Math 126: Resources
- WINTER 2012
Numerical analysis and numerical methods
- Some
disasters attributable to bad numerical computing.
- Application examples. Laplace BVP:
SEM electron gun
design,
molecular
electrostatic energy surface
(linearized Poisson-Boltzmann equation is modified Helmholtz),
Stokes BVP coupled to surface motion:
fluid dynamic simulation of millions of vesicles.
Helmholtz BVP:
radar
scattering from aircraft,
electron micrograph of photonic crystal.
Laplacian eigenmodes:
car interior
acoustic resonance,
quantum
chaos.
- Intro to floating-point
number system by Cleve Moler (founder of Mathworks).
- Numerical Analysis
Digest email discussion group archive.
- Runge phenomenon demo applet from Scott Sarra at Marshall.
- J. P. Boyd, "Fourier and Chebyshev Spectral Methods" (2001, Dover),
available here
- D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd Edition (Springer, 1998). [CK]
This has a good summary of scattering theory, potential theory, and
numerical methods for scattering.
-
W. H. Press, S. A. Teukolsky, W. H. Vetterling, and B. P. Flannery,
Numerical
Recipes in C, 2nd Edition,
available here as PDF files online, also worth buying for your shelf.
Good
all-round general overview of numerical methods, with lots of practical
tips,
good intuitive explanations, aimed at users. Not very rigorous,
up-to-date or complete on PDEs (sometimes this book annoys
numerical analysists, but physicists like it).
- Anna-Karin Tornberg's notes on Gaussian quadrature
- Kress '91
paper on Helmholtz scattering, with singular product Martensen-Kussmaul style quadratures described to get spectral accuracy for D + i.eta.S in 2D.
- Marcos Capistran Ocampo's thesis on inverse 2D
obstactle scattering, with Yu Chen (NYU, 2003).
- G. Still's paper
on Moler-Payne theorem and generalizations, Numer. Math. (1988).
- Time Betcke's thesis (Oxford,
2005), a great tutorial, with many new results, on the MPS for
eigenvalue problems.
- My paper with Timo on MPS for interior BVP, using fundamental
solutions basis functions: MFS
paper.
My preprint (SIAM J. Numer. Anal.) on MPS for eigenvalue problems.
- My paper with Andrew Hassell on fast computation of eigenmodes using BIE and the Neumann-to-Dirichlet map.
- Fast Multipole Methods:
-
A Short
Course on Fast Multipole Methods by Rick Beatson and Leslie Greengard
-
P.-G. Martinsson's course notes on
fast algorithms, as of March 2011.
-
Gumerov course
notes
-
A Fast Algorithm for Particle Simulations, L. Greengard and V. Rokhlin, J. Comput. Phys. 135, 280-292 (1987). The O(N) algorithm. Has proofs of multipole truncation errors for 2D Laplace kernel. Errata: Equation (2.10) has a divide by zero instead of
by l. Also note that their monopole convention is log z not log 1/z, which changes the sign of the last term in (2.10).
-
A Fast Adaptive Multipole Algorithm for Particle Simulations
by J. Carrier, L. Greengard, and V. Rokhlin, SIAM J. Sci. Stat. Comput. 9, 669 (1988). Quad-tree goes to variable depth while still provably O(N).
- Demmel's course notes on Barnes-Hut treecode, from 1996.
-
L. Ying et al, J. Comput. Phys. 196, 591-626 (2004) has
a nice review of FMM including in App. B the translation operators
for 2D Laplace kernel
PDEs, integral equations, and analysis
- Geoffrey Hemion's notes
on complex analysis.
- Paul Garabedian, Partial Differential Equations (1964),
is `functional analysis-free', simple and clear,
has integral equations, compactness, minimax eigenvalues, Weyl's Law
proof.
- I. Stakgold, Boundary value problems of mathematical physics,
Vols. 1 & 2
(New York, Macmillan, 1967-68; SIAM republished corrected editions
under Classics in Appl. Math #29, 2000). Berry/Cook seems to have 3
copies of the 1967 editions. Great general introduction to
PDEs, Greens functions, spectral theory,
Schwartz distribution theory, potential theory,
scattering theory, applications.
He also wrote the relevant
Green's Functions and Boundary Value Problems,
2nd Edition (Wiley, 1998).
- F. B. Hildebrand, Methods of Applied Mathematics, 2nd edition
(1965). Although old-looking, Chapter 3 on integral equations is a very clear
introduction.
- My Math 46
Introduction
to Applied Math course, good for introduction to integral equations and
Fourier transforms. Also see Logan course book used.
- R. Kress, Linear Integtral Equations, 2nd Edition
(Applied Mathematical Sciences vol. 82, Springer-Verlag, 1999).
Rather formal functional-analytic
background, potential theory, integral operators, numerical methods,
more detail than we'll need. Excellent. On reserve at Berry/Cook
Library.
- D. Colton and R. Kress,
Inverse Acoustic and Electromagnetic Scattering Theory, 2nd
Edition,
(Springer, 1998), is a good summary of (`forward') scattering theory
and numerical methods.
- D. Colton and R. Kress,
Integral Equation Methods in Scattering Theory (Wiley, 1983).
Classic on boundary integral equations, formal style which proves
everything from the ground up. Berry/Cook has 1 copy.
-
Scattering theory overview by Tilo Arens.
-
Rainer Kress' forward and inverse scattering analysis lecture notes:
go to Teil 1-3 under Inverse Scattering.
- Volume of
d-dim ball, for Lecture 18.
Coding tips
- Always write on paper what you want to achieve, and then write out a pseduocode, before you start typing.
- Try to put all user-adjustable parameters once at the top of your
program (script). Make everything work from these parameters.
- Write brief comments to explain what non-obvious lines do.
- Break down tasks which are repeated into subroutines (in MATLAB
these are called functions). Write out the interface (inputs and
outputs) to your function on pencil and paper before you start to code it up.
Document your function as you go.
- For every function, write a test routine that goes along with it and
verifies it does what it is supposed to do, on at least one known
test case.
LaTeX
This is the typesetting package essentially all mathematicians use.
You will use it for projects, and, I hope, homeworks.
- Here is our department's LaTeX resources.
- If you have a UNIX account (you probably do as a grad student; ask
your sysadmin) or linux OS
then you can use a standard text editor and the latex command.
If you have Mac OSX or Windows you need to install a LaTeX
distribution, as described here.
- Please let me know if you are stuck with installing LaTeX.
Sarunas Burdulis (our sysadmin, behing the math office) may also be able to help better than I.
- Once you have it installed,
here are some
simple sample files that you can edit for your homework.
They produce PS which you can convert to a PDF file.
- Here is a quick guide to all the math symbols and brackets, etc. You will need to
\usepackage{amssymb,amsmath} to get some of these symbols.
- Beamer is a great latex package in which to make slides for your talks.
(Although I still use Prosper).
Here's my toy files: a.tex which uses EPS figure
fig.eps. It produces this PDF.
Web authoring
- Get yourself a website account
hosted at Dartmouth if you don't already have one
(you probably do if you're a grad student; ask your department sysadmin).
- Dave Raggett's simple
HTML guide is all you could ever need.
- Don't forget to Reload the page in your browser to check that the
updates you have made worked.
- Here
is the simplest webpage I ever made (`View Page Source' on your
browser).
It contains text and one link; you could modify it and replace
the address in the
link by href="figure1.ps" if
figure1.ps is a file in the
same directory, for instance a figure file printed to file from Matlab.
