In Winter 2015 as a final project for my advanced topics in numerical linear algebra course with Jack Poulson, I implemented a general multifrontal solver with nested-dissection ordering in C++ for sparse linear systems.
For my final project in a course on the top 10 algorithms of the 20th century, I implemented a recycled-subspace variant of the LSMR algorithm for solving linear systems. Unfortunately, recycling did not seem to make much of a difference for my test problems, but I believe there could still be something here.
For our final project in large-scale numerical optimization with Michael Saunders (Spring 2013), Austin Benson and I created complex implementations in FORTRAN of the popular LSQR and LSMR algorithms for solving linear systems. These are based heavily off of the original real-arithmetic implementations from the Stanford Systems Optimization Laboratory and can be found here (LSQR) and here (LSMR).
The purpose of this project, from my Winter 2013 numerical optimization course with Walter Murray, is to assess the optimal control and minimum time necessary to bring a spacecraft employing simple solar sail technologies from the Earth’s orbit around the sun to the orbit of Mars and back again. The problem is modified as a two-dimensional system in polar coordinates about the sun and Newton’s laws are assumed to be sufficient to model the required kinematics. To format the minimum-time kinematics problem as a standard optimization problem, we solve a sequence of sub-problems where the final time is held fixed and a feasible trajectory is calculated. We consider the objective function to be the squared difference between the final point of the trajectory and the desired final state, and use a penalty function to account for the nonlinear constraints given by Newton’s laws. The optimal control is subject to box constraints. Solving this optimization problem is done with an active-set BFGS method employing a line-search satisfying the strong Wolfe conditions.
For our senior design project, Emir Salih Magden and I worked with Professor Tom Vandervelde of Tufts Renewable Energy and Applied Photonics Labs to create a product to characterize the material quality of a semiconductor, for use in a research environment. Using computer-controlled linear actuators, a tunable band-pass filter, lenses, mirrors, and an optical sensor, we created a device controlled by LabView to move different points of a semiconductor sample into the path of a laser and measure the resulting photoluminescence.
Inverting the heat equation is a problem of great interest in the sciences and engineering, particularly for modeling and monitoring applications. For my final project for my class in inverse problems at Tufts, I looked at the heat equation defined on some domain containing a point heat source at a known location. Assuming the magnitude of the heat source to be an unknown function of time and given noisy transient measurements at other points in the domain, I looked into the use of the conjugate gradient method to solve the adjoint problem and recover the heat source magnitude function.
UPDATE: due to popular demand, here is the piece of Matlab code not included in the write-up.
MNtoI = @(m,n,M,N) (n-1)*M+m;
From left to right: V. Minden, L. Clegg, D. Brady, S. MacLachlan.
In 2010, I competed for the first time in the COMAP Mathematical Contest in Modeling, a contest in which teams of undergraduates are given four days to model, simulate, and write a report about a real-world problem revealed to teams on the first day of the competition. With Dan Brady and Liam Clegg, I created a discrete model for quantitative criminology applications — given a set of spatio-temporal points representing crimes in a spree, predict the times and locations of future crimes. Our entry, From Kills to Kilometers, won the designation of “Outstanding” as well as an external prize from INFORMS.
In 2011, I competed again with a slightly different team. With Stephen Bidwell and Liam Clegg, I developed a model for generating snowboard halfpipe designs, with the goal of maximizing attainable vertical air. Our entry, Pipe Dreams, was awarded Honorable Mention.