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.
The Compact Muon Solenoid (CMS) experiment is a large particle physics detector built on the Large Hadron Collidor at CERN. At this point, so many users are working on this experiment that there is a strong need not to mine both experimental data and also data about users and user activity. CS MEng students may be interested in some data mining projects out of Cornell physics related to these projects; these involve some very interesting systems problems as well as potentially exercising ideas in deep learning. If interested, please contact Valentin Kuznetsov or Peter Wittich for further details – but you can likely talk me into being your CS field representative.
Web Dashboards for Expensive Optimization
For an ongoing project on software for optimizing expensive functions, I would like to have a web dashboard similar to the Dask.distributed web interface, probably built using the Bokeh visualization library.
Fast DNN+LR Projections
For a topic mining application (joint work with David Mimno), I am looking for a way to quickly project sparse matrices to the closest (low rank) doubly non-negative matrix – that is, a matrix that is both positive definite and elementwise non-negative. The catch is that the algorithm should ideally never form a full dense representation. I have some ideas about how to do this, though they are not completely fleshed out.
For this project, you should probably have a good linear algebra class and prior exposure to gradient-based optimization methods. And you’ll need to remember your multivariate calculus.
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 maturirity to take on this project.
Resonances in Quantum Graphs
A quantum graph is a graph with differential equations on the edges and compatibility conditions at the nodes. I am interested in the case of infinite quantum graphs, where a local irregular region (a scatterer) is surrounded by a regular lattice extending off to infinity. I want to build a tool that will compute the spectral properties of such an object – bound states and resonance poles – given a description of the far-field lattice structure and the local irregular region.
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. But if you don’t have these courses and think the project sounds fun anyhow, do come talk to me!
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.
Model Reduction of Buckling Lattices
Three-dimensional printing lets us make elaborately microstructured materials, but our understanding of how to model these microstructures is still young. The mechanics of microtruss structures can be analyzed by mostly-standard methods when the strains are small and the truss structure is periodic; but the picture becomes more complex even at small strains when the truss is not quite periodic, and it becomes very complicated at large strains when the individual elements can buckle. I have some ideas about how to use a type of model reduction to simulate the behavior of such structures quickly, and would like to try them out – but given the background required, and given that I don’t have a code to do it at hand, I would be delighted to have some help first putting together a code to do the analysis in the simple case.
You should have a course in numerical computing to tackle this project. Prior exposure to finite element analysis or structural analysis will also be helpful.
Response Surfaces for Structure Prediction
GASP is a Genetic Algorithm for Structure and Phase Prediction that “predicts the structure and composition of stable and metastable phases of crystals, molecules, atomic clusters and defects from first-principles.” An alternative to using genetic algorithms to understand the energy landscape is to use surrogate methods (aka response surface methods), but initial experiences suggest that this won’t work well unless the response surface is carefully chosen. I’d like to try a response surface approach that incorporates some basic chemical knowledge (e.g. the knowledge that potential energy blows up if you try to put two atoms en the same place) in order to get a better result.
This probably isn’t a very good project for most CS MEng students, but if you’re coming from a background where building an optimizer on top of a DFT code sounds fun, come talk to me!