LIST OF SOME MATH 126 PROJECT IDEAS. Barnett 2/15/12
Interpolation & Quadrature (without BIE):
* Work through more advanced examples from Trefethen's ATAP (Approximation
Theory and Approximation Practice), not necessarily using Chebfun.
Could be party expository project.
* Derive quadrature weights for Clenshaw-Curtis, compare this against Gaussian
quadrature, and read Trefethen's paper on this comparison. Use potential
theory to answer the question he poses about charges on the ellipse at the
end.
* Understand and code up Rokhlin scheme for Gaussian nodes in O(N).
Quadrature for BIE:
* Presentation and investigation of quadrature schemes for singular
functions on the real line (Rokhlin, Gimbutas, Alpert, Martinsson),
or for 3D patches (Bruno-Kunyansky).
* Understand and Implement product quadrature
for log-singularities such as the S operator. Young-Martinsson's
scheme for quadrature for Helmholtz equation DLP.
* Investigate how varying the parametrization of a curve affects the
accuracy (convergence rate, etc) in the global periodic trapezoid
rule for the Laplace BVP. Can you devise an (automatic?)
reparametrization that maintains high accuracies even when the curve
approaches itself very close?
* Implement adaptive schemes for a BVP using 16th-order Gaussian panels
which refine when the coefficient decay is not sufficient on a given panel.
* Implement Kress's (Martensen-Kussmaul) spectral quadrature for log
times analytic kernels, use to get spectral convergence in the
DLP+SLP exterior scattering problem.
* Implement Alpert quadrature for SLP+DLP in Helmholtz, check convergence
rates, review uses in literature. [hard, and done in MPSpack].
BIE Applications:
* Code up Dirichlet problem in the exterior of a large number of
smooth closed curves. Use this to compute fluid flow through, or
electrostatic polarization of, this exterior domain, via an external field
of the form u_inc(x,y) = x.
* Fluid dynamics: Stokes flow around multiple smooth 2D obstacles,
spectral accuracy. BONUS: Use force to drive viscous motion and
evolve in time?
* Investigate numerically `creeping waves' or other high-frequency wave
phenomena beyond the geometric optics approximation. (Eg, diffraction).
Do numerical studies vs frequency k, compare to theory
* Investigate the inverse scattering problem, ie, use fitting to far-field data
to iteratively adjust a smooth curve defined by a few Fourier coefficients.
How complex a shape can you solve for, given noise in the far-field data?
See Ocampo thesis. [Done by Peng Peng Yu, 06, was fun]
* Implement Neumann BCs for Laplace or Helmholtz BVP, interior or exterior.
Either use Kress '95 spectral discretization of hypersingular BIE,
which would be interesting to see how the condition number grows with N,
or use right-preconditioning methods of Bruno-Turc or Greengard-Gimbutas
and Kapur-Rokhlin quadrature.
This would open up the following:
* model a point source driving an acoustic horn in 2D.
* other sound-hard scattering problems.
* Sound-soft (unrealistic) acoustic horn design, and optimization of
far-field pattern.
* Solve an applied problem of your choice which requires integral equations.
* Implement boundary integral method for other linear PDEs from heat flow or
fluid dynamics (modified Helmholtz equation, heat equation, Stokes),
review applications in the literature.
* Implement transmission scattering (scattering from a dielectric
material with interior wavenumber differing from the exterior). Use
hypersingular-cancelling layer representations inside and outside
(as in Rokhlin 1983), eventually with a singular quadr scheme such
as Kapur-Rokhlin, Alpert, or Kress. [Getting quadr right is
hard. This is all done in MPSpack so you could use for comparison.]
Use for plasmonics.
* Make concave objects with pockets to trap scatt waves, look for
narrow resonances, match with WKB prediction for lifetimes (using
gaussian cross-section, geom optics description).
[Advanced]
Numerical or Functional Analysis:
* Func analysis project: present and explain the proof of convergence of the
Nystrom method for 2nd-kind Fredholm IEs with cpt operator. (This requires
Anselone's theory of collectively cpt ops, as presented by Kress).
Laplacian Eigenvalue Problems:
* Compute a bunch of Laplacian eigenmodes and eigenvalues via svd of
I-2D vs k. Study their statistics and compare against random matrix
theory predictions (involves a little reading about quantum chaos).
* Study the eigenvalues of I-2D and use their flow to create a fast algorithm
for finding Dirichlet eigenmodes by measuring their rates of angular change.
Background reading of similar method: Tureci-Schwefel, J. Phys. A, 2007.
Evaluation of Potentials:
* Use adaptive Gaussian quadrature on the Nystrom interpolant to make
an interior Laplace BVP solver that evaluates accurately at any
points up to the boundary. See Helsing's work on this, review some
of the literature. Compare against current research of
Barnett-Greengard-O'Neil-Kloeckner-Epstein. [See Nguyen-Barnett poster].
Generalizations of what we've done:
* Hyperbolic geometry: understand and implement eigenmodes or scattering
on the pseudosphere (constant negative curvature), either via BIE on
the sphere, or MPS. Use Aurich-Steiner 1993 as a reference for BIE on
Poincare disc.
* Spherical geometry: solve interior BVPs for Laplace or Helmholtz on
the sphere. Involves changing kernel and understanding new effects.
* Implement anything from the course in a smooth 3D domain,
demonstrating convergence rate and discussing scaling of
computational effort. Use methods from Colton-Kress book for smooth
deformations of the sphere. Or, settle for low-order convergence.
* Handle domains with corners in boundary integral methods:
read about and test some
quadrature rules adapted for corner singularities (eg, Atkinson,
Kress books, Kress 1991 review), and compare against piecewise
(dyadic) Gaussian quadrature in the style of Bremer/Rokhlin/Greengard.
* Code a Fast Multipole Method with one (or more!) levels to evaluate
the self-interaction of N points in R^2 with the Laplace kernel, in
O(N^(4/3)) or better. Note the naive algorithm is O(N^2). (You will
already code a O(N^(3/2)) method in HW, via src-to-multipole
expansion for points falling in boxes). To get this scaling you will
need need translation operators and local expansions.
* Model an acoustic `Helmholtz resonator' (Neumann boundary conditions,
exterior wave scattering for a nearly-closed cavity shape) relevant
to music or architecture, study corrections from the predicted frequency.
Less relevant projects relating to method of particular solutions
(alternative to integral equation methods):
* Present Vekua's theory of PDEs in the complex plane (Henrici review).
[Advanced]
* Compute the logarithmic capacity of the unit square to machine
precision. (Will involve either boundary integral or MPS type
methods, careful consideration of quadrature or basis sets).
[Has been done by Yong Su '09.]
* Try MPS or MFS for wave scattering or interior BVPs, study
convergence, and efficiency, relative to that of boundary integral methods.
* Study the convergence rate of the MPS with a single singular corner,
and relate to complex analytic properties of the solution, based on
T. Betcke's thesis and publications.
* Compute Laplacian eigenmodes on a polyhedron or hyperbolic manifold
by matching value and derivatives on the edges, as in the MPS.