Research
I am mostly interested in physically motivated problems in applied mathematics. Some areas I am currently working on include active cloaking strategies, accelerated optimization algorithms, and inverse problems in imaging.
Active Thermal Cloaking
In general, the goal of cloaking is to make an object indistinguishable from the medium that contains it. I am interested in "active" cloaking, which achieves this via specially designed sources. In practice, these sources may be realized by the use of thermoelectric pumps. A benefit of active cloaking is it also allows for mimicking, making an object appear as a different object from the perspective of thermal measurements. Active cloaking is also attractive in the case of thermal cloaking since it is tunable, allowing for cloaking in the transient domain. Furthermore, a "passive" cloak (a cloak that achieves cloaking by surrounding the object with an exotic material) would need many layers.
Mathematically, we have demonstrated a strategy to realize active thermal cloaking. The idea is largely based on Green identities, which allow the reproduction of "nice enough" solutions to homogenous PDEs inside (or outside) bounded domains by only controlling the boundary of the domain. A portion of our contribution is to give a sufficient condition for reproducing solutions in the exterior of bounded domains. This work is a collaboration with Sébastien Guenneau, Fernando Guevara Vasquez, and Maxence Cassier. So far, this collaboration has resulted in this publication which appears in the Proceedings of the Royal Society A. We have been working on relaxing an assumption necessary for this Green identity cloak namely, that the sources need to surround the object. This led to some additional mathematical results that pertain to other physical applications beyond temperature that may be of interest. This work was recently published in the Philosophical Transactions of the Royal Society A.
Thermal noise imaging
Hybrid inverse problems use coupled physics to relate boundary measurements to some unknown internal parameters of a body. These problems can be split into two steps: (1) relating measurements to an internal functional and (2) recovering information about the unknown parameters from measurements of the internal functional. We propose a novel kind of hybrid inverse problem for recovering the conductivity of a body by inducing currents using thermal noise.
This work is collaboration with Fernando Guevara Vasquez and China Mauck, a preprint is available here.
Optimization Algorithms
I recently took a course with Bao Wang titled "The Mathematics of Data Science." This has led to a collaboration between Bao, Fernando, and myself where we are working on some new results for optimization algorithms (ex. gradient descent). There are two directions that I am interested in pertaining to optimization. The first is finding conditions that give certain theoretical guarantees for the convergence of optimization algorithms (as well as their continuous counterparts). The second is improving models for particular applications to give better (or faster) convergence.