Math 116: Resources
- FALL 2008
Numerical analysis and numerical methods
- Some
disasters attributable to bad numerical computing.
- Application examples. Laplace BVP:
SEM electron gun
design,
molecular
electrostatic energy surface.
Helmholtz BVP:
radar
scattering from aircraft,
photonic
crystal guiding calculation,
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).
- J. P. Boyd, "Fourier and Chebyshev Spectral Methods" (2001, Dover),
available here
-
L. N. Trefethen, Spectral Methods in Matlab (SIAM, 2000).
Beautiful key ideas of differentiation, interpolation, quadrature
on grids, with exponential convergence. Elegant 1-page Matlab codes.
-
R. Kress, Numerical Analysis, Graduate Texts in Mathematics, no.
181
(Springer, 1998). Thorough introduction, with Kress'
usual clear but dense style of building up everything with proofs.
Good for interpolation and quadrature.
-
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
- 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
(to appear, SIAM J. Numer. Anal.) on MPS for eigenvalue problems.
- Fast Multipole Methods: Gumerov course
notes,
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
- 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.
- 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.
Quantum chaos, semiclassical analysis
Coding
Web authoring
- Get yourself a website account
hosted at Dartmouth
if you don't already have one.
- Dave Raggett's simple
HTML guide is all you could ever need.
- Don't forget to Reload the page in your browser to check updates you
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.
