**Rewrite the code for parabolic PDEs to use multigrid to solve the elliptic problems and generally clean up the code.**This is one-person project.**Solve an image segmentation problem.**Use 1 and 2 to compute solutions to "Threshold dynamics for the piecewise constant Mumford-Shah functional", by Selim Esedoglu and Yen-Hsi Richard Tsai,*J. Comput. Phys.***211**(2006), pp. 367--384. This is a two-person project.**Apply domain decomposition techniques.**Block the matrix operator so that each triangle in the coarse triangulation leads to its own local operator. The object of this project is speed, not functionality---you should be able to cut the solution time by at least 1/3, and perhaps parallelize it. It's a one- or two-person project, depending on how much you want to achieve.**Add a reaction term to the parabolic code.**There is code in`mmoc.scm`to solve a convection-diffusion equation using the "modified method of characteristics" of Douglas and Russell; if you're interested in this kind of stuff you can add a reaction term (modelling chemistry, say) to this code. (Note added in later: the code in`mmoc.scm`doesn't work, fixing it is a project.)**Write time-stepping code for second-order hyperbolic problems.**Use either the time-stepping and initial data of Chapter 13 of Larsson and Thomee or Section 9.11 of Claes Johnson's "Numerical Solution of Partial Differential Equations by the Finite Element Method" with alpha=1/2 and gamma=1/2. This is a one-person project.**Exploit symmetry in matrix representations.**The`Sparse-matrix`and`Sparse-matrix-operator`classes exploit the sparsesness of a matrix`A`, but not its symmetry. Define two new classes`Symmetric-parse-matrix`and`Symmetric-sparse-matrix-operator`that exploit the symmetry of`A`by calculating and storing and using only the lower-triangular elements. See how much faster you can make the code.**Write code to solve the "obstacle problem".**Use the Douglas-Rachford iteration method as described in the paper*Splitting algorithms for the sum of two nonlinear operators*by P. L. Lions and B. Mercier, SIAM J. Numerical Analysis, (16) 1979, pages 964-979. A one-person project. (There used to be code for the obstacle problem in the system, but it was nonsense and I removed it.)**Finite difference schemes.**This is a multi-step project. It would need to be coordinated among several people. Among the subprojects:- Find a way to describe the domains you want to handle; probably just rectangles or unions of rectangles. Discretize these domains; refine the discretizations.
- Calculate the finite difference spaces; build the matrix multiplication operator based on the coefficients.
- Build injection and projection operators for multigrid applied to the finite difference problems.

**Write code to solve the linear, time-dependent Schrödinger equation.**This is a one-person project. This problem is solved several orders of magnitude faster using multigrid rather than conjugate gradient to solve the elliptic problems that arise in Crank-Nicolson time stepping, so you'll need to use multigrid for this project. See this (messy) description of time-stepping.**Parallelize the code.**There is wrapper code to use MPI with Gambit, and one can block the matrix multiplications so that computations on all coarse triangles are done in parallel. A two-person project.**Implement Dirichlet boundary conditions.**Just for u=0 on the boundary of the domain. A one-person project.**Note:**Not an appropriate project, it's already coded.**Implement periodic boundary conditions on a square.**A one-person project.