Morning Session:11:35 -12:25 pm

Recent Advances in Statistics and Applied Math

1. Hannah Powell: hpowell9@charlotte.edu  (in-person)

Title: Reduced Models for Dynamical Systems

2. Phuong (Mai) Nguyen: pnguye45@uncc.edu (in-person)

Title: Inverse scattering without phase: Carleman convexification and phase retrieval via the Wentzel–Kramers–Brillouin approximation

3. Kody Angell and Cove Pesterfield: kangell1@charlotte.edu, bpester1@charlotte.edu (poster in-person, continue the discussion in lunch)

Title: Oh NAR! The Importance of Exogenous Inputs in Autoregressive Neural Networks

1 Title: Reduced Models for Dynamical Systems

Abstract: Modeling resilience in natural systems is a major subject in the field of ecology. Studying this resilience helps ecologists better understand these systems as a whole, and possibly prevent population collapse. One method of modeling resilience of a species is to model how its population changes based on one variable parameter that represents some changing environmental condition. This model, while computationally simple, does not capture the complexities of real systems. Models of ordinary differential equations have been developed in response to this, based on a system of interacting species. These models are detailed but are also more computationally complex, and their behavior is not easy to predict. Researchers developed a reduced model that characterizes the complexities of the multidimensional system of equations as one ordinary differential equation based on a single parameter. This work aims to extend work already done in previous research by applying similar techniques to systems of stochastic differential equations and ultimately to produce a reduced model that is able to predict the behavior of a stochastic system.

2. Title: Inverse scattering without phase: Carleman convexification and phase retrieval via the Wentzel–Kramers–Brillouin approximation

Abstract: This study addresses the challenging and interesting inverse problem of reconstructing the spatially varying dielectric constant of a medium from phaseless backscattering measurements generated by single-point illumination. The underlying mathematical model is governed by the three-dimensional Helmholtz equation, and the available data consist solely of the magnitude of the scattered wave field. To address the nonlinearity and severe ill-posedness of this phaseless inverse scattering problem, we introduce a robust, globally convergent numerical framework combining several key regularization strategies. Our method first employs a phase retrieval step based on the Wentzel--Kramers--Brillouin (WKB) ansatz, where the lost phase information is reconstructed by solving a nonlinear optimization problem. Subsequently, we implement a Fourier-based dimension reduction technique, transforming the original problem into a more stable system of elliptic equations with Cauchy boundary conditions. To solve this resulting system reliably, we apply the Carleman convexification approach, constructing a strictly convex weighted cost functional whose global minimizer provides an accurate approximation of the true solution. Numerical simulations using synthetic data with high noise levels demonstrate the effectiveness and robustness of the proposed method, confirming its capability to accurately recover both the geometric location and contrast of hidden scatterers.

3. Poster Presentation Title: Oh NAR! The Importance of Exogenous Inputs in Autoregressive Neural Networks

Abstract: Laser damage induced at blue wavelengths in pigmented biological tissues results in complex interactions between photothermal and photochemical mechanisms. Traditional modeling approaches rely on the Arrhenius equation or zero-order kinetic models which fail to accurately capture experimental data due to their inability to represent the specific nature of biological damage processes. These conventional models cannot adequately account for complex interactions between multiple damage mechanisms occurring simultaneously. Here we applied a nonlinear autoregressive model with exogenous inputs (NARX) to characterize photochemical and photothermal damage in biological tissues under controlled laser light exposure and varying wavelengths. Different kernel types were explored and the optimal result with the lowest root mean squared error (RMSE) was selected. The NARX framework, adapted from Wang et. al., (2022) incorporated memory effects, nonlinear dynamics,and external input variables, enabling better representation of complex biological responses to laser irradiation and capturing both immediate and delayed damage responses. The work incorporates a  range of different parameters, enabling comprehensive spectral analysis of cellular damage mechanisms. This approach adapted to the complex differences between damage induced by different wavelengths. The resulting computational framework offered a powerful tool for evaluating phototherapy efficacy with important implications for understanding the interplay between damage types.