Computational and Applied Mathematics (CAM) Seminar

 

Fall, 2006

 

CAM seminar  usually runs from 2:00-3:00PM, Wednesday.

 

(Link to CAM seminar talks in other years.)

(Link to Numerical Analysis Group Webpage)

 

 

Date

Speaker/Institution

Title

 

 

 

Aug 23

Organizing meeting

 

Aug 30

Prof. Volker Mehrmann (Guest speaker),

Technische Universitat, Berlin

Optimal control of differential-algebraic equations (abstract)

Sept 6

Ralph Byers

Dirty Secrets of the TI83 and TI84 (abstract) (more information)

Sept 13

Erik Van Vleck

The error in orthogonal integration (abstract)

Sept 20

Weishi Liu

On a class of singular perturbation problems with turning points (abstract)

Sept 27

Chuan-Tau Lan (Guest speaker), J. L. Constant Professor, KU

Identification of a nonlinear dynamical model from noisy test data (abstract)

Oct 4

Weizhang Huang

A mathematical theory for anisotropic mesh adaptation (abstract)

Oct 18

Hongguo Xu

Krylov matrix and matrix functions (abstract)

Oct 25

Myunghyun Oh

A brief survey on nonlinear stability (abstract)

Nov 1

Xianping Li

Effect of regularization on numerical simulation of phase change problems (abstract)

Nov 8

Milenna Stanislavova

Nonlinear stability and instability for Hamiltonian systems with symmetries (abstract)

Nov 29

Ventsislav Ivanov

Chaotic Synchronization (abstract)

Dec 13

Dr. Jingtang Ma (Guest speaker), Simon Fraser University

Convergence of Moving Mesh Methods for Time-Dependent PDEs (abstract)

 

 

 

1. Volker Mehrmann, Optimal control of differential-algebraic equations.

 

It has been an open problem for quite some time to extend the classical theory of optimal control (e.g. the optimality conditions and the maximum principle) to differential-algebraic equations (DAEs). Some applications where these problems arise will be presented. Then the classical optimal control theory for ordinary differential equations will be reviewed and it will be demonstrated by some simple examples that this theory cannot be extended directly to DAEs of arbitrary index.

 

It is then shown that via a behavior approach and a reformulation of the DAE one can derive the optimality conditions and a maximum principle. The resulting optimality boundary value problems can then be reinterpreteted in terms of the original systems.

 

The results and appropriate numerical methods are illustrated by applying them to a motor controlled pendulum.

 

 2. Ralph Byers, Dirty Secrets of the TI83 and TI84

 

        or

 

IF (E/D = 2)   and (E-D = 0);          THEN GROAN

IF (A-100 = 0) and (A-.99-.01 not= 0); THEN CRY

IF (B <= 2)    and (2B  > 4);          THEN SCREAM

 

Disclaimers:

 

        o This is primarily *not* a talk about rounding errors.

 

        o It is not a research talk. 

 

        o It is accessible to anyone with a sound background in

          decimal arithmetic.

 

        o Frustrating experience with a TI83 or TI84 calculator is

          recommended but not required background.

 

        o It is a short talk of no more than 30 minutes.

 

The TI83 and TI84 are among the most popular calculators on campus. Hundreds of unsuspecting undergraduates succeed with help from these calculators despite (or perhaps because of) some hidden ``features'' in the built in arithmetic.  Students, instructors, engineers, scientists and hobbyists write complex TI83/TI84 programs.  Nevertheless, it is difficult to calculate differences A - B and remarkable difficult to correctly distinguish A>B from A=B from A<B.

 

This non-judgmental talk tells some (but probably not all) of the hidden secrets of the TI83 and TI84 arithmetic and sets forth some TI83/TI84 issues and pitfalls.

 

More information:

 

       1) Dirty Secrets Writeup in PDF format (here).

 

        2) ASCREAM TI83/TI84 calculator program .TXT file (here).

           This is the program I showed in the seminar demonstrating some oddities in TI83/TI84 arithmetic.

 

        3) ASCREAM TI83/TI84 calculator program .83p file (here).

 

        4) DIGIT14 TI83/TI84 calculator program .TXT file (here).

           This program stores all significant digits in variable X as a string in str0.

 

        5) DIGIT15 TI83/TI84 calculator program .83p file (here).

 

 

       There are two ways to get the following programs into a calculator.

 

       The easy, inexpensive way:

              Copy the programs from someone who has them on their calculator already, e.g., the instructor. 

              Use the link cable that came with the calculator and follow the instructions in Chapter 19

              of the guidebook 

 

       The difficult, expensive way:

              You may use a web browser to download the programs to a PC, then use  linking software

              And special cables to copy the programs into your calculator. Of course, TI sells link cables and

              distributes link software through many academic bookstores.

              Alternatively, you may download ``free'' software and/or make your own cables.

 

3. Erik Van Vleck, The error in orthogonal integration.

 

Consider the matrix differential equation Q' = Q*S(Q,A), t>0, Q(0) orthogonal, where S(Q,A) is a specific skew-symmetric matrix function that depends on the dependent variable Q and a given bounded matrix function A(.). In this talk we outline a forward error analysis for the numerical approximation of Q(.) that is valid on infinite time intervals.  Key ingredients are the uniqueness of the QR factorization of a matrix, a recently developed backward error analysis that takes advantage of the orthogonal structure in the problem, and integral separation.

 

4. Weishi Liu, On a class of singular perturbation problems with turning points.

 

Phase portrait is the ultimate goal for understanding nonlinear dynamics. If we couldn't obtain phase portraits for ALL nonlinear systems (this is a safe assumption), we try to do so for some SPECIAL and IMPORTANT classes; if we couldn't get GLOBAL phase portraits (this is also a safe assumption), we try to get LOCAL ones.

 

Singularly perturbed problems (with and without turning points) are a class of special but important systems. We will try to get a rough sketch of local phase portraits for an even more special class of singularly perturbed systems with turning points.

 

5. C. Edward Lan, Identification of a nonlinear dynamical model from noisy test data

 

A fighter aircraft suffered uncommanded rolling motion, called wing drop, during flight tests at transonic speeds. The instability was not identified during the design phase through the conventional wind-tunnel testing. Therefore, a new dynamic wind-tunnel test program with a free-to-roll test rig was initiated. The main purpose of our project is to identify possible dynamic instabilities of the full-scale aircraft by examining the roll dynamic characteristics of the models in the tunnel. To improve the tunnel balance data caused by model vibration, a model-based filtering method is developed and applied. The corrected rolling moment coefficients are then modeled through a fuzzy logic algorithm as functions of: angle of attack, sideslip, roll angle, etc. A wing drop theory is proposed and verified based on the values of the relative nonlinear aerodynamic stiffness in the rolling equation of motion, Wing drop is predicted if the relative aerodynamic stiffness becomes zero, or changes sign. This is equivalent to the vanishing of frequency in the vibration theory. It is found that among those test runs examined only the preproduction F-18E model encounters wing drop conditions, while the AV-8B model develops only the wing rock motion (i.e. limit-cycle oscillations). It is also found that wing rock motion in the tunnel on a free-to-roll test rig is mainly caused by the nonlinear damping effect of the time rate of sideslip angle, not the traditional roll damping due to roll rate.

 

6. Weizhang Huang, A mathematical theory for anisotropic mesh adaptation.

 

In this talk I will present a mathematical theory of anisotropic mesh adaptation for the numerical solution of partial differential equations. Two principles, equidistribution and alignment, will be shown to be necessary and sufficient for a complete control of the size, shape, and orientation of mesh elements. Applications and numerical results will be presented to demonstrate that the developed equidistribution and alignment principles are useful tools for use in designing reliable and robust algorithms for adaptive anisotropic mesh generation.

 

7. Hongguo Xu, Krylov matrix and matrix functions.

 

Let A be an n-by-n matrix and b be a column vector. The matrix [b,Ab,...,A^{n-1}b] is a Krylov matrix of A. Krylov matrix is a basic tool in numerical linear algebra. It appears in numerical solution of large scale system of linear equations and numerical eigenvalue problem. Can Krylov matrix do more? We will give a positive answer in this talk. We will show that when all the eigenvalues of A has a single Jordan block, f(A) can be formulated by a Krylov matrix. Here f(t) can be a polynomial, a rational function, an exponential, or just an analytic function. The formulation may reduce the cost for computing f(A) from at least O(n^4) to O(n^3).

 

8. Myunghyun Oh, A brief survey on nonlinear stability.

 

For a spectrally stable traveling wave, we consider the stability of the wave for the full partial differential equation. We survey available methods and difficulties of them. We introduce the pointwise semigroup method which has been used for traveling waves in conservation laws.

 

9. Xianping Li, Effect of regularization on numerical simulation of phase change.

 

Regularization is commonly used in numerical simulation of phase change problems. The choice of regularization method and the parameter used therein affects greatly the accuracy of solution. A numerical study of the issue will be presented in this talk.

 

10. Milenna Stanislavova, Nonlinear stability and instability for Hamiltonian systems with symmetries.

 

The study of nonlinear stability of coherent structures, such as soliton solutions, is very important and numerous analytical techniques have been developed. From the spectrum of the linearization one can typically produce stability results for dissipative systems and only nonlinear instability results for conservative systems. I will describe the available global techniques for proving orbital stability (instability) results in the case of Hamiltonian systems with symmetries. The main focus will be on particular examples of PDEs for which one can apply the abstract theory with or without modifications.

 

11. Ventsislav Ivanov, Chaotic Synchronization.

 

The first usage of the logistic map can be traced back to Pierre Francois Verhulst, a Belgian mathematician with a quest for refining the Malthusian demographic model. Later usages include modeling the surplus produced by species in the presence of population limiting factors such as finite resources and death. Much interest in the map's chaotic regions, formed by multiple period doubling bifurcations, has produced its latest usage for generating random numbers. This talk will present the current state and results generated by a chaos analysis tool aimed at analyzing the behavior inside the map's chaotic regions. The goal of the research is to exploit this apperant randomness, design a secure digital communication channel (corresponding to the map's biotic potential value) and effectively transmit sensitive information across arbitrary communication mediums.

 

12. Jingtang Ma, Convergence of Moving Mesh Methods for Time-Dependent PDEs.

 

I will present the recent progress on the convergence of the moving mesh methods for time-dependent PDEs. Two typical PDEs whose solutions contain small layers and behave blow-up will be discussed. Brief history and open questions will also be outlined in this talk.

 

This page is maintained by Weizhang Huang.