brittle fracture, the most widely accepted theories are
based on Griffith's criterion and limited to the propagation of an
isolated, pre-existing crack along a given path. Extending Griffith's
theory into a global minimization principle, while preserving its
essence, the concept of energy restitution in between surface and bulk
terms, G. Francfort and J.-J. Marigo proposed a new formulation for the
brittle fracture problem. The basis of their model is the minimization
of a total energy with respect to any admissible displacement and crack
field. The main advantage of this approach is to be capable of
predicting the initiation of new cracks, computing their path, and
accounting the interactions betwenn several cracks, in two and three
space dimensions. Of course, this has a price both theoretically and
numerically. In particular, in order to acheive global minimization with
respect to any crack set, one has to devise special numerical methods.
After briefly reviewing the issues of brittle fracture mechanics, I
will present the Francfort-Marigo model. I will rapidly describe some
elements of its analysis, and present a numerical approximation based on
the properties of Gamma-convergence. I will derive necessary optimality
condiions with respect to the global time evolution, and show how to use
them in a minimization algorithm. Then, I will present some extensions
of the original model, accounting for body forces (under some
restrictions) or thermal loads, and describe how to adapt the numerical
implementation. I will illustrate my talk with several large scale two
and three dimensional experiments.
For more information, e-mail ma-chair@wpi.edu.
October 2006
Computational Mechanical Analysis for Aortic Dissection
Friday, 10/13/2006 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Feng Gao (School of Information Science Japan Advanced
Institute of Science and Technology)
ABSTRACT: The aorta is the largest artery that delivers blood from the heart to the
rest of the body, and many cardiovascular diseases often occur on the
aortic arch. Aortic dissection is a pathological state in which a tear
develops in the intima layer of the aorta. Blood enters at the site of
the tear, separates the layers of the aorta, and spreads the dissection.
Identifying the factors causing aortic dissection may provide useful data
for inclusion in the design of prevention systems.
The aorta is a large-caliber vessel that consists of three distinct
layers: a tunica intima, a tunica media, and a tunica adventitia. A three
layered aortic arch model was constructed based on the aorta structure. A
loosely coupled method was used to study the complex mechanical
interaction under blood flow in a three-layered aortic arch model. The
complex mechanical interaction between blood flow and wall dynamics in the
layered aortic arch model was simulated. This study investigated the flow
field in the aortic arch, showed the wall displacement and aorta
deformation, determined wall stresses distribution on aortic arch and
across aortic wall. The stress distributions were continuous in the
one-layered model, while for the three-layered model, the stresses are
much higher in the media than that in the intima and adventitia.
Arterial stiffness is an important determinant of cardiovascular risk, and
patients with aortic aneurysm are prone to have aortic dissection. We
therefore also evaluated the influence of wall stiffness and aortic arch
aneurysm on wall stress distribution in the layered aortic arch model. The
result showed increasing the medial stiffness caused an increase in wall
stress in media and arch aneurysm caused high wall stress in aortic wall.
The numerical layered aortic model may prove useful for studying
biomechanical analysis and the pathogeneses of aortic dissection.
For more information, e-mail ma-chair@wpi.edu.
Towards an Understanding of Nonlinear Electrochemical Transport
Friday, 10/27/2006 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Kevin Chu (Princeton University)
ABSTRACT: Charge transport plays a crucial role in the response of many colloidal,
micro-fluidic, micro-electrochemical and biological systems subjected to
applied electric fields. As a result, studying the electrochemical transport
properties of these systems is an important first step towards understanding
their overall response to applied electric fields. To avoid the nonlinear
coupling between the electric field and ion concentration fields, analysis of
charge transport problems has traditionally relied on simple circuit models
which are only valid in the weak field regime. Unfortunately, to understand
the behavior of systems in strong applied fields (which are easy achieved at
the micro- and nano-scale), we are forced to face the full nonlinear charge
transport equations. In this talk, we will discuss recent advances in our
theoretical understanding of the nonlinear charge transport equations in the
thin double-layer limit. Our main results include a general formulation of
of "surface conservation laws" for diffuse boundary layers and
characterization of the structure of the electric and ion concentration fields
around spherical, metallic (i.e. highly-polarizable) colloid particles.
Along the way, we will discuss various practical issues associated with
numerical solution of the nonlinear governing equations.
For more information, e-mail ma-chair@wpi.edu.
November 2006
Integrative Analysis of Genomic Data
Friday, 11/3/2006 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Peter J. Park, Ph.D. (Childrens Hospital, Harvard Medical
School, Assitant Professor Harvard-Partners Center for Genetics and
Genomics, Associate Director of Bioinformatics)
ABSTRACT: Several new technologies have enabled generation of biological
data in an unprecedented scale. Examples of such data are genome
sequences, gene expression data, protein interactions maps, and RNA
interference screens. A central question confronted by scientists in
many areas of biology and medicine now is how to derive biological
insights by computational analysis of large data sets. I will describe
some of these data sets and give examples of how integration of multiple
data sources can lead to a new biological understanding. No biology
background will be assumed.
Information on the speaker: Peter studied applied mathematics at Harvard
and Caltech. His doctoral thesis in 1999 was on numerical analysis of
partial differential equations with Prof. Thomas Hou. Seeking a more
interactive and rapidly evolving field, he did his postdoctoral fellowship
in the Biostatistics Department at the Harvard School of Public Health,
where he was introduced to molecular biology and genomics. He then moved
to Children's Hospital Boston in 2001 as an instructor at the Medical
School and has been there since, becoming an assistant professor in 2006.
In addition to research, he teaches a statistics course at the School of
Public Health and serves as a guest lecturer in a number of courses. He is
Associate Director of Bioinformatics at the Harvard-Partners Center for
Genetics and Genomics and a member of the Affiliated Faculty at the
Harvard-MIT Division of Health Sciences and Technology.
He frequently serves as a referee for journals, including Science, PNAS,
Bioinformatics, Nucleic Acids Research, Genome Biology, PLoS Computational
Biology, PLoS Biology, PLoS Medicine, BMC Bioinformatics, BMC Genomics,
BMC Biology, Genomics, Journal of the Royal Statistical Society,
Biometrics, Bulletin of Mathematical Biology, BioTechniques, Pacific
Symposium for Biocomputing, Statistics in Medicine, Lifetime Data
Analysis, Physiological Genomics, FEBS Letters, Expert Review of Molecular
Diagnostics, In Silico Biology, International Journal of Cancer, American
Medical Informatics Association Symposium.
For more information, e-mail ma-chair@wpi.edu.
The Mathematics of Automobile Cams and Valve Motion
Friday, 11/10/2006 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Ron Mosier (Applied Mathematician, DaimlerChrysler, AG (Retired))
ABSTRACT: The cam is the most mathematical part of the automobile. The shape of the cam causes the automobile's valves to move with a motion prescribed by the cam designer, whose task can be made easier by the knowledge and use of differential geometry, linear algebra, complex analysis, spline theory, and, of course, the calculus. Still, for whatever reason, most books and manuals for engineers still teach cam design using myriad numbers of special cases, estimates, and seldom any mathematics of a level higher than trigonometry. With only one exception, splines are never mentioned.
This talk will be in two parts. The first part will illustrate the use of fundamental differential geometry in cam analysis. Then we will show how simple linear algebra and working in the complex plane can make cam shape determination very easy. The second part will show how Chrysler group designs valve motion using shape preserving splines. Until now, Chrysler has kept the design methods of valve motion secret. This talk will be the first public description.
The first part of the talk will be quite easy and accessible to most undergraduates and all graduate students. The second part will be somewhat more difficult. The faculty will easily follow all of the talk and teachers will be able to find numerous easy, but practical, examples to illustrate topics in the their classrooms.
It is hoped, but not promised, that handouts will be available.
For more information, e-mail ma-chair@wpi.edu.
Harold J. Gay Lecture Series--Fractal Roughness: Beautiful, damn hard, and surprisingly useful
Friday, 11/10/2006 3:00 PM-4:00 PM
Olin Hall, 107
SPEAKER: Benoit Mandelbrot (Yale and Pacific Northwest National Laboratory)
ABSTRACT: Some fractals imitate mountains, clouds, stock markets, and many other aspects of nature and culture. Others yield wild and wonderful new patterns that a child can draw but great masters struggle or fail to understand. All are shapes that look the same from any distance, far away or close by. Since time immemorial, some have been used by great artists. A hundred years ago, mathematicians called them monsters and an excuse to split from physics. Nowespecially since my 1975 term fractalthey help heal this split. Fractal geometry helps mathematics and the sciences to cross a long avoided boundary between the smooth and the rough. Partial differential equations must allow very rough solutions. Man's basic sensation of roughness can now be measured intrinsically by fractal dimension, first step to being mastered. An introduction to fractal geometry with updates on some current developments including finance.
For more information, e-mail ma-chair@wpi.edu.
Asymptotic analysis of utility-based hedging strategies for small number of contingent claims
Friday, 11/17/2006 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Mihai Sirbu
ABSTRACT: In the framework of incomplete financial models the role of hedging
strategy is to provide the optimal trade-off between risk (error of
replication) and return. We study the linear approximation of
utility-based hedging strategies for small number of contingent
claims. We show that this approximation is actually a mean-variance
hedging strategy under an appropriate choice of a numeraire and a
risk-neutral probability. The presentation is based on joint work with Dmitry Kramkov.
For more information, e-mail ma-chair@wpi.edu.
December 2006
Harold J. Gay Lecture Series--Multidimensional Conservation Laws - More Questions than Answers
Friday, 12/1/2006 11:00 AM-12:00 PM
Bartlett Center
SPEAKER: Barbara Keyfitz (Fields Institute and University of Houston)
ABSTRACT: The analysis of quasilinear hyperbolic partial differential
equations presents a number of challenges. Although equations of this
type are important in a number of applications, ranging from high-speed
aerodynamics, through magnetohydrodynamics, to multiphase flows important
in industrial technology, there is little theory against which even to check
the reliability of numerical simulations.
Development of a theory for conservation laws in a single space variable has
led to remarkable advances in analysis, including the theory of compensated
compactness and the study of novel function spaces. Recently, a number of
groups have begun to approach multidimensional systems via self-similar
solutions.
In this talk, I will give some history of the development of conservation
law theory, including an indication of why the applications are important.
I will describe some of the recent results on self-similar solutions, and
the interesting results in analysis that they involve. Finally, I will
outline some of the paradoxical questions that remain.
For more information, e-mail ma-chair@wpi.edu.
Harold J. Gay Lecture Series--From Collisionless Shocks to Integrable Systems
Wednesday, 12/6/2006 11:00 AM-12:00 PM
Salisbury Laboratories, 104
SPEAKER: Cathleen Morawetz
ABSTRACT: Collisionless shocks have been studied twice. First in the 1950s
they were proposed as a mechanism for heating up a controlled nuclear fusion
machine for creating energy. But their mathematical structure was an open
question. Now such a shock has been observed by Voyager 2 in its travels
through space. The lecture will first describe how collisionless shocks
occur in the solar system .Then we will examine in a simple model what we
mean mathematically by a collisionless shock and why its structure is a
puzzle. Finally we look at how these investigations led to the study of
completely integrable systems of partial differential equations.
For more information, e-mail ma-chair@wpi.edu.
Thin Films of Martensitic Materials
Friday, 12/8/2006 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Marian Bocea (North Dakota State University)
ABSTRACT: The shape memory effect is an important manifestation of the martensitic phase transformation, a phenomenon observed in various metallic alloys, ceramics, and biological systems. In view of their technological applications as microactuators, thin films of martensitic materials utilizing the shape memory effect have been intensively studied in recent years. The key analytical issue in this context is to rigorously understand how effective thin film models stem from three-dimensional Nonlinear Elasticity via dimensional reduction. After giving an overview on the martensitic phase transformation and the shape memory effect, I will motivate the need for dimensional reduction, and I will revisit the derivation of Bhattacharya and James' theory of martensitic thin films in the absence of interfacial energy contributions to the total bulk energy.
For more information, e-mail ma-chair@wpi.edu.
The depolymerization which drives nematode sperm motility
Friday, 12/15/2006 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Charles Wolgemuth (University of Connecticut Health Center)
ABSTRACT: In this talk I will discuss a model for how contractile force is generated in crawling nematode sperm cells and use this model to describe the biomechanics of the translocation of these cells. Nematode sperm are a good model system for studying cellular crawling as their primary function is motility. The cytoskeleton of nematode sperm is composed of Major Sperm Protein (MSP) rather than actin. MSP forms apolar filaments and there do not appear to be molecular motors that associate with it. Therefore, it is believed that the cytoskeletal meshwork of MSP is the sole force-producing agent driving motility. Recent experiments suggest that depolymerization of the MSP cytoskeleton can produce contractile force. I will describe a model that explains how depolymerization of the cytoskeleton (which can be modeled as an elastic meshwork immersed in a fluid) can produce a contractile force. This model agrees well with in vitro experiments on the contraction of MSP fiber bundles. Then, I will apply this model to the crawling of nematode sperm cells. We find that the model predicts that larger cells crawl faster, which has been observed experimentally. In addition, the model makes definite predictions about how the size, adhesion, and permeability of the membrane affect crawling. I will discuss these results and describe some experiments that we have been doing to test them. Finally, coupling of this model to a stick-slip mechanism for adhesion provides an explanation for the periodic contractions that are observed in other crawling cells.
For more information, e-mail ma-chair@wpi.edu.
January 2007
Quasi-static evolution for damage
Friday, 1/19/2007 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Adriana Garroni (University of Rome)
ABSTRACT: We consider a variational model for damage proposed by Francfort and
Marigo. This energy based model is nonconvex and in the minimization procedure
microstructures can be produced. A relaxed incremental problemthat accounts for
irreversibility can be defined and, by means of time discretization, a relaxed
quasi-static evolution can be obtained. We give an alternative model for damage
based on a threshold criterion. We prove that an 'energy based' solution is also
a 'threshold' solution. As a byproduct we also obtain that local minimizers for
the energy based model are actually global minimizers.
For more information, e-mail ma-chair@wpi.edu.
Pricing Equity Swaps in an Economy with Jumps
Friday, 1/26/2007 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Mia Hinnerich (Stockholm School of Economics)
ABSTRACT: Empirical evidence confirms that asset price processes exhibits jumps
and that asset returns are not Gaussian. Hence pricing formulas for
equity swaps that rely on the assumption of a Gaussian economy may
potentially be wrong. We provide a pricing model for equity swaps, when
all the asset price processes in the economy are allowed to jump. The
market is driven by a general marked point process as well as by a
standard multidimensional Wiener process. In order to obtain closed-form
solutions of the swap values, we assume that all parameters in the asset
price processes are deterministic, but possibly functions of time. We
derive swap values using martingale methods and the technique of
convexity corrections rather than using replicating portfolios. Our
results are an extension of the results of Liao & Wang (2003). The
martingale method is the key that enables the extension to
jump-diffusions. We find that the pricing formula for quanto equity
swaps does indeed change when we allow for jumps in the economy. We
provide a simple numerical illustration of how to quantify the pricing
errors that occurs if the true process is a jump-diffusion but the
Gaussian model is used instead. We find that the pricing errors can be
of economically significant size.
For more information, e-mail ma-chair@wpi.edu.
Parameter Estimation for Stochastic PDE's with Multiplicative Noise
Tuesday, 1/30/2007 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Igor Cialenco (University of Southern California)
ABSTRACT: We will investigate a parameter estimation
problem for some parabolic SPDEs driven by multiplicative noise.
Some Maximum Likelihood Estimators of the parameter,
based on finite-dimensional approximation of the solution will be
presented, and consistency and asymptotic normality of these
estimators will be discussed.
Finally we will show how this problem can be applied to fixed income
market (modeling forward rates).
For more information, e-mail ma-chair@wpi.edu.
February 2007
Optimal Life Insurance Purchase, Consumption, and Portfolio under an Uncertain Life
Friday, 2/2/2007 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Jinchun Ye (Base-2 Capital, Chicago)
ABSTRACT: A continuous-time model is developed for determining a wage earner's optimal strategies for dividing lifetime income between the purchase of life insurance, consumption, and risky investment. The wage earner, whose lifetime is uncertain, seeks to maximize the expectation of (1) the utility of consumption while still alive and working, (2) the utility of the bequest upon premature death, and (3) the utility of the size of the estate upon retirement (if he or she lives that long). This talk will focus on how to analyze the proposed model by a dynamic programming approach and a martingale approach and how to numerically solve the relevant Hamilton-Jacobi-Bellman equation by a Markov chain approximation approach. Numerical examples will be presented in an effort to explore economic implications.
For more information, e-mail ma-chair@wpi.edu.
No Arbitrage without Semimartingales
Tuesday, 2/6/2007 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Hasan Sayit (University of California-Santa Barbara)
ABSTRACT: Absence of arbitrage is a basic equilibrium condition in financial mathematics. However, the fundamental theorem of asset pricing (Delbaen and Schachermayer (1994)) asserts that any process that is not a semimartingale is not consistent with this condition. A typical example of such processes is Fractional Brownian Motion, a process that has drawn much attention in recent years in financial mathematics due to its long-range dependence property.
In this talk, we show that with suitable restrictions on allowable trading strategies, one has no arbitrage in settings where the traditional theory would admit arbitrage possibilities. In particular, price processes that are not semimartingales are possible in our setting, for example Fractional Brownian Motion.
For more information, e-mail ma-chair@wpi.edu.
Hyperbolic Systems with Dissipation
Friday, 2/9/2007 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Cleopatra Christoforou (Northwestern University)
ABSTRACT: I will begin with an introduction to the theory of hyperbolic conservation laws that arise in Continuum Physics, emphasizing the major challenges, construction methods and most recent developments. The second part of the talk will be devoted to global entropy weak solutions of systems of conservation laws with dissipation constructed via the vanishing viscosity method. I will present results of existence, uniqueness and stability for systems with (i) dissipative source and (ii) fading memory. Last, I will show that the vanishing viscosity method is an effective tool to establish the relaxation limit.
For more information, e-mail ma-chair@wpi.edu.
Nonlinear Mechanical Property of Tracheal Cartilage and its Collapse Behavior
Tuesday, 2/20/2007 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Zhongzhao Teng (University of Zaragoza, Spain)
ABSTRACT: Despite being the stiffest airway of the bronchial tree, trachea nonetheless undergoes significant deformation under physiological pressures, which affects the flow in the airway and biological functions of lung. An analysis of the interaction requires a fundamental knowledge of mechanical property of tracheal rings and further study on its collapse behavior. In this talk, a mathematical model for obtaining the mechanical property of tracheal cartilage will be presented as well as the further collapse simulation. Experimental results for determination of the material properties of trachea will be displayed. I will demonstrate (a) the current model can capture the nonlinear material property of the cartilage very well, (b) the linear model under-estimates the tracheal stability.
For more information, e-mail ma-chair@wpi.edu.
Mathematical Study of Brain Tumor Therapies and Initiation of Brain Tumor Stem Cells
Friday, 2/23/2007 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Jianjun Paul Tian (The Ohio State University)
ABSTRACT: Glioma is the most serious malignant brain tumor. In order to improve the efficacy of therapies, it is important to understand its progression with therapies and its genesis. In this talk, I will first present our effort in understanding of glioma progression with different therapies in terms of mathematical models. The first model is about virotherapy of glioma, which is a free boundary problem with five nonlinear partial differential equations. Virotherapy is a promising treatment for malignant solid tumors, and it is now in animal experimental stage. In order to treat human glioma by virotherapy, it is critical to understand all factors involved in the therapy. Our model finds an important factor of the therapy, burst size of virus, and the effect of immunosuppression drug cyclophosphamide in animal experiments. The model prediction has been verified by experimental results. The second model is about radiotherapy plus chemotherapy after surgical resection, which is a two-component free boundary problem. After surgery, the tumor progression depends on the degree of resection and radiation, and a particular drug. We use human data to estimate parameter values, and the model can predict the mean survival times of patients who undergo different protocols of treatments. The third part of the talk is about initiation of brain tumor stem cells. This work is ongoing, and our focus now is on two molecular signals SHH and GABA in subventricular zone niche of adult neural stem cells. Hopefully, we can reconcile the contradiction that GABA has inhibitory effect both on stem cell proliferation and on the velocity of cells moving within the niche and that of speed of migration, and we can test hypothesis that stem cells will become cancer stem cells once they leave niche signal control.
For more information, e-mail ma-chair@wpi.edu.
March 2007
Elliptic problems in perforated domains with nonlinear boundary conditions: homogenization by the unfolding method
Friday, 3/30/2007 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Doina Cioranescu (Universite Pierre et Marie Curie (Paris 6))
ABSTRACT: We introduce the unfolding operator for functions defined on periodically perforated domains as well as a boundary unfolding operator. As applications, we study first the homogenization of some elliptic problems with a Robin condition on the boundary of the holes, proving convergence and corrector results. Then, we extend these results to the case of nonlinear boundary conditions on the boundary of the perforation. The use of unfolding operators avoids the introduction of extensions for functions defined on oscillating perforated domains.
For more information, e-mail ma-chair@wpi.edu.
April 2007
The strange term in nonlinear elliptic PDE's via the unfolding method
Monday, 4/2/2007 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Alain Damlamian (Universite Pierre 12 Val de Marne)
ABSTRACT: The well-known "Strange term" arising at the limit when considering solutions of Dirichlet problems in domains with periodically distributed small holes (with homogeneous Dirichlet condition) is extended in this presentation to the non-linear monotone Leray-Lions framework via the unfolding method. The strange term involves a maximal monotone graph whose expression is obtained in terms of limits of unfolded operators.
For more information, e-mail ma-chair@wpi.edu.
Digital Feedback Regulation and State Estimation Techniques for Nonlinear Systems
Friday, 4/6/2007 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Nikolaos Kazantzis
Abstract: The present work proposes a new approach to the digital feedback regulator synthesis problem for nonlinear systems. The problem under consideration is formulated and addressed in the context of functional equations theory, and a rather general set of necessary and sufficient conditions for solvability is derived. Within the proposed regulator synthesis framework, the time-constants/dynamic modes of the closed-loop system dynamics (controlled system) are favorably assigned, and process regulation is attained in the presence of adverse disturbances that drive the system far from the reference equilibrium point. Furthermore, the problem of designing nonlinear digital state estimators for the accurate dynamic reconstruction of unmeasurable state variables is addressed within a similar framework of analysis. It is demonstrated that the proposed nonlinear state estimator can be designed such that the estimation error, or the mismatch between the state estimate and the actual state variable, decays to zero with an assignable rate/speed of convergence. Finally, the proposed nonlinear digital feedback regulator synthesis method, as well as the state estimator design method, are applied to a biological reactor that exhibits steady-state multiplicity. The control objective is to regulate the reactor at the middle unstable steady state that is considered to be the operationally favorable one, by manipulating the dilution rate. Simulation studies have been conducted to evaluate the performance of the proposed nonlinear feedback regulator and state estimator, as well as to illustrate the main design aspects. It is shown that the nonlinear feedback regulator and state estimator clearly outperform the standard linear ones, especially in the presence of adverse conditions, under which linear regulation at the unstable steady state is not always feasible and linear state estimation unsatisfactory.
For more information, e-mail ma-chair@wpi.edu.
Implicit and Split Numerical Solutions for a Reaction-Diffusion System Modeling Radiation Diffusion with a Fusion Heat Source
Friday, 4/13/2007 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Carol S. Woodward (Center for Applied Scientific Computing, Lawrence Livermore National Laboratory)
Abstract: Modeling radiation diffusion processes has traditionally been accomplished through inaccurate and nonscalable solution methods based on decoupling and linearizing the model equations. We present an algorithm for the fully implicit solution of radiation diffusion with material energy coupling and a fusion heat source. Our method fully converges the nonlinear solution, but uses lagged coefficient information when developing the Krylov basis and preconditioner.
Numerical results will be shown illustrating the benefits of fully implicit solution methods using Newton-Krylov techniques as well as the parallel scalability of the method. In addition, we present results exploring the relative costs of the most common operator splitting and implicit methods for coupled diffusion and reaction systems applied to our system. We will discuss the practical application of recently developed stability theory (Ropp & Shadid, 2005) on these systems and point out both advantages and disadvantages of split and implicit strategies.
This work was performed under the auspices of the U.S. Department of Energy by University of California Lawrence Livermore National Laboratory under Contract W-7405-Eng-48.
For more information, e-mail ma-chair@wpi.edu.
Characterization, Modeling and Management of Inferential Risk, Data Quality Risk and Operational Risk in Survey Procedures
Friday, 4/20/2007 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: John Eltinge (Bureau of Labor Statistics)
Abstract: This paper explores some conceptual and methodological issues that are important in the design, operation and analysis of large-scale government surveys. We view the design of survey procedures (including initial planning, sample design, data collection, inference and dissemination) as a mixture of optimization and risk management efforts in the presence of constraints and incomplete information. This in turn suggests several potentially rich areas for research in mathematical and applied statistics.
Five topics receive principal attention, beginning with some relatively well-defined technical issues and then expanding to several broader topics related to data quality and risk management. First, a review of the goals, constraints and risk profiles of survey practice suggests a spectrum of potential approaches to survey work, ranging from rigorously predetermined survey procedures at one extreme to highly exploratory analyses of previously collected data at the other extreme. Classical randomization-based procedures are arguably compatible with a mandate for predetermined methodology. Nonetheless, these procedures have limitations arising from efficiency issues, the presence of nonsampling error, and prospective inferential interest beyond the finite population that was sampled. These limitations lead to review of a second class of approaches to the analysis of survey data, based on models for survey variables, auxiliary variables and nonsampling error processes.
Third, we use the framework of risk management to explore six dimensions of survey data quality suggested in Brackstone (1999): accuracy (incorporating all of the components of error considered in standard models for total survey error), timeliness, relevance, interpretability, accessibility and coherence. Fourth, we expand our discussion of risk management by considering operational risk, i.e., the risk that one or more steps in a survey procedure may not be carried out as specified. Finally, we note that work with large-scale surveys will involve a mixture of statistical science and statistical technology, and we suggest that the literature on adoption and diffusion of technology can offer important insights into the distribution of expectations, utility functions and behaviors of large survey organizations, data analysts and other data users.
For more information, e-mail ma-chair@wpi.edu.
Nonlinear Elasticity: Liquid Crystal Elastomers and Gels
Friday, 4/27/2007 11:00 AM-12:00 PM
Stratton Hall, 203
Speaker: Maria-Carme Calderer
Abstract: Since the celebrated existence theorems in nonlinear elasticity by John Ball in 1977, many applications have emerged in the study of complex material phenomena involving diffusiveless phase transformations. This talk examines recent applications of calculus of variations to study liquid crystal elastomers and gels. In the former, we explore the significantly different properties of the energy that result from the interaction of the anisotropy of liquid crystals with the nonlinear elastic properties of polymers. In particular, we analyze the physical and mathematical significance of the "soft elastomer modes." Indeed, the basic signature of liquid crystal elastomers is their capability of experiencing large deformations along principal stretch directions parallel to certain nematic liquid crystal preferencial axis. In studies of gels, we examine how the combination of the chemistry and network elasticity opens up to exciting new applications in the field of pharmacology.
For more information, e-mail ma-chair@wpi.edu.
May 2007
Quasistatic evolution problems in plasticity with softening
Friday, 5/18/2007 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Gianni Dal Maso (SISSA, Trieste, Italy)
ABSTRACT: In plasticity theory the term softening refers to the reduction of the yield stress as plastic deformation proceeds. We deal with this problem in the quasistatic case, in the framework of small strain associative elastoplasticity. The presence of a nonconvex term due to the softening phenomenon requires the extension of a variational framework proposed by Mielke to the case of a nonconvex energy functional. In this problem the use of global minimizers in the corresponding incremental problems is not justified from the mechanical point of view. We analize a different selection criterion for the solutions of the quasistatic evolution problem, based on a viscous approximation. In view of the nonconvexity of the problem, taking the limit as the artificial viscosity parameter tends to zero leads to a weak formulation of the problem in a space of Young measures. Moreover, since the growth exponent of the energy is one, we need a suitable notion of generalized Young measure in order to deal with concentration effects. Finally, the classical notion of total variation of a time-dependent function on a time interval has to be extended to time-dependent families of Young measures. This enables us to define, in this generalized context, a notion of dissipation, which plays a crucial role in Mielke's variational approach. Some examples show that smooth initial data may lead, after a critical time, to a Young measure solution with concentration phenomena.
These results have been obtained in collaboration with Antonio DeSimone, Maria Giovanna Mora and Massimiliano Morini.
For more information, e-mail ma-chair@wpi.edu.