Publications
Summary
Generally speaking, I am interested in problems related to deep learning, numerical partial differential equations, computational number theory, and computational electromagnetics. As a graduate student, I developed algorithms to solve diffraction problems in periodic media, as well as methods for calculating zeros of the Riemann zeta function.
In recent years, I’ve immersed myself in applying deep learning to real-world mathematics and industry, actively exploring and experimenting with neural network designs. My aim is to pioneer novel theories and algorithms to benefit computational scientists.
Published in Journal of Scientific Computing, 2024
The scattering of electromagnetic radiation by a layered periodic diffraction grating is an important model in engineering and the sciences. The numerical simulation of this experiment has been widely explored in the literature and we advocate for a novel interfacial method which is perturbative in nature. More specifically, we extend a recently developed High–Order Perturbation of Surfaces/Asymptotic Waveform Evaluation (HOPS/AWE) algorithm to utilize a stabilized numerical scheme which also suggests a rigorous convergence result. An implementation of this algorithm is described, validated, and utilized in a sequence of challenging and physically relevant numerical experiments.
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Published in SIAM Journal on Mathematical Analysis, 2023
The scattering of linear waves by periodic structures is a crucial phenomena in many branches of applied physics and engineering. In this paper we establish rigorous analytic results necessary for the proper numerical analysis of a class of High–Order Perturbation of Surfaces/Asymptotic Waveform Evaluation (HOPS/AWE) methods for #numerically simulating scattering returns from periodic diffraction gratings. More specifically, we prove a theorem on existence and uniqueness of solutions to a system of partial differential equations which model the interaction of linear waves with a periodic two–layer structure. Furthermore, we establish joint analyticity of these solutions with respect to both geometry and frequency perturbations. This result provides hypotheses under which a rigorous numerical analysis could be conducted on our recently developed HOPS/AWE algorithm.
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Published in UIC Electronic Thesis and Dissertation (ETD) System, 2022
This thesis presents rigorous analytical and numerical results necessary for the numerical analysis of a class of High–Order Perturbation of Surfaces/Asymptotic Waveform Evaluation (HOPS/AWE) methods in a laterally periodic two–layer structure. Numerical simulations of scattering returns from periodic diffraction gratings are crucial to a large number of applications in physics and engineering, and the work presented here examines methods for numerically modeling scattering returns from such structures. The strategies presented in this thesis represent the results of our efforts towards the dual goals of 1) Proving a theorem on the existence and uniqueness of solutions to a system of partial differential equations which model the interaction of linear waves in periodic layered media and 2) Developing a numerical algorithm to record scattered energy through a novel interfacial method that is perturbative in nature. The first of our goals is established through classical methods based on the theory of Sobolev spaces and regular perturbation theory. The proof involves several rigorous analyses, and we formulate the scattering problem in terms of Dirichlet–Neumann Operators which are computed using the Transform Field Expansion methodology. A novelty of our approach is the joint analyticity of solutions with respect to both geometry and frequency perturbations. The theory itself is then validated through our second goal which is the development of a joint HOPS/AWE algorithm. For this, we develop a special class of interfacial numerical algorithms that are well–suited to periodic diffraction problems. Our algorithm calculates the Reflectivity Map, $R$, which measures the response (reflected energy) of a periodically corrugated grating structure as a function of its illumination frequency. Moreover, we present a series of challenging and physically relevant numerical experiments to validate the scattering results expected by our algorithm. Forthcoming research will focus on extending the proof of analyticity to additional parameters relevant to the geometry of the structure, increasing the complexity of the structure through generalizing our results to any finite number of layered interfaces, implementing parallel programming techniques to handle multilayered surfaces, and reducing the computational cost of our HOPS/AWE algorithm. The analysis of multilayered periodic structures with numerous perturbation parameters will be an area of substantial interest for practitioners in the electromagnetic and engineering communities.
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Published in University of Michigan Deep Blue System, 2015
This project focuses on exploring different computational methods of the Riemann Zeta function. The zeta function is formally defined as $\zeta(s)$ and the Riemann Hypothesis states that all non-trivial zeros of $\zeta(s)$ lie on the critical line where $\Re(s) = 1/2$. Modern research and advanced algorithms have not been able to disprove the Riemann Hypothesis. More recently, a variation of the Odlyzko-–Schönhage algorithm has verified the Riemann Hypothesis up to $10^{13}$ zeros. This project is composed of two major programs which calculate functional values and zeros of the Riemann zeta function. The first program calculates $\zeta(s)$ for any value of $s$ provided $s\in\mathbb{R}$ through what is known as the Cauchy-Schlömilch transformation. The second program uses the well known Riemann–Siegel formula. Prior to the work done inside the Odlyzko–Schönhage algorithm, this was the method of choice for finding zeros of $\zeta(s)$. The algorithm is heavily dependent on locating sign changes of the Riemann-Siegel $Z$ function in order to find zeros on the critical line. Loosely speaking, one can derive this formula from a direct relationship that was originally found by Siegel which states that $Z(t) = e^{i \theta(t)} \zeta\left(\frac{1}{2}+it\right)$.
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