Graduation Semester and Year

Spring 2026

Language

English

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics

Department

Mathematics

First Advisor

Dr. Souvik Roy

Second Advisor

Dr. M. Farooq Wahab

Third Advisor

Dr. Andrzej Korzeniowski

Fourth Advisor

Dr. Gaik Ambartsoumian

Fifth Advisor

Dr. Hristo Kojouharov

Abstract

In this work, we present two mathematical frameworks for signal denoising and cardiovascular risk prediction. In the first part of this dissertation, we use an optimization-based approach for signal denoising in scientific applications. We formulate the denoising problem as the minimization of an objective functional that combines a weighted data-fidelity term accounting for heteroscedastic noise with an L1-regularization term that promotes sparsity in the reconstructed signal. We use the Sequential Quadratic Hamiltonian (SQH) method to solve the optimization framework, which is further combined with Perona–Malik anisotropic diffusion. For signals with significant baseline drift, we apply the BEADS algorithm as a preprocessing step. We prove the existence and uniqueness of the minimizer. The proposed methods are validated on Raman spectroscopy, thermogravimetric (TG), and X-ray diffraction (XRD) data. The results show that our methods effectively reduce noise while preserving peak shapes and integrated signal areas.

In the second part of this dissertation, we present a four-variable ordinary differential equation (ODE) model that describes changes in heart rate, systolic blood pressure, diastolic blood pressure, and sleep recovery for cardiovascular risk prediction. We perform preprocessing, parameter estimation, and sensitivity analysis. We then develop two composite risk scores based on mean arterial pressure (MAP) and heart rate (HR) dynamics and evaluate them on eight patients. The results show that both MAP and HR scores increase with disease severity and match the clinical classifications for all patients. These results demonstrate that a dynamic ODE-based framework can represent physiological relationships and may improve personalized cardiovascular risk assessment.

The main mathematical contributions and novelty of this work are organized into two parts. For the signal denoising part, we propose a new optimization framework that couples L1-sparsity regularization with anisotropic diffusion filtering, and we design an SQH-based scheme that combines them through different sequential strategies, with the approach tested on several real noisy signals. For the cardiovascular risk prediction part, we develop the four-variable ODE system to describe how the physiological variables influence one another over time, set up a preprocessing framework that converts raw high-frequency recordings into one-day averaged profiles for time-series analysis, fit the model on real test cases, and derive the two risk score formulas based on heart rate (HR) and mean arterial pressure (MAP).

Keywords

Optimization-based signal processing, Signal denoising, Optimization framework, L1-sparsity regularization, Sequential Quadratic Hamiltonian (SQH) method, Perona–Malik anisotropic diffusion, Heteroscedastic noise, BEADS algorithm, Baseline correction, Raman spectroscopy, Thermogravimetric analysis (TGA), X-ray diffraction (XRD), Cardiovascular risk prediction, Ordinary Differential Equations (ODEs), Mean arterial pressure (MAP), Heart rate (HR) dynamics, Sensitivity analysis, Latin Hypercube Sampling–Partial Rank Correlation Coefficient (LHS-PRCC), Parameter estimation, Composite risk scores

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Available for download on Wednesday, May 10, 2028

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