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.