CURRENT LIST OF Math116 PROJECT IDEAS - please add! (let me know...)


* Implement anything from course in 3D, test convergence, make nice
  slice plots

* Corners in BIE: guassian quadrature rules on intervals meeting at
  corner.  study non-compactness, form of density there. Use latter
  part of Kress 1991 review, graded mesh. Compare against naive
  uniform grid which ignores corners, also piecewise gaussian
  quadrature, Rokhlin/Greengard.

* Try MPS or MFS for wave scattering or interior BVPs, study
  convergence and corners.

* Implement optical/acoustic dielectric scattering (2 materials with
  different wavenumbers, match layer representations inside and
  outside, Rokhlin 1983).

* Dielectric bound modes.

* Matt's piecewise homogeneous coefficient Laplace equation (in a
  box?): fwd problem for current affected by a circular change in
  coeff.

* Compare BIE for interior eigenvalue problem to MPS.

* Compare geometric optics against wave scatt, study some
  geometric wavenumber-independent BIE variants (Chandler-Wilde).

* 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).

* Measure scar strength for a single UPO of known Lyapunov exponent,
  vary it.

* Inverse problem: use map from incident to far field pattern to
  invert for shape, using a basic few-param representation of shape
  (like Yu Chen's PhD student).

* Hyperbolic geometry: understand and implement eigenmodes or scattering
  on the pseudosphere.

* Model an acoustic `Helmholtz resonator' (Neumann BC, exterior wave
  scattering with reentrant shape).

[* Fast Multipole Method (FMM) - probably not realistic]

(Yu Chen's 2-object separation factorization).