Applied PDE Seminar (MAP6939(01))
Department of Mathematics, Florida State U., Fall 2009
Tuesday 3:30-4:20, LOV104
This page is maintained by
Dr. Xiaoming WANG
We are interested in partial differential equations and their applications in a very broad sense.
Local and guest speakers will present talks on related topics. Speakers are encouraged to provide motivation
as well as technical details to facilitate graduate student learning.
Please let the organizer know if you are interested in giving a talk in the seminar.
All are welcome.
-
- Date: Tue. Aug. 25, 2009
Speaker: Organizational meeting
- Date: Tue. Sept. 1, 2009
Speaker: Dr. Xiaoming Wang, FSU
Title: Scalar linear stochastic differential equations
Abstract: We present explicit solution formulas to generic
scalar linear stochastic differential equations. We will also derive formula for the moments of the equation. The long time behavior of these moments can be computed explicitly under suitable physical assumptions. Possible application to the study of climate will be mentioned. The audience is expected to know Itô's formula. The calculation may lead to a publishable master's thesis.
- Date: Tue. Sept. 8th, 2009 Special time and place: 4:00PM, College of Engineering Bldg., 2525 Pottsdamer St., Rm A226 (joint with Engineering seminar)
Speaker: Dr. William L. Oberkampf, Consulting Engineer, Austin, Texas
Title: Model Validation and Predictive Capability
Abstract: The issue of model validation has received significantly
increased
attention during the last decade. Here, we take a restricted view of model
validation - meaning the quantitative assessment of model accuracy by means
of comparison of model results with experimentally measured results. This
interpretation of model validation is in contrast to the common viewpoint
that validation is the processes of adjusting model parameters in order to
obtain improved agreement with experimental data. We refer to this latter
process as model calibration or model updating. In this talk, a method is
discussed that quantitatively estimates model accuracy in the face of
available
experimental data. This type of procedure, usually referred to as the
computation of a validation metric, can be used to characterize the
disagreement between the quantitative predictions from a model and relevant
empirical data. A quantitative comparison of model and measurement can
utilize
either deterministic values from each, a statistic from each of the
respective
probability distributions, or the complete probability distributions of
each. This
talk discusses a validation metric that compares the complete
distribution from
both the model and the experimental data. The metric can also address
the case
where only one experimental measurement is obtained. That is, one can use a
pooling technique for multiple system response quantities expressed in
different
units and dimensions in order to assess model accuracy given only one
set of
experimental data. The proposed metric has several desirable properties
that
should make it practically useful in engineering, including
objectiveness in evaluation
of model accuracy, robustness to uncertainty in both the model and the
measurements,
and retaining the dimensional units of the data that are compared. The
metric can also
be used as a basis estimating model accuracy when the model is used to
extrapolate
to conditions for which direct experimental observations are not
available. For the
case of extrapolation of the model to some new intended use, not only
does one
have the contribution to uncertainty due to the validation metric, but
also uncertainties
associated with the new application. Using a simple example problem, we
compute
the validation metric and show how it can be used in predictive capability.
- Date: Sept. 15 (postponed to 22), 2009
Speaker: Drs. Xiaoming Wang and Bill Hu, FSU
Title: Some mathematical issues related to karst aquifer
Abstract: We present the so-called coupled continuum pipe-flow model for flows in karst aquifer. We will investigate the well-posedness as well as accurate and efficient numerical approximation of the model. Potential problems with the original formulation by geologist, possible fix and related numerical issues will be discussed. There are many open problems to be solved which may be suitable for Master and/or Ph.D. thesis for both computationally oriented or theoretically oriented students.
- Date: Sept. 29, 2009
Speaker: Dr. Xiaoming Wang
Title: Boundary layer associated with the Darcy-Brinkman-Boussinesq system
Abstract: We demonstrate that there exists a boundary layer of thickness proportional to the square root of the Darcy-Brinkman number in the infinite Darcy-Prandtl number Darcy-Brinkman-Boussinesq system for convection in fluid saturated porous media. There are many problems remaining which may be suitable for master or Ph.D. thesis for theoretically oriented students, or large scale computation.
- Date: Oct. 6, 2009
Speaker: Dr. Xiaoming Wang
Title: Boundary layer associated with the
Darcy-Brinkman-Boussinesq system II
Abstract: This is a continuation of the talk from the previous week.
- Date: Oct. 13, 2009
Speaker: No talk. Organizer out of town.
- Date: Thurs. Oct. 15, 2009 Special time and place: 3:15PM, LOV204B
Speaker: Dr. Dilek Dustegor
Center for Advanced Power Systems
Florida State University
Title: Towards a Smarter Grid: a computational tool for fault detection and
location
Abstract: The electric power grid, though a critical infrastructure, is
getting old and unable to meet challenges caused by its increasing size,
complexity, and integration. There is a recognized need for the development
and implementation of a new power delivery - the Smart Grid. The transition
from the current state to the smart grid requires interdisciplinary effort
in many areas and, particularly, R&D in system theory, protection,
resilience, reliability, and modeling & simulation applicable to large-scale
complex integrated systems. During my presentation, advanced computational
analytical tools will be presented for the fast and accurate detection, and
location of faults in a smart grid. These tools can also be used for the
strategic placement of sensors in the grid at the design stage, hence
improving existing and developing new topologies to maximize the robustness
of power grids and other network-like infrastructures.
- Date: Oct. 20, 2009
Speaker: Dr. Qingshan CHEN, FSU
Title:
The Hille-Phillips-Yosida Theorem and its application in
geophysical fluid dynamics
Abstract:
The classical Hille-Yosida Theorem is a tool to establish the
well-posedness of certain time dependent PDEs. In this talk I will
introduce a special form of that theorem, so called
Hille-Phillips-Yosida Theorem. Its application to a geophysical fluid
dynamical problem will be demonstrated. But before this, I will
introduce the required functional analysis background, and work out a
couple of very simple examples. This talk should be accessible to people
who have had some exposure to basic functional analysis and PDE theory.
- Date: Oct. 26, 2009
Speaker: Dr. Sanjeev Srivastava, CAPS, FSU
Title: Complexity Assessment for an All Electric Ship
Abstract: There is no universal formulation of complexity or complex systems. Many authors have commented on and discussed the multiplicity and the variety of definitions of complexity for several kinds of systems including living cells, brain, immune system, financial market, ecosystem, and human population. Usually, in social and biological systems the complexity is characterized by high dimensionality, distributed nature of resources and information, emergence, and self-organization proprieties. Even more variance in definition exists for nonlinear dynamical systems; additional complexity can arise as result of parameter sensitivity, sensitivity to initial condition, or due to a major disturbance. Some simple examples are iterated numerical systems (the well studied logistic map) which have wide ranging behaviours (periodic, chaotic, etc.) depending on system parameters. Such nonlinear sensitivity to parameters is common in engineering systems. One such example is auxiliary systems found in all electric ships. These auxiliary systems consists of various sub-systems that are very closely coupled. These sub-systems have their own controls, which might compete with each other. Also, these controls are generally designed to operate over a linear region but the system itself can easily go into non-linear regions, resulting in severe fault or disruption.
The main objective of our research is to assess the complexity dynamics in a context of nonlinear systems with chaotic behaviour, and extend the results of the assessment to large scale systems, such as an electrical ship auxiliary system. To achieve this goal, we propose to use sensor derived information uncertainty computed from Shannon entropy, as a basis to measure the sensed system's complexity. In order to estimate the system complexity dynamics, currently we are investigating two methodologies. The first one is based on statistical entropy computation, and the second one uses the nonlinear time series analysis techniques from existing chaos theory. It is envisioned that the study of complexity issues will give a new way to design the system and the controller. These considerations at the design stage will act as a guideline to help the designer evaluate the system parameters and the interconnections that facilitate stable system operation and minimize the uncertainties. Moreover the online monitoring of the complexity will facilitate evaluations of the ship system's capability to perform its mission.
- Date: Nov. 10, 2009
Speaker: Dr. Xiaoming HE, FSU
Title: Bilinear Immersed Finite Elements (IFE) For Interface Problems
Abstract: The traditional finite element method needs a body-fitting mesh to solve a
partial differential equations with interface. This restriction leads to
many drawbacks and it is necessary to use a mesh independent of interface
for many applications. This presentation discusses a bilinear immersed
finite element (IFE) for solving interface problems on a mesh independent of
interface. The bilinear IFE space is a nonconforming finite element space.
The error estimates for the interpolation of a Sobolev function indicate
that this space has the usual approximation capability expected from
bilinear polynomials. Then this space is implemented to Galerkin method,
finite volume element (FVE) method and discontinuous Galerkin (DG) method.
Our numerical examples and convergence analysis show that these methods have
the optimal convergence rates. In order to improve the efficiency of the DG
method with bilinear IFE, we develop a selective DG method, which apply DG
formulation wherever needed, but Galerkin formulation everywhere else.
- Date: Nov. 3, 2009 (Coincide with 2009 Sir James Lighthill Lecture )
Speaker: Dr. Alexandre Chorin, UC Berkeley
Title: Non-Bayesian Particle Filters
Abstract: Filtering and data assimilation are indispensable tools in engineering, weather forecasting, and other areas where one has to make predictions on the basis of uncertain models supplemented by a stream of uncertain data. In nonlinear problems such filtering can be excessively laborious. I will present a scheme, related to chainless sampling that tames the amount of labor. Its main features are a representation of each new probability density by a set of functions of Gaussian variables (a distinct function for each sample and each step), and a resampling based on normalization constants. Examples will be given.
- Date: Nov. 17, 2009
Speaker: Dr. Xiaoming Wang
Title: Unconditionally stable time approximation of gradient systems
Abstract:
Many scaling laws for important physical gradient
systems
are
revealed through long time behavior. This poses special difficulty
for
numerical simulation since classical discretization may not preserve
the
long time stability of the original gradient system. Here we present a
generalization of Eyre's idea of convex-concave decomposition of the
energy functional. In particular we present several unconditionally
stable
and convergent numerical schemes for the gradient system
characterizing
thin film epitaxy.
There are several interesting related problems which could lead to thesis at the master or Ph.D. level.
- Date: Nov. 24, 2009
Speaker: Dr. Oliver Steinbock, FSU (Chem)
Title: Rotating wave solutions in excitable reaction-diffusion systems
- Date: Dec. 1, 2009
Speaker: Dr. Wei YANG, FSU (Chem)
Title: Quantitative simulation of complex biomolecular systems: The Generalized
Ensemble Approach
- Date: Friday Dec. 4th, 2009 (joint with DSC seminar)
Speaker: David Ambrose, Drexel University
Title: Free-Surface Problems in Irrotational Fluids
Abstract: We discuss proofs of well-posedness of free-surface problems
in irrotational fluids in both two and three dimensions. The problems
to be considered are the vortex sheet with surface tension, the water wave,
and Darcy flow. In two dimensions, the proofs of well-posedness use
ideas from the numerical work of Hou, Lowengrub, and Shelley; these ideas
include using an arclength parameterization of the interface and using
geometric dependent variables rather than Cartesian coordinates. The
analysis for the three-dimensional problems requires extending these
ideas. If time allows, new numerical results for three-dimensional
flows, using ideas from the analysis of the three-dimensional problems,
will be discussed. This includes joint work with Nader Masmoudi and with
Michael Siegel.
- Date: TBA
Speaker: Vivian (Guifang) Zhou, FSU
Title: Fractional differential equations
Please contact Xiaoming Wang
if you have any question regarding this seminar.