2024 Faculty and Student Summer Talks

In this paper, we apply estimated characteristic functions for censored two-sample location-scale model checks. The proposed test relies heavily on Stute and Wang's Kaplan-Meier integrals for the characteristic function as well as the plug-ins for means and variances.  As would be expected with plug-ins, however, the test is not asymptotically distribution free. Bootstrap is employed to obtain critical values needed for the testing. Numerical studies validate the proposed testing method.

 

In this talk, we present a derivation to describe the dynamics of a thin oil film covering a quasi-static water drop both under the excitement of a MHz-frequency SAW propagating through the underlying substrate. It has been shown in experiments that if the displacements caused by such a wave are in a given range (low to moderate values), then the oil film (on the order of hundreds of micrometers) will be translated along the direction of propagation while the water drop will slightly deform, however, remain stationary (with fixed contact lines). Beginning from the assumption that both phases individually satisfy the Stokes flow equations, we are able to follow the general outline to derive the thin-film equation to get a modified version of that for both phases. In this way, we reduce the problem from solving for velocities and pressures of both phases to just solving fourth-order PDEs for the height profiles of the two phases. The talk concludes by looking at numerical results of the case of a rigid obstacle with a brief comparison to experiments.

 

Numerical stability is a critical property for a time-integration scheme.  In the context of Runge-Kutta methods applied to stiff differential equations, A-stability is one of the most basic and practically important notions of stability. Dating back to the work of Dahlquist, it has been known that A-stability is equivalent to the Runge-Kutta stability function satisfying a particular convex feasibility problem. Specifically, up to a transformation, the stability function lies in the convex cone of \emph{positive functions}. In recent years, sum-of-squares optimization and semidefinite programming have become valuable tools in developing rigorous certificates of stability in dynamical systems. Therefore, it is natural to employ these convex optimization tools for the purpose of rigorously certifying A- and A(α)-stability in Runge-Kutta methods.

Two distinct convex feasibility problems defined by linear matrix inequalities are introduced. The first approach employs sum-of-squares programming applied to the Runge-Kutta $E$-polynomial, making it applicable to both A- and A(α)-stability. The second approach refines the algebraic conditions for A-stability, as developed by Cooper, Scherer, Türke, and Wendler (CSTW), to incorporate the Runge-Kutta order conditions. The theoretical enhancement of the algebraic conditions facilitates the practical application of the refined conditions for certifying A-stability within a computational framework.

The E-polynomial and CSTW methodologies are utilized to obtain rigorous stability certificates for several implicit Runge-Kutta schemes proposed in the literature. Specific attention is given to certifying the implicit Runge-Kutta schemes utilized in the SUite of Nonlinear and DIfferential/ALgebraic equation Solvers (SUNDIALS).

Membrane filtration is an important and ubiquitous process in industrial applications, and there is a growing body of mathematical models that capture this complex process. Previous theoretical work models the internal structure of membrane filters as a network of cylindrical pores whose radii are drawn from a uniform distribution, with fouling modeled as an adsorption process; i.e. the gradual accretion of fouling particles on the inner walls of the pores. Simulation-based approaches are used to measure membrane filter performance, using metrics such as total throughput and accumulated foulant concentration. In the present work, we investigate the correlation between the performance of these networks and their topological properties, in order to discover optimal pore topologies for membrane filter design. We use persistent homology as our principal tool for quantifying topological features, where the radii of a network’s pores are represented by a collection of two-dimensional points known as a persistence diagram. The data encoded in these persistence diagrams are then statistically correlated with the performance metrics, particularly with total throughput. Favorably strong correlation between total throughput and filter topology will be presented.

 

Neurotransmitter release is caused by the entry of calcium ions into the cell in response to a depolarizing pulse ("action potential"), and their subsequent binding to calcium sensor proteins that fuse neurotransmitter-containing vesicle with the synaptic membrane. The probability (efficiency) of such fusion may transiently change from pulse to pulse, either increasing (short-term facilitation) or decreasing (short-term depression). Facilitation and depression are collectively termed short-term synaptic plasticity (STSP or STP).

In this project we investigate STP by modeling calcium dynamics inside the synapse, focusing on the role of calcium buffers in STP phenomenon. This involves numerical solution of the corresponding reaction-diffusion PDE system describing the evolution of calcium and buffer concentrations. It had been shown previously that facilitation may result from the depletion of buffer inside the synaptic terminal (“facilitation by buffer saturation”), or from the dislocation of buffer from the membrane upon calcium binding (“facilitation by buffer translocation”). However, we find other, previously unreported dynamic behaviors in the case of calcium buffers with more than one calcium binding site, such as calmodulin or calretinin. Our goal is to fully explore the dependence of short-term calcium dynamics on calcium buffering properties, focusing on buffers characterized by cooperative calcium binding. We will also discuss possible extensions of this work, including extensions to other fundamental calcium-dependent cell mechanism.

In this talk we examine the stability and spreading of a water drop under the influence of vertical oscillations. We start by introducing the thin film model which includes inertial effects and briefly go through the derivation. We will then discuss some simulation results and how this dynamic system can be described by a surrogate static system modeled by the Young-Laplace Equation. Finally we will investigate the different forces acting on the drop in order to gain some insight into the drop’s spreading dynamics. This can be useful in pattern formation and coating of surfaces where control over the spreading of fluids is important.

In this talk, we will explore the rhythmic patterns in music, focusing on Taal and Clave, which represent isochronous and non-isochronous cyclic rhythms, respectively. We will discuss the significance of these rhythmic structures and their properties, including beat generation and synchronization continuation. The primary objective of this study is to develop a hybrid model that combines biophysical and artificial neural network (ANN) techniques to analyze and replicate these rhythmic patterns.

Our approach involves enhancing the Beat Generator (BG) model, originally based on the Hodgkin-Huxley framework, to handle both isochronous and non-isochronous synchronization. This enhancement is achieved by incorporating the Tapping Neuron (TN), a Morris Lecar model, which uses frequency information from the BG and counting values from the ANN. The model determines the cycle length, including taps and rests, with the rests learned and classified by the ANN.

We will present methods showing how the hybrid model generates voltage-time course traces of non-isochronous and cyclic rhythms. The BG model's ability to replicate isochronous sound files and perform beat tracking will be shown, highlighting its alignment with external stimuli. Further, we will illustrate the mapping of Clave to Taal counting, the classification of Clave types by the ANN, and the hybrid model's replication of non-isochronous cyclic rhythms. Results on Son Clave forecasting, peak location, and the differentiation between 4/4 and 12/8 Clave will also be discussed.

Polynomial interpolation is a ubiquitous tool in numerical analysis. We present a method for deriving collocation points that attain faster convergence in integration and interpolation errors than standard tensor-product Chebyshev points in 2 and 3 dimensions, with ready generalization to higher dimensions. The natural polynomial basis associated to the points is efficient in "Euclidean" degree, which is the relevant measure for interpolating or integrating a generic analytic function. In addition to this efficiency advantage, there are natural algorithms based on the fast Fourier transform for working with the points; the conversion of point values to polynomial basis coefficients (and vice versa) can be performed in O(n log n) time. We compare the results of interpolating and integrating with these points to other "reduced" sets of points, including the "Padua" points.

Understanding electromagnetic wave behavior has many applications such as detecting defects in manufacturing metamaterials and nanophotonic crystals. In order to study this, we will discuss the direct scattering problem of a locally perturbed infinite layer in which Maxwell’s Equations reduces to the Helmholtz Equation. We will introduce the Floquet-Bloch transform to convert the system of equations into a quasi-periodic system in two spatial variables before applying a volume integral equation and FFT to solve this problem. Ongoing work will include expanding to Maxwell’s Equations and solving the inverse problem.

In spatial point processes intensity estimation, traditional methods like kernel estimators and point process models (PPM) have been effective in capturing linear correlations between points. However, they fall short when dealing with non-linear correlations. Deep learning models, such as Neural Networks, Autoencoders, and Variational Autoencoders (VAE), offer a promising alternative to address these limitations due to their inherent properties and settings. These models are widely used and acknowledged for their flexibility and capability to handle complex, non-linear relationships and allows it to maintain the flexibility of traditional models, such as bandwidth selection in kernel estimators and feature selection in PPM.

Block Bootstrap in Phylogenetic Tree Estimation

In phylogeny estimation, Felsenstein’s bootstrap support values are the most common approach to assess the reliability and robustness of phylogenies estimated from sequence data. Bootstrap support values are based on resampling with replacement the sites of a reference alignment until obtaining a bootstrap alignment
of the same length. Then, bootstrap trees are estimated using the same inference method. Finally, the support for every branch on the reference tree is measured as
the bootstrap proportion (BP) of pseudo-trees containing that branch. Importantly, this bootstrap method assumes that the columns in the alignment are independent and identically distributed. However, this assumption is unrealistic and is often violated in practice due to insufficient generality, leading to the omission of some evolutionary cases where certain sites are correlated with others. Therefore, we suggest that block bootstrap, a method different from the traditional one, is a suitable model that should be considered instead. This project aims to explore new possibilities for estimation in phylogenetic trees by applying block bootstrap methods with several substitution models.

In this talk, I will present the numerical simulation of the instability in a two-layer flow. We will examine the wave evolution using the VOF (Volume Of Fluid) method. Furthermore, the interface instability results from the Chebyshev spectral collocation method and finite difference method are presented to validate our outcomes. Finally, we will investigate the effect of stabilizing and destabilizing parameters on the growth rate.

We study several dielectric particles suspended in a dielectric fluid in stokes flow under a uniform applied electric field. Depending on physical parameters and field strength, a single particle can have either a stable or unstable solution. In the unstable case, the solution bifurcates into a class of solutions where the polarization is antiparallel to the applied field, and the particles rotate about an axis perpendicular to the applied field. We present the relevant boundary integral equations, and compare our results with those found in the literature.