Welcome to MATH 745 - TOPICS IN NUMERICAL ANALYSIS

Fall 2012 Edition
Time & Place: Tuesdays 12:00-13:00 and Fridays 10:30-12:30 in HH/312




Instructor: Dr. Bartosz Protas
Office: HH 326, Ext. 24116
Office hours: Tuesdays 13:30-14:30 and Fridays 13:00-14:00, or by appointment

Announcements:
  • Project reports are due by midnight on Friday, December 21. They should be should be submitted electronically to the instructor.

  • Information about requirements for the project reports has been posted here

  • Solutions to Homework Assignment #2 are already posted.

  • Solutions to Homework Assignment #1 are already posted.




    Outline of the Course:

     The course will focus on techniques for numerical solution of Partial Differential Equations (PDEs). The objectives of the course are essentially twofold: first, provide students with an understanding of the deeper mathematical foundations for certain classical numerical methods which they should already be familiar with, and, secondly, introduce students to more advanced numerical methods for PDEs. The course will address both theoretical aspects, such as error and stability analysis, as well as certain implementation issues. The presented methods will be illustrated using well-known PDEs from mathematical physics. The specific topics that will be discussed include (optimistic variant):

    1) Critical Review of Finite--Difference Methods
         a) Discretization of differential operators; incorporation of boundary conditions
         b) Accuracy and conditioning of numerical differentiation
         c) Advanced numerical differentiation (complex step derivative, Pade schemes, compact finite differences)
    2) Review of Approximation Theory
         a) Functional analysis background (Hilbert spaces, inner products, orthogonality and orthogonal systems)
         b) Best approximations
         c) Interpolation theory
    3) Spectral methods for PDEs
         a) Differentiation in spectral space
         b) Fourier and Chebyshev methods; fast transforms (FFT)
         c) Application to nonlinear problems (pseudo--spectral methods, dealiasing)
    4) Multiresolution methods for PDEs
         a) Orthogonal wavelets
         b) Discrete wavelet transform (DWT)
         c) Multiresolution representation of functions

    Primary Reference:
         a) L. N. Trefethen, Spectral Methods in Matlab, SIAM, (2000).

    Supplemental References:
         b) K. Atkinson and W. Han, Theoretical Numerical Analysis: A Functional Analysis Framework, Springer (TAM 39), (2001)
         c) J. P. Boyd, Chebyshev and Fourier Spectral Methods, Second Edition (Revised), Dover, (2001).

    In addition to the above references, sets of lecture notes and example MATLAB codes will be made available to students on the course webpage.

    Prerequisites:

     Numerical Analysis at the undergraduate level (including numerical methods for ODEs and PDEs), Partial Differential Equations, basic programming skills in MATLAB

    Grades:

     The final grades will be based on
         a) two 20 min quizzes (2 x 10% = 20%),
         a) two homework assignments (2 x 10% = 20%),
         b) a take-home final project (60%).

    The tentative quiz and homework due dates:
         i) Quiz #1 - Friday, October 26
         ii) Quiz #2 - Friday, November 30
         iii) Homework Assignment #1 - Tuesday, October 2 (distributed) / Tuesday, October 9 (due)
         iv) Homework Assignment #2 - Tuesday, October 30 (distributed) / Tuesday, November 6 (due)

    I reserve the right to alter your final grade, in which case, however, the grade may only be increased.

    Academic Integrity:

     You are expected to exhibit honesty and use ethical behaviour in all aspects of the learning process. Academic credentials you earn are rooted in principles of honesty and academic integrity.

    Academic dishonesty is to knowingly act or fail to act in a way that results or could result in unearned academic credit or advantage. This behaviour can result in serious consequences, e.g., the grade of zero on an assignment, loss of credit with a notation on the transcript (notation reads: "Grade of F assigned for academic dishonesty"), and/or suspension or expulsion from the university.

    It is your responsibility to understand what constitutes academic dishonesty. For information on the various types of academic dishonesty please refer to the Academic Integrity Policy,. The following illustrates only three forms of academic dishonesty:
         1) Plagiarism, e.g., the submission of work that is not one's own or for which other credit has been obtained.
         2) Improper collaboration in group work.
         3) Copying or using unauthorized aids in tests and examinations.

    Important Notice:

     The instructor and university reserve the right to modify elements of the course during the term. The university may change the dates and deadlines for any or all courses in extreme circumstances. If either type of modification becomes necessary, reasonable notice and communication with the students will be given with explanation and the opportunity to comment on changes. It is the responsibility of the student to check their McMaster email and course websites weekly during the term and to note any changes.