Jean-Luc Thiffeault's Homepage

Math 715: Methods of Computational Mathematics II: Spring 2014


Lecture Room: B211 Van Vleck Hall
Lecture Time: 12:05–12:55 MWF
Lecturer: Jean-Luc Thiffeault
Office: 503 Van Vleck
Phone: (608)263-4089
Email: jeanluc@[domainname],
where [domainname] is math point wisc point edu
Office Hours: Wed and Thu 11:00–11:45

Description

Topics: numerical linear algebra, finite element methods, boundary integral method.

We will begin by covering a number of issues in numerical linear algebra, such as matrix decomposition theorems, conditioning and stability in the numerical solution of linear systems, and iterative methods. With these tools in hand, we will proceed to discuss the finite element method, continuous and discontinuous Galerkin methods, multigrid methods, and error estimates. We will also cover boundary element and boundary integral techniques for the numerical solution of PDEs recast into integral form.

Incoming students should be comfortable with programming (in C++, Fortran, Python, or Matlab, or...), should have completed coursework in linear algebra, and should have familiarity with solving partial differential equations.

Textbooks

There is no official course textbook. You may be interested in visiting the following supplementary texts, which will be placed on reserve in the Math Library (Van Vleck B224):

Homework/Grading

The final grade will be determined by scores on homework assignments, which will be both analytical and computational in nature. Feel free to discuss the homework with each other, but you are required to code/write up your own solutions. Your work and results should be communicated clearly: points will be deducted for lack of clarity!

Homework Sets

Homework set #1 (Due 2/17/2014): HW #1

Homework set #2 (Due 3/03/2014): HW #2

Homework set #3 (Due 3/2431/2014): HW #3

Homework set #4 (Due 4/21/2014): HW #4

Homework set #5 (Due 5/12/2014): HW #5

Schedule of Topics

lecture date pages topic
1 01/22 T&B 3–24 Norms
2 01/24 T&B 25–31 The Singular Value Decomposition
3 01/27 T&B 32–47 More on SVD; Projectors
4 01/29 T&B 48–62 QR Factorization & Gram–Schmidt
5 01/31 T&B 69–76 Householder Triangularization
6 02/03 T&B 69–92 Least Squares; Condition Number
7 02/05 T&B 93–96 Condition Number
8 02/07 T&B 97–107 Floating Point Arithmetic
9 02/10 T&B 108–120 Accuracy and Stability
10 02/14 T&B 121–128 Stability of Backsubstitution; LU Decomposition
11 02/17 T&B 147–162 LU Factorization
12 02/19 T&B 163–175 Stability of LU; Cholesky Factorization
13 02/21 T&B 175–188 Cholesky Decomposition; Eigenvalues
14 02/24 T&B 190–207 Rayleigh Quotient; Inverse Iteration
15 02/26 T&B 207–215 Rayleigh Quotient Iteration ; QR Algorithm
16 02/28 T&B 215–233 QR Algorithm with Shifts; Other Algorithms
17 03/03 T&B 234–240 Computing the SVD
18 03/05 Jacobi & Gauss–Seidel Iteration
19 03/07 T&B 250–265 Arnoldi Iteration for Eigenvalues
20 03/12 Discussion of Homework Problem
21 03/12 T&B 266–275 GMRES
22 03/14 T&B 293–298 Conjugate Gradient Method
23 03/24 T&B 298–301 Convergence of CG Method; Intro to Elliptic Problems
24 03/26 Johnson 14–18 Three equivalent problems
25 03/28 Johnson 18–22, 26–29 Finite Element Method with Piecewise Linear Functions; FEM in 2D
26 03/31 Johnson 29–35 FEM in 2D (cont'd); Hilbert Spaces
27 04/02 Johnson 35,38,40–42 Hilbert Spaces; Natural vs Essential Boundary Conditions
28 04/04 Johnson 50–57 Lax–Milgram Theorem; Error Estimate
29 04/07 Hilldale Lecture at WID
30 04/09 Johnson 57–62 Some examples of applications of Lax–Milgram
31 04/11 Johnson 67–81 Finite-Element Spaces
32 04/14 Johnson 67–81 More examples of FE Spaces
33 04/16 Johnson 84–92 Error estimates
34 04/18 Johnson 84–92 Error estimates (cont'd; proof)
35 04/21 Johnson 101–104 Physical example: elasticity
36 04/23 Johnson 106–107 Physical example: Stokes flow; Mixed methods
37 04/25 Johnson 146–159 FEM for parabolic PDEs: discontinuous Galerkin method
38 04/28 Johnson 167–196 FEM for hyperbolic PDEs
39 04/30 Johnson 214–223 Boundary Integral Method
40 05/02 Johnson 224–230 Numerical solution of Fredholm integral equations; Boundary Element Method
41 05/05 BHM 7–27 Intro to Multigrid
42 05/07 BHM 31–43 Intro to Multigrid (cont'd)