University of Utah
Department of Mathematics


Math 5740   Mathematical Modeling
M W F     9:40-10:30 am    JWB 208

Instructor:   Elena Cherkaev
Office:   LCB 206   ph: 581-7315 
        email:  elena@math.utah.edu
Office hours:  M 10:30am-11:30am
                           and by appointment
http://www.math.utah.edu/~elena/


Texts:

 Industrial Mathematics: A Course in solving real-world problems, Avner Friedman and Walter Littman,  SIAM, 1994
Computational methods for inverse problems, Curtis R. Vogel, SIAM, Frontiers in Applied Mathematics, 2002
 MATLAB programs from this book can be downloaded from  http://www.math.montana.edu/~vogel/Book/
 Instructor notes

Project 1:   Salt Lake Tribune article on air pollution

Text for project 2: Introduction to Climate Models

Project 3 :   Test  Images   Matlab notes
 

Course description:  The course will discuss mathematical models arising from real-world applications. The emphasis will be placed on formulating a well-posed problem, studying stability of its solution, as well as methods of regularizing the problem in case when it is ill-posed.

 

 

The course is addressed to senior undergraduate and graduate students in mathematics, science, and engineering. This class fulfills one of the requirements for the Computational Engineering and Science (CES) Program here at the University. Enrollment in the program is necessary to obtain CES Certificate or MS credit. - If you are interested in learning more about the CES program, please visit www.ces.utah.edu or contact Karen Feinauer (karenf@cs.utah.edu, 585-3551).

 Prerequisites: Calculus and Differential Equations

               


      


Tentative Course Outline and Homework Schedule

 

Part I
Jan 7 -   Jan21

Air quality modeling problem,  Friedman, Littman, Ch. 2.

Advection-diffusion equation, von Neumann stability criterion, conditionally and unconditionally stable
numerical schemes.

 

Part II 
Jan 23-
   Feb 11

 Problem in electron beam lithography, Friedman, Littman, Ch. 3

 The Cauchy problem for the heat equation, Fourier series, the heat equation forward and backward in time, Cesaro summability of Fourier series.

 Hw 1 – Feb 1

Part III
Feb 13 - March 31

 Population dynamics problem

 

 Hw 2 –  Feb 29

Part IY
March 7-
  March 31

 Atmospheric optics problem. Vogel, Ch. 1,2,3.

 Discrete model,
SVD decomposition. Ill-posedness. Regularization by filtering, truncated SVD, Tikhonov filter. Integral equation model. The Picard condition.

 Hw 3 – March 28

Part Y 
March31 -  Apr 23

 Image de-blurring problem. Vogel, Ch. 4,5

 Statistical estimation
theory. Parameter identification problem. Variational
regularization methods. Landweber iterations. Iterative regularization methods.

 Hw 4 – Apr 18

 


Homeworks:   Homework assignments will be given for each section we discuss. For the homeworks you will need to use/write short programs using your favorite computer language or system - Matlab, Fortran, or C.
Projects:  There will be three projects during the course. Grades:  Class grade is based on handed in homeworks and completed projects.
 Holidays:

Martin Luther King Jr. Day holiday

Monday, January, 21

Presidents’ Day holiday

Monday, February, 18

Spring break

Monday - Friday,  March, 17 - 22