PDE/APPLIED MATH SEMINAR (Sponsored by NSF DMS 0610149)

TUESDAY 3:10pm-4:00pm RH 308

Coordinator: Hakima Bessaih (RH210)

Spring 2008

January 29th 2008: Roberto Camassa, UNC at Chapel Hill.

Title: Enhanced diffusion in a class of time varying shear flows.

Abstract: From drug delivery in human lungs to planetary distribution of heat and chemicals, transport and mixing are omnipresent features of fluid flows. This talk will look into some of the mathematical issues behind enhancement of diffusion at large scales fueled by small scale dynamics. Diffusion enhancement plays a crucial role in understanding phenomena ranging from wind chill effect, spread of airborne contaminants, and sub-grid parametrizations for weather forecasting. The case of a very simple flow, where the fluid velocity field has a shear component subject to a time-oscillatory "cross-wind," will be examined in detail. We derive new rigorous asymptotic formulae for the enhanced diffusion induced by the temporally varying shear flow, in the limit of vanishing Strouhal number ("slow" wind oscillations), and infinite Peclet number ("small" diffusivity). These formulae document a surprisingly rich structure in the parametric dependence of the enhanced diffusion on Peclet and Strouhal numbers, characterized by regions with peculiar scaling and monotonicity behaviors.

February 12th 2008: Lynn Bennethum, University of Colorado Denver.

Title: Introduction to Modeling Flow and Deformation Processes in Porous Media.

Abstract: The study of flow through porous media is one in which mathematicians have been at the forefront of developing the model and associated numerical techniques/analysis.
The study of deformation of porous media, on the other hand, has historically been primarily developed within the field of geomechanics. In this talk we examine the deterministic models typically used for modeling flow and deformation of porous media. We examine the assumptions and the interplay between the modeldevelopment and the experiments performed. As time permits we'll examine how the models can be expanded to account for swelling porous media, such as clay (or expansive) soils.

February 19th 2008: Dimitri J. Mavriplis, Department of Mechanical Engineering, University of Wyoming.

Title:Adjoint-Based Sensitivity Analysis for Computational Fluid Dynamics.

Abstract: Adjoint-Based approaches are powerful techniques for computing sensitivities of a variety of simulation problems. The talk will cover recent work in the development, formulation and implementation of adjoint-based sensitivity analysis methods for computational fluid dynamics problems. First, a methodical approach for formulating and solving the discrete adjoint equations for fluid flow problems wiil be given. The use of these sensitivity values for aeodynamic shape optimization problems will then be demonstrated. Next, adjoint-based a posteriori error estimation techniques will be discussed. The formulation of goal-based error estimates for CFD problems will be given, both for spatial error for steady-state problems and temporal error for unsteady problems. These error estimates will then be used to drive an adaptive process where the error in the output functional of interest is reduced through mesh or time step refinement.

February 26th 2008: Peter Polyakov, University of Wyoming.

Title: Euler formula for PDEs as an application of a Radon transform.

Abstract: In the talk I will describe the earlier work of G. Henkin and myself on the complex Radon transform. Then I will describe our latest result generalizing the previous work and providing a description of the space of solutions of a system of linear PDE with constant coefficients in terms of d-bar closed residual currents.

March 4th 2008: Vladimir Ajaev, Southern Methodist University, Texas.

Title: The Effect of Evaporation on Moving Contact Lines.

Abstract: When a liquid droplet is spreading on a horizontal solid substrate, the line where its surface comes into apparent contact with the substrate is referred to as a moving contact line. Imposing proper physical boundary conditions at such contact lines is essential for an accurate description of the overall dynamics of the droplet surface, regardless of the numerical method used to solve the equations for liquid flow inside the droplet. While isothermal contact lines have been studied extensively, the effect of evaporation on contact line motion has not been fully understood. We propose a mathematical model that describes the coupled effects of viscous flow, evaporation, and surface tension in the vicinity of a contact line. The case of a thin axisymmetric droplet is used to illustrate the model and carry out comparison with experimental data. The approach is then applied to other situations such as the growth of dry patches in evaporating liquid films and the coating of a heated solid surface with a film of viscous liquid.

March 18th 2008: No seminar (Spring Break!)

March 25th 2008: Long Lee, University of Wyoming.

Title: Complete integrable particle methods and the recurrence of initial states for a nonlinear shallow-water wave equation.

Abstract: We propose an algorithm for an asymptotic model of shallow-water wave dynamics in a periodic domain. The algorithm is based on the Hamiltonian structure of the equation and corresponds to a completely integrable particle lattice. In particular, ``periodic particles'' are introduced in the algorithm for waves travelling through the domain. Each periodic particle in this method travels along a characteristic curve of the shallow-water wave model, determined by solving a system of nonlinear integro- differential equations. We introduce a fast summation algorithm to reduce the computational cost from $O(N^2)$ to $O(N)$, where $N$ is the number of particles. With the aim of providing a test of the algorithms, we scale the shallow-water wave equation to make it asymptotically equivalent to the KdV equation in the form studied by Zabusky and Kruskal in their seminal 1965 paper. Using the fast summation algorithm and the set of scales from the scaling analysis, we investigate interaction of solitons and recurrences of initial states for the shallow-water wave equation in periodic domains. Comparison of the recurrence behavior between the shallow-water and the KdV equation is provided. Finally we introduce a particle and an integral method for the shallow-water wave equation in finite intervals with homogeneous boundary conditions.

April 1st 2008: Patrick Jenny, ETH Zurich.

Title: A Brief Introduction to Probability Density Function (PDF) Modeling.

Abstract: Opposed to other modeling approaches, in PDF methods the evolution of a joint probability density function is considered. Typically, this results in a higher level of closure; at a higher computational cost, however. While PDF methods were originally developed for turbulent reactive flow simulations, the approach is much more general and it is of potential value for a wide range of further research areas including flow and transport in porous media, light scattering, and non-equilibrium gas dynamics. The goal of this course is to provide enough background for a qualified judgement on the suitability and value of the PDF modeling approach for a given problem. It will be explained how to derive PDF transport equations and for turbulent reactive flow it is shown how to model the unclosed terms."

April 8th 2008: Bengt Fornberg, University of Colorado at Boulder.

Title: Radial Basis Functions for Solving PDEs - Some Recent Developments.

Abstract: Radial basis functions (RBFs) originated in the 1970s as a method for interpolating scattered data. More recently, both our knowledge about RBFs and their range of applications have grown tremendously. They easily generalize to multiple dimensions, handle irregular domains, and can be spectrally accurate both for interpolation and for solving PDEs. We will discuss some key properties of RBF interpolants and also a couple of computational algorithms for RBFs which bypass ill-conditioning issues in the particularly interesting case of relatively flat basis functions. The standard way to numerically implement RBFs is merely a potentially ill-conditioned algorithm for a genuinely well-conditioned problem. These algorithms can be very advantageous when using RBFs to solve PDEs.

April 15th 2008: Snehalata Huzurbazar, Department of Statistics University of Wyoming.

Title: An Introduction to Bayesian Hierarchical Modelling

Abstract: Recent advances in computational capabilities have made feasible applications of Bayesian methodology for data analysis. Traditionally, statistical modelling of data has relied heavily on empirical models based on methods such as regression, and it has been difficult to incorporate complicated theoretical models (eg. pde's) for the processes generating the data. Bayesian hierarchical models make it possible to take into account modelling of the data as well as modelling of the processes generating the data. Such modelling can also address problems such as measurement errors, missing data, etc. which arise with real world data, and problems such as uncertainty about the theoretical model, as well as incorporate our prior understanding of the processes. I will provide an introduction to BHMs, illustrate the methodology with an application published by Wikle (2003) and end with some applications in progress from my current research.

April 22nd 2008: Man-Chung Yeung, Department of Mathematics, UW.

Title: An analysis of ML(n)BiCGSTAB.

Abstract: ML(n)BiCGSTAB is a Krylov subspace method for the solution of linear systems.It is a natural generalization of BiCGSTAB from the single starting Lanczos to the multiple starting Lanczos. An analysis indicates that the residuals of the approximate solutions in ML(n)BiCGSTAB can be defined in some other ways (other than the definition in Yeung and Chan’s 1997 paper). Some new definitions of the residuals result in new versions of ML(n)BiCGSTAB which have cheaper memory requirements and can converge faster. However, they also tend to diverge from the corresponding true residuals more easily. Numerical experiments will be presented to illustrate this phenomenon.

April 29th 2008: Chandana Wijeratne, Department of Mathematics, UW.

Title: Stochastic differential equations driven by a fractional Brownian motion.

Abstract: Stochastic processes are well-known for modeling financial markets, biology, turbulence and other life phenomena. A particular one is the fractional Brownian motion (fBM). Stochastic differential equations driven by a fBM will be studied. Here fractional calculus will be used since the usual Ito calculus is not applicable because the fBM is not a semimartingale. In particular the integration with respect to a fBM will be explained.

Fall 2007

December 4th 2007: Michael K. Stoellinger, University of Wyoming. (POSTPONED)

November 27th 2007: Kenneth R. Driessel, Iowa State University, A Convergent Flow That Computes the Best Positive Semi-Definite Approximation of a Symmetric Matrix.
Abstract: We shall work mainly in the space of square real symmetric matrices with the Frobenius inner product. Consider the following problem:

Problem: (Best positive semi-definite approximation.) Given an n-by-n real symmetric matrix A, find the positive semi-definite matrix that is closest to A.

We shall discuss the following differential equation in the space of symmetric matrices:

X' = (A-X)X^2 + X^2(A-X) .

In particular, we shall discuss the following result:

Theorem.
Let A have eigenvalues which are distinct and nonzero. Let the initial value X(0) be a scalar matrix rI where r is positive. If rI-A is positive definite then the solution X(t) of the differential equation converges to the positive semi-definite matrix which is closest to A.

(Note that the conditions on A are generic.) This result shows that the differential equation can be used to solve the best positive semi- definite approximation problem.

November 20th 2007: No seminar!!! (Thanks Giving's break)

November 13th 2007: S. S. Sritharan, University of Wyoming, The Feyman-Kac Formula: A Gateway to Applied Mathematics.

Abstract: Probabilistic representation of the solution of deterministic partial differential equations, known as the Feynman-Kac formula, is at the heart of many of the modern applications in mathematical science such as Monte Carlo techniques, option pricing in financial market, optimal stopping, control theory, nonlinear filtering, stochastic differential games, to name a few. In this talk we will start from the very basic examples such as Brownian and Poisson processes and indicate how the subject establishes major links between different branches of mathematics and enables us to understand such apparently disparate subjects in a unified but concrete way.

November 6th 2007: Victor Ginting, University of Wyoming, A Stochastic Homogenization for Reservoir Simulation.

Abstract: Uncertainties in reservoir flow characterization come from many sources, among which are the absolute permeability, boundary conditions, and source term. Hence, it is imperative not only to make the best prediction of the flow characterization but also to quantify the uncertainties inherent in that prediction. Common practice is using Monte Carlo simulation from which statistical properties of the flow can be obtained.
In this talk, I shall describe a stochastic homogenization procedure for typical reservoir simulations that makes use of the notion of stochastic moment equations. Assuming that the source of uncertainty is from the permeability, these equations are derived using perturbative expansion of both the permeability and the pressure governing the flow. The stochastic moment equations are then used to compute the expected value of the pressure, as well as the pressure covariance and cross-covariance of the pressure and permeability. The performance of this method shall be shown through several numerical examples, comparing its results with Monte Carlo simulation

October 30 2007: Greg Lyng, University of Wyoming, Viscous Profiles for Gas Dynamics and Combustion: Existence and Stability.

Abstract: The primary focus of this largely expository talk will be a description of the use of Geometric Singular Perturbation Theory, as initiated by Fenichel (J. Diff. Eqs. 1979), to prove the existence of viscous profiles for gas dynamics and the Navier-Stokes equations for a reacting gas mixture. The first result recovers an argument originally due to Gilbarg (Amer. J. Math. 1951) while the second result is due to Gasser and Szmolyan (SIAM J. Math. Anal. 1993).Finally, we will indicate how the geometric information in the constructed profiles carries stability information, as in Lyng & Zumbrun (Arch. Rational Mech. Anal. 2004).

October 23 2007: Hakima Bessaih, University of Wyoming, On Stochastic Shell Models.

Abstract: Shell Models are some of the most interesting examples of artificial models of fluids that capture some of their properties like Kolmogorov power law. We will address several descriptions and results of the stochastic models. The long time behavior will be studied (invariant measures and attractors). Moreover, a relationship with their linear contrepart will be studied through a parameter and the continuous dependence with respect to this parameter will be proved.

October 16 2007: Francisco Solis, CIMAT Mexico, Weighted Power Means Discrete Dynamical Systems: Fast Convergence Properties.
Abstract: We study families of discrete dynamical systems obtained by using iteration functions given by weighted power means in an attempt to understand the role of hyper-rapid convergence in almost linear maps. Our interest resides in concepts related to the velocity of convergence. We provide an ordering on these families regarding their dependence on parameters.

October 9 2007: Greg Duane, University of Wyoming and NCAR, Chaos Synchronization in PDE Systems for Data Assimilation and for Detecting Relationships Between Scales.
Abstract: A pair of loosely coupled chaotic systems can commonly be made to synchronize, despite sensitive dependence on initial conditions, because
"conditional Lyapunov exponents" are negative. The synchronization, phenomenon, first investigated in ODE systems, has been extended to PDE
systems such as those describing the climate system. Data assimilation can be viewed in this context as the task of synchronizing two systems,
one representing ``truth" and the other representing ``model", that are coupled loosely in one direction. The synchronization approach to data
assimilation is shown to be equivalent to standard approaches (3dVAR and Kalman filtering) under a certain linearity assumption, by considering
a stochastic system of two PDEs coupled through a noisy channel, and optimizing the coupling for maximum synchronization. The synchronization approach
also provides a natural way to treat nonlinearities.

Model error can also be treated naturally in the synchronization view: It can be proved that if two identical systems can be made to synchronize, then
parameter adjustment rules can be added to the dynamical equations so as to synchronize both parameters and states in a pair of systems with
mismatched parameters. This adaptive model approach is useful in situations where ensembles, used in the customary Kalman filtering approach, are too
computationally expensive.

Scale-limited coupling is effective for synchronization because of slaving or partial slaving of the small scales, as would define an inertial manifold or
approximate inertial manifold, respectively. It is suggested that the existence of a slow manifold, defined by variables that are simply independent
of small-scale, fast processes, might also be detected via partial synchronization. The suggestion is illustrated using a Hamiltonian system,
from particle physics, that predicts the occurrence of coherent structures in the early universe.

October 2 2007: Benito Chen, University of Wyoming, Mathematical Modeling of Bioremediation of Trichloroethylene in Aquifers.

Abstract: Trichloroethylene (TCE) is a very common contaminant of groundwater. It is used as an industrial solvent and is frequently poured into the soil. There exist bacteria that can degrade TCE. In contrast with most cases of bioremediation, the bacteria that degrade TCE do not use it as a carbon source. Instead the bacteria produce an enzyme to metabolize methane. This enzyme can degrade other organics including TCE. In this paper we model in situ bioremediation of TCE in an aquifer by using two species of bacteria: one that forms biobarriers to restrict the movement of TCE and the second one to reduce TCE. The model includes flow of water, transport of TCE and the nutrients, bacterial growth and degradation of TCE. Nonstandard numerical methods are used to discretize the equations. Some results are presented.

September 25 2007: Meng Xu, University of Wyoming, The peculiar mathematical structure of Non-Newtonian flows.
Abstract. In this talk, I will be focusing on the mathematical structure of Non-Newtonian flows. An analysis of difference between Newtonian and Non-Newtonian flows will be described mathematically. The main point is to derive the constitutive equation for the UCM fluid from the Fokker-Planck Equation and explain the type of the equations governing viscoelastic flows. Some examples of Non-Newtonian flows in nature will also be given. In the end, I will list my research goals in the future study of Non-Newtonian fluid dynamics.

September 18 2007: Dan Stanescu, University of Wyoming, Shock wave micro-structure in inviscid gas flow.

Abstract. In this second talk, after we spend a few minutes to define functions on the hyperreals as well as differentiation and integration, we'll use this knowledge to study the micro-structure of shock waves. In 1949, Morduchow and Libby found the exact entropy distribution across the shock layer of a VISCOUS heat-conducting gas. They observed that, unlike the other flow variables that behave monotonically, the entropy reaches a maximum in the layer. The shock layer, where viscous and heat-conducting effects are important, is very thin (of the order of several mean free paths); for all practical purposes it can be assimilated with a discontinuity. We'll show that if we associate different Heaviside functions (with their jumps located within the same infinitesimal interval) with the flow variables, the governing equations of INVISCID gas flow give the RELATIONS between these functions. We'll also prove that the same INVISCID equations do predict the maximum in entropy within the infinitesimal layer. However, the INVISCID equations are not able to define the SHAPE of the functions; additional physical information is needed, and this is only provided by a viscous analysis.

September 4 2007: Dan Stanescu, University of Wyoming, On Nonstandard Analysis.

Abstract. Leibniz tried to develop calculus using the idea of infinitesimals, but failed in his attempts to establish a firm mathematical theory based on them. Although infinitesimals have been used ever since, only in the late 1950's did A. Robinson succeed to develop such theory using results in mathematical logic. Our purpose is to use it for the study of fluid flow discontinuities. Multiplication of generalized functions can then be handled consistently in a relatively easy fashion using only elementary calculus, since they become smooth functions over an infinitesimal interval. One thus obtains an alternative to Colombeau's or Rosinger's theories of multiplication of distributions.

In this first seminar on the topic I will discuss the construction of a nonstandard (or hyperreal) number system from sequences of real numbers.
We'll later use this knowledge to study inviscid gas dynamics and obtain an interesting result concerning the entropy jump in a shock wave, which turns out to be non-monotone as opposed to jumps in specific volume, velocity and pressure. The presentation will be geared toward graduate students; I will start from basics and proceed at a pace that should
allow anyone to follow through easily.