Math 116: Numerical methods for PDEs and waves
Alex Barnett, Fall
2008, Tu and Th 10:00-11:50am
2D scattering from sound-soft obstacle
Rapid progress in computer power and numerical algorithms in
recent decades has revolutionized science and technology.
The Laplace equation (describing steady-state diffusion, heat flow,
electrostatics) and Helmholtz equation (linear waves, acoustics,
electromagnetics, optics, quantum) are linear PDE boundary value
ubiquitous in modeling the real world. They may be solved numerically by
recasting the problem onto the boundary; this is more efficient at short
wavelengths (and easier to code) than standard discretization methods.
will build codes, analyse their errors, and later
explore phenomena in wave scattering
and quantum chaos (short-wavelength asymptotics).
You will learn some of the deep mathematics required to understand
the success and efficiency of modern algorithms.
Admin: Office hours M 3-4pm, F 2-3 pm.
Our course TA and coding coach is Jon Brown, who runs
the X-hour 3pm Wed.
Consider the following
Matlab classes, if you like
learn in a
group (the 1st is a bit late to help, but the 2nd looks useful)
- Introduction to Matlab: Oct 8 4-6pm. Berry Instructional Center,
- Programming With Matlab: Oct 16 4-6pm, Berry Instructional Center,
Lecture notes, topics, readings,
Lecture notes are my own rough scribbles; brackets [NLA] etc refer to
books, as listed in the syllabus
- Week 1: Overview linear
PDEs. Linear systems, norms, singular value decomposition. [NLA] Ch. 1-5.
- Week 2:
Condition number, floating-point representation, stability. [NLA Ch.
Interpolation on intervals. [NA] Ch. 8.1
- charges_polynomial: shows contour
lines of size of log of monic polynomial qn+1 in C for
two node spacings
- charges_equilibrate: relaxation
animation towards equilibrium for charges on an interval
- Week 3:
Quadrature, convergence rate given by nearest singularity, error bounds
and proving convergence [NA]
- Week 4:
Periodic quadrature [NA] Ch. 8.2, 9.4.
Integral Equations, Nystrom method. [Atkinson] Ch 1.1.-1.3, [NA] Ch. 12.3.
- Week 5:
Functional analysis and compact operators [NA] Ch. 12.1-2.
Laplace's equation, Green's representation formula. [LIE] Ch. 6.1
Worksheets: laplace (Greens identities
and checking the fundamental solution).
- Week 6:
Potential theory, jump relations, boundary integral formulations.
Interior and exterior Laplace BVP.
[LIE] Ch. 6
- Code for HW6 #2,3: dlp_bvp_simple.
Note the use of vectorization to fill Nystrom matrix with no loops.
- Week 7:
Helmholtz scattering problems.
Avoiding interior resonances: Brakhage-Werner-Leis-Panich method.
Dirichlet eigenvalues of domains, Method of Particular Solutions.
- Week 8:
Error analysis of MPS. [Still, Barnett, Betcke papers]
Minimax characterization of eigenvalues, Weyl's law (theorem) for
eigenvalue counting asymptotics [GAR] Ch. 11.
Worksheets: eigbnds (baby Moler-Payne proof)
- Week 9:
Low-rank approximation of kernels, Fast Multipole
In-class project presentations.
- Code for low-rank demonstration of 2D Laplace kernel: lowrank.
Project topic ideas list. Final writeups due 9am Wed Dec 10.
Course grades based on 60% HW, 40% project.