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README.md

Final Projects

Policy and Schedule

The final project can be either a computational project or a survey paper. Computational projects may be done either solo or in a group of two (a two-person project should have larger scope than a solo). Survey papers must be solo.

Some suggested topics are below. You are of course free to come up with other topics yourself. Please submit a one-page project proposal (via GradeScope) to let me know what your topic will be, and who if anyone you will be working with, by Monday, May 17.

Final project reports will be due on Thursday, June 10.

I expect the final reports to be approximately ten pages, though they may be longer if there are lots of figures or plots and tables of experimental results. If you are doing a computational project, please present your results both as tables and as plots. Generally speaking, decide what conclusions you are going to argue for, and then make plots or diagrams that will make the evidence for your case as clear as possible.

Please submit your report to GradeScope as a single PDF file.

Examples of computational projects:

  • Investigate using the iterative symmetric indefinite linear solvers MINRES and SYMMLQ together with shift-and-invert Lanczos to find Fiedler vectors. (The online templates collection includes C and Matlab code.)

  • Use the BTER software from Sandia (or another graph generator) to create several groups of test graphs, each group having "similar" structure and a wide range of sizes. For each group determine how the Fiedler value scales with n. Can you create a group to match a desired scaling (e.g., O(1/n) or O(sqrt(n))?

  • Use a big parallel computer (or the cloud) to compute Fiedler values and Fiedler vectors for the largest connected component of all the undirected graphs in the SuiteSparse collection (or as many as possible). Save the results for possible inclusion in the collection.

  • Experiment with combinatorial preconditioning.

  • Experiment with a randomized Kaczmarz linear solver.

  • Experiment with a randomized incomplete factorization linear solver.

  • Implement something from an application you know about in any area of science and engineering.

  • Compare both the quality of results and the computational cost of spectral clustering on some test problems, using each of the three matrices L, N, and A.

Examples of survey projects:

  • Any single application area you are interested in.

  • Multigrid approaches to Laplacian linear solvers and eigensolvers.

  • History and experience with Kaczmarz projection methods in non-Laplacian applications.

  • Eigenvalues of random graphs (under various definitions).

  • Random walks on graphs and relationship to Laplacians.