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Published in *arXiv preprint*, 2020

An investigation into the dynamics of roll solutions at the zigzag boundary of the planar Swift-Hohenberg equation. Done as a part of an REU at Ohio University during the summer of 2019.

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** Published:**

**Abstract**: Given the inability to directly observe the conditions of a fault line, inversion of parameters describing them has been a subject of practical interest for the past couple of decades. To resolve this under a linear elasticity forward model, we consider Bayesian inference in the infinite-dimensional setting given some surface displacement measurements, resulting in a posterior distribution characterizing the initial fault displacement. We employ adjoint-based gradient computation in order to resolve the underlying partial differential equation constrained optimization problem and take care to leverage both dimensionality reductions in the parameter space and the low-rank nature of the resulting posterior covariance, owing to sparse measurements locations, to do said computation in a scalable manner.

** Published:**

**Abstract**: We consider sensitivity analysis of Bayesian inverse problems with respect to modeling uncertainties. To this end, we consider sensitivity analysis of the information gain, as measured by the Kullback-Leibler divergence from the posterior to the prior. This choice provides a principled approach that leverages key structures within the Bayesian inverse problem. Also, the information gain reduces to the Bayesian D-optimal design criterion in the case of linear Gaussian inverse problems. However, the derivatives of the information gain are not simple to compute, nor are finite differences always possible let alone scalable. To solve half the puzzle, in this talk we present the method of computing eigenvalue sensitivities for implicitly defined linear operators appearing in PDE-constrained optimization problems. Specifically, we consider eigenvalue sensitivities of the so-called data misfit Hessian and its preconditioned counterpart. We start with simple examples and work our way up to the expressions in the information gain. Our approach relies on adjoint based methods for gradient and Hessian computation. The resulting expressions for the sensitivities will be exact and can be computed in a scalable manner.

** Published:**

**Abstract**: We consider sensitivity analysis of Bayesian linear inverse problems with respect to modeling uncertainties. To this end, we consider sensitivity analysis of the information gain, as measured by the Kullbackâ€”Leibler divergence from the posterior to the prior. This choice provides a principled approach that leverages key structures within the Bayesian inverse problem. Also, the information gain admits a closed-form expression in the case of linear Gaussian inverse problems. The derivatives of the information gain are extremely challenging to compute. To address this challenge, we present accurate and efficient methods that combine eigenvalue sensitivities and hyper-differential sensitivity analysis that take advantage of adjoint based gradient and Hessian computation. This results in a computational approach whose cost, in number of PDE solves, does not grow upon mesh refinement. These results are presented in an application-driven model problem, considering a simplified earthquake model to infer fault slip from surface measurements.

Graduate course, *New York University, Department of Computer Science*, 2017

In the fall semester for both 2017 and 2018, I was a tutor and TA for the fundamental algorithms course, taught by Alan Siegel, at New York University.

Undergraduate course, *New York University, Department of Computer Science*, 2018

In the spring semester for both 2018 and 2019, I was a tutor and TA for the basic algorithms course, taught by Alan Siegel, at New York University.

Undergraduate course, *New York University, Department of Computer Science*, 2020

In the spring semester of 2020, I worked as an undergraduate tutor for the numerical computing course, taught by Margaret Wright.

Undergraduate course, *North Carolina State University, Department of Mathematics*, 2020

In the fall semester of 2020, I taught a recitation section and graded for the Elements of Calculus Course.

Undergraduate course, *North Carolina State University, Department of Mathematics*, 2021

In the spring semester of 2021, I taught a recitation section and graded for the course Calculus II taught by Patrick Sprenger.

Undergraduate course, *North Carolina State University, Department of Mathematics*, 2021

In the first 6-week summer semester of 2021, I taught all recitation sections and graded for the course Calculus II taught by Ashley Tharp.