Numerical Analysis and Approximation II

Math 702: Numerical Analaysis and Approximation II

Instructor  Course Section                     Time              Room
------------------------------------------------------------------------
Eric Olson  Math 702 Numerical Analysis II     TR 9-10:15am      AB206

Course Information

Instructor:
Eric Olson
email:
ejolson at unr edu
Office:
Tuesday and Thursday 12 noon DMS 238 and by appointment.
Homepage:
http://fractal.math.unr.edu/~ejolson/702

Supplemental Texts:

Justin Solomon, Numerical Algorithms: Methods for Computer Vision, Machine Learning and Graphics, CRC Press, 2015.

J.W. Thomas, Numerical Partial Differential Equations: Finite Difference Methods, Springer Verlag, 2010.

David Kincaid, Ward Cheney, Numerical Analysis: Mathematics of Scientific Computing, 3rd Edition, American Mathematical Society, 2002.

J Stoer, R Burlisch, Introduction to Numerical Analysis, 3rd Edition, Springer, 2002.

William Press, Saul Teukolsky, William Vetterling, Brian Flannery, Numerical Recipes, 3rd Edition, Cambridge University Press, 2007.

Roger Peyret, Spectral Methods for Incompressible Viscous Flow, Springer, 2002.

Evans, Blackledge and Yardley, Numerical Methods for Partial Differential Equations, Springer, 2000.

Evans, Blackledge and Yardley, Analytic Methods for Partial Differential Equations, Springer, 1999.

Announcements

[19-May-2019] Answers for Programming and Homework 2

I have created a solution key for the combined programming and homework assignment number 2. Even if you received full credit for your work, please compare my solutions to yours to note any differences in approach or presentation.

[15-May-2019] Solution Key for Homework 1

I have created a solution key for homework one. Even if you received full credit for your work, please compare my solutions to yours to note any differences in approach or presentation.

[14-May-2019] Final Exam

Our final exam will be held in the usual classroom AB206 from 7:30 to 9:30am on Tuesday, May 14.

[13-May-2019] Solution Key for Project 1

I have created a solution key for programming project one. Even if you received full credit for your work, please compare my solutions to yours to note any differences in approach or presentation.

[07-May-2019] Navier-Stokes Part III

Here is the Fortran file continuing our code for solving a lid-driven flow for the Navier-Stokes equations in a two-dimensional domain.

[02-May-2019] Exam 1

There will be an exam in class on May 2 to prepare you for the final exam. I've added a few topics to the list we discussed in class. Tentatively, the following topics are under consideration:
  1. A proof of the orthogonality lemma for the discrete Fourier transform.

  2. A proof of the inversion theorem for the discrete Fourier transform

  3. A derivation of the conquer-and-divide step for constructing a fast Fourier transform of size N=2n.

  4. An analysis showing the time complexity of the fast Fourier transform is O(n logn).

  5. An algorithmic description of the Gauss–Newton method for solving a non-linear optimization problem.

  6. A proof of the Poincare inequality

    ∫0L |f(x) − V|2dx ≤ L2 ∫0L |f '(x)|2dx   where   V= L−1 ∫0L f(x) dx.

  7. A proof that the velocity profiles of the viscous Burgers equations exponentially flatten out as

    ∫0L |u − V|2 ≤ M e−αt    where    V= L−1 ∫0L u0.

  8. A proof of Taylor's theorem which says

    f(x+h) = f(x) + h f '(x) + (h2/2!) f ''(x) + ··· + (hn/n!) f(n)(x) + Rn
    where
    Rn = ∫xx+h ((x+h-t)n/n!) f(n+1)(t) dt.

  9. An explanation with proof how to simplify the geometric series

    αp + αp+1 + ··· + αq    where    p<q.

  10. A proof using Taylor's theorem that the finite difference approximations have the following orders:

    f '(x) ≈ (f(x+h)-f(x-h))/(2h) + O(h2)
    and
    f ''(x) ≈ (f(x+h)-2f(x)+f(x-h))/h2 + O(h2).

[30-Apr-2019] Navier-Stokes Part II

Here is the Fortran file continuing our code for solving a lid-driven flow for the Navier-Stokes equations in a two-dimensional domain.

[25-Apr-2019] Navier-Stokes Part I

Here is the Fortran file beginning our code for solving a lid-driven flow for the Navier-Stokes equations in a two-dimensional domain.

[25-Apr-2019] Programming and Homework 2

Programming and Homework Project 2 on solving the viscous Burgers equation is is now available. This combined assignment will be worth 40 points. The data files and project description may be found here.

[20-Apr-2019] Homework 1

Homework 1 covers the fast Fourier transform. Please work the problems at the end of the handout and turn these in by April 20. Here are the supporting files.

[18-Apr-2019] Successive Over-relaxation Method

Here are the Fortran files used in class to test the SOR method for solving the linear system Ax=b.

[08-Apr-2019] Fluid Flow

We will be working through this material using finite-differences to approximate solutions to the Navier–Stokes equations.

[04-Apr-2019] Aliasing

Here are the Matlab files to show the effect of aliasing and a simple way to dealias the convolution used to compute the nonlinear term in the the viscous Burgers equations.

[02-Apr-2019] The Viscous Burgers Equation

Here are the Matlab files using Euler's and the RK2 method to solve the viscous Burgers equations.

[26-Mar-2019] Euler's Method

Here are the Matlab files using Euler's method to solve the dissipative wave equation.

[12-Mar-2019] Solving the Linear Wave Equation

Here are the Matlab files using the Fourier transform to solve the dissipative wave equation.

[07-Mar-2019] Fourier Transform in Fortran

Here are the Fortran files used in class for the Parallel Fourier Transform.

[08-Mar-2019] Programming Project 1

Programming Project 1 on solving non-linear least squares is is now available. The data files and project description may be found here.

[28-Feb-2019] Fourier Transform in Fortran

Here are the Fortan files used in class to demonstrate the fast Fourier translform.

[26-Feb-2019] Fourier Transform in Matlab

Here are the Matlab/Octave files used in class to demonstrate the fast Fourier translform.

[07-Feb-2019] Non-linear Optimization

Here are the Matlab/Octave and Maple files used in class to perform non-linear optimization using the Gauss-Newton least squares method.

[05-Feb-2019] Frequency Modulation

Here are the Matlab/Octave files used in class to create data sets resulting from non-linear frequency modulation with random noise.

Additional Resources

Grading

    2 Homework Assignments    20 points each
    2 Programming Projects    20 points each
    1 Midterm                 30 points
    1 Final Exam              50 points
   ------------------------------------------
                             160 points total

Homework and Exams




Final Exam

The final exam will be held on 
Tuesday, May 14 from 7:30-9:30am in AB206. 


Equal Opportunity Statement

The Mathematics Department is committed to equal opportunity in education
for all students, including those with documented physical disabilities
or documented learning disabilities. University policy states that it is
the responsibility of students with documented disabilities to contact
instructors during the first week of each semester to discuss appropriate
accommodations to ensure equity in grading, classroom experiences and
outside assignments.


Academic Conduct

Bring your student identification to all exams. Work independently on
all exams and quizzes. Behaviors inappropriate to test taking may disturb
other students and will be considered cheating. Don't talk or pass notes
with other students during an exam. Don't read notes or books while taking
exams given in the classroom. 
You may work on the programming assignments in groups of two if desired.
Homework may be discussed freely. If you
are unclear as to what constitutes cheating, please consult with me.



Last updated:
Thu Jan 31 08:55:25 PST 2019