I have more ideas than time, and am happy to share! If you are interested in these projects, email me and we'll talk about it. If you have a project idea of your own, we can talk about that, too.
Continuation Methods for Optimization on Surrogates
Surrogate methods for global optimization of expensive functions alternately sample the expensive function, update a model (a surrogate to the expensive function), and use the model to determine where to sample next. This last step frequently involves an inner global optimization problem that may be somewhat expensive in its own right. In this project, we propose to use a parameter continuation approach to solve this global optimization on the surrogate through a sequence of optimizations on smoother and simpler surrogates.
You should ideally have a course in numerical computing to take on this project.
Preconditioning Radial Basis Functions with Clustered Centers
Radial basis function interpolation is a standard method of interpolating data in more than one space dimension. It has many attractive features, but the problem can become unstable when posed in the normal basis, particularly for very smooth basis functions or when the basis function centers are clustered. Recently-developed methods address the former case, but the latter is more of a concern when RBF interpolants are used for optimization, as is the case in some of my projects. I have an idea for a formally equivalent method that gets around this stability issue by a preconditioning strategy, and would like help implementing it.
You should have a course in numerical computing and some mathematical maturity to take on this project.
Evans Functions and Nonlinear Eigenvalue Problems
The Evans function is a Wronskian-type construction that plays a role similar to the characteristic polynomial in the unbounded-domain spectral problems that arise in the analysis of 1D solitary wave stability. This construction has been used computationally to good effect for some problems; however, some preliminary results (and analogy to more conventional eigenvalue problems) suggests that this should be less numerically stable than an alternate method based on a nonlinear eigenvalue computation. I have coded variants of this before, and now have some nice theory to back up my favored algorithm. But I would love to have help in doing a systematic comparison between my favored algorithm and methods based on the Evans function. Ideally, I would also love to have a numerical error analysis of both methods, too.
You should have a course in differential equations and a course in linear algebra to tackle this project, and ideally a course in analysis and a course in numerics.