Math 6390, Fall 2013

Topics in Mathematics: Scientific Computing

John Zweck

Course Materials

Syllabus
Background Material (A list of topics I hope you have seen before)

Lecture Notes

1. Numerical Methods for Unconstrained Optimization
2. Solvers for Nonlinear Systems
3. ODE solvers I
4. ODE solvers II
5. ODE solvers III
6. Computational Methods for Dynamical Systems
7. The Three Body Problem
8. Brief Comments on Finite Precision Arithmetic (Related to a code debugging exercise we did in class)
9. Random Number Generation
10. Probability I
11. Probability II
12. Sampling From Nonuniform Distributions I
13. Sampling From Nonuniform Distributions II
14. The Monte Carlo Method
15. Linear Dispersive Waves
16. The Nonlinear Schroedinger Equation I
17. The Nonlinear Schroedinger Equation II
18. The Split-Step Fourier Method for Numerical Solution of NLSE I
19. The Split-Step Fourier Method for Numerical Solution of NLSE II
20. NLSE with Noise
21. Introduction to the Finite Element Method
22. The Finite Element Method: Linear Lagrange Elements
23. Introduction to FEniCS
24. Introduction to Python (Demo Codes)
25A. Implementation of 1D Finite Element Method Using Python Classes [pdf]
25B. Implementation of 1D Finite Element Method Using Python Classes [python code]
25. Finite-Element Modeling of Trace Gas Sensors I
26. Finite-Element Modeling of Trace Gas Sensors II

Projects

General Advice for Scientific Computing Projects

MATLAB Introduction (not for credit)
Mini-project 0: Implementation of Newton's method for unconstrained optimization (Due Wed Sept 11)
Project 1: Three ODE problems (Due Wed Oct 2)
Project 2: The restricted planar three-body problem (Due Fri Oct 25)
Project 3: Sampling from Non-Uniform Distributions (Due Wed Nov 13)
Project 4: Monte Carlo Simulations for the Nonlinear Schroedingier Equation with Noise (Due Mon Dec 2)
Project 5: Finite-Element Modeling of a Trace Gas Sensor (Due Thurs Dec 19 at 10am [NO EXCEPTIONS])

MATLAB Code for Demonstration in Class

Matlab demo codes and results generated using publish.
Tar-gzip distribution of the Matlab demos codes.

MATLAB Tutorials

Here are two good MATLAB tutorials. The first teaches basic MATLAB porgramming skills, while the second is a more advanced.

Mathworks Matlab Tutorial
Interactive Matlab Course from Eindhoven Technical University

Installing FEniCS

FEniCS Installation

Python Resources

Python on wikipedia
Python Demo Codes
The Python Tutorial
Numerical Python (NumPy)
SciPy is a Python-based ecosystem of open-source software for mathematics, science, and engineering. Anaconda is a free Python distribution for Scientific Computing that includes over 100 of the most popular Python packages for science, math, engineering, data analysis. Unfortunately (at least on a Mac) it is not compatible with FEniCS.
Python source code for Langtangen's book "A Primer on Scientific Programming with Python


FEniCS Demos

Several of you have had trouble using FEniCS plot command. Here is a work around that works nicely (at least for linear Lagrange elements).

A function, u, that is defined at the vertices of an unstructured 2D triangular mesh and is assumed to be linear on each triangle of the mesh can be represented using a set of points (x,y,z) where the points (x,y) are the vertices of the mesh and the z values are defined by z=u(x,y). Such a set of M points can be stored in an Mx3 matrix, XYZ. In addition, we need to know what the triangles in the mesh are. To specify K triangles we can use a Kx3 matrix, Tri, the k-th row of which is a triple of numbers in the range 1,2,3,...,M that specifies the indices of the rows of the matrix XYZ that form the vertices of the k-th triangle.

The python script linked to below generates the matrices XYZ and Tri for a FEniCS Function, u, defined on a Mesh. The matlab script reads in these matrices and uses matlab's trisurf function to plot the function, u.

Python and matlab scripts to use Matlab's trisurf function to plot a FEniCS Function defined using an unstructured mesh.

NOTE: If the FEniCS plot function does work for you, you can save your figure using the commands

viz=plot(u,interactive=-true)
viz.write_ps("filename",format="pdf")

See the FEniCS Tutorial for more info.