# Projects

This is a collection of course projects and the like relatively tangential to my primary research. Enjoy.

## Graduate

### Multifrontal solver

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.

### Recycled LSMR

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.

### Complex Krylov methods

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).

### Optimal Solar Sailing

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.

## Undergraduate

### Photoluminescence Characterization of Solar Cells

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.

### Inverse Heat Conduction

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;

### Mathematical Contest in Modeling

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.