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The conference will take place on Saturday, December 10 from 8:50 am - 5:00 pm. There are four sequential sessions of 80 minutes each. The speakers will have 15 minutes to talk with 5 minutes for questions and transitions.
| Saturday | 08:50 am - 09:00 am | Opening Remarks |
|---|---|---|
| 09:00 am - 10:20 am | Presentation Session I | |
| 10:20 am - 10:50 am | Break (refreshments provided) | |
| 10:50 am - 12:10 pm | Presentation Session II | |
| 12:10 pm - 02:00 pm | Lunch | |
| 02:00 pm - 03:20 pm | Presentation Session III | |
| 03:20 pm - 03:40 pm | Break (refreshments provided) | |
| 03:40 pm - 05:00 pm | Presentation Session IV |
Here is a more detailed program that specifies the scheduled time for each talk.
| 09:00AM | Reduction to smooth maps for the global dynamics of a vibro-impact energy harvester |
| Lanjing Bao (Georgia State University) | |
| Vibro-impact dynamic pairs appear in multiple engineering applications including vibro-impact harvesters. Such devices can harvester energy from vibro-impact motion of a ball moving freely within a driven cylindrical capsule with two dielectric elastomer membranes that cover both its ends. When the membranes are impacted and deformed by the ball, the capacitance change generates an extra charge to be harvested. Due to the complexity of the non-smooth motion, there is a lack of mathematical approaches to analyzing the global dynamics of such vibro-impact systems. In this talk, I will present a computational method for reducing the non-smooth dynamics into smooth maps which represent the evolution of the system’s states from one impact to another. These maps allow for a semi-analytical study of the system’s global dynamics via an auxiliary map approach for estimating the solutions’ basins of attraction. Our results may provide practical guidelines for designing such a vibro-impact energy harvester with a desired globally stable dynamical regime that can maximize the energy gain. This is a joint work with Profs. Igor Belykh (GSU), Rachel Kuske (Georgia Tech), and Daniil Yurchenko (University of Southampton, UK). |
| 09:20AM | SAV Ensemble Algorithms for the magnetohydrodynamics equations |
| John Carter (Missouri University of Science and Technology) | |
| We develop two linear, second-order accurate, unconditionally stable ensemble methods with shared coefficient matrix across different realizations and time steps for the magnetohydrodynamics equations. The viscous terms are treated by a standard perturbative discretization. We employ the Generalized Positive Auxiliary Variable method to discretize nonlinear terms, resulting in linearity of the algebra equation for the scalar variable, provable positivity of the scalar variable, and flexibility in handling complex boundary conditions. Artificial viscosity stabilization that modifies the kinetic energy is introduced to improve accuracy of the GPAV ensemble methods. |
| 09:40AM | A partitioned method for the solution of fluid-structure interaction and ROM implementation |
| Amy de Castro (Clemson University) | |
| Partitioned methods for the solution of fluid-structure interaction (FSI) problems are often the preferred approach as they allow the reuse of existing codes for each subdomain that take into account its unique mathematical and physical properties. We present a partitioned method for FSI problems based on a monolithic formulation of the problem which employs a Schur complement equation for a Lagrange multiplier. This algorithm eradicates the need for iterations between the fluid and structure subdomains and instead allows them to be decoupled and solved separately at each time step. The Lagrange multiplier, representing an approximation of the interface flux as well as the fluid pressure, serves as a Neumann boundary condition for each sub-problem, allowing for the fluid and structure to be solved independently at each time step. To reduce computational costs, we consider implementing a reduced order model (ROM) for one or both subdomains. The Schur complement method developed for the original FSI scheme can be utilized to couple a reduced order model with either a full order model or a reduced order model on the other subdomain. We show numerical results demonstrating the method’s capability to capture each type of coupling. |
| 10:00AM | Low regularity integrators for semilinear parabolic equations with maximum bound principles |
| Cao Kha Doan (Auburn University) | |
| This talk is concerned with conditionally structure-preserving, low regularity time integration methods for a class of semilinear parabolic equations of Allen-Cahn type. Important properties of such equations include maximum bound principle (MBP) and energy dissipation law; for the former, that means the absolute value of the solution is pointwisely bounded for all the time by some constant imposed by appropriate initial and boundary conditions. The model equation is first discretized in space by the central finite difference, then by iteratively using Duhamel’s formula, first- and second-order low regularity integrators (LRIs) are constructed for time discretization of the semi-discrete system. The proposed LRI schemes are proved to preserve the MBP and the energy stability in the discrete sense. Furthermore, their temporal error estimates are also successfully derived under a low regularity requirement that the exact solution of the semi-discrete problem is only assumed to be continuous in time. Numerical results show that the proposed LRI schemes are more accurate and have better convergence rates than classic exponential time differencing schemes, especially when the interfacial parameter approaches zero. |
| 10:50AM | Effects of scaffold elasticity and nutrient depletion on cell proliferation in a tissue engineering scaffold pore |
| Haniyeh Fattahpour (Georgia State University) | |
| A tissue engineering scaffold contains pores lined with cells that allow nutrients-rich culture medium to pass through to encourage cell proliferation. Several factors contribute to tissue growth, such as the amount of nutrients in the feed, the elasticity of the scaffold, and the characteristics of the cells. These effects have been examined separately in several studies; however, in this work, they are examined concurrently. This work consists of the following objectives: (i) the development of a mathematical model describing nutrient fluid dynamics, scaffold elasticity, and cell proliferation; (ii) solving the model and simulating the cell proliferation process; (iii) developing a reverse algorithm based on the desired geometry of the final tissue to determine the initial scaffold configuration. |
| 11:10AM | Convolutional Neural Network for Phase Field Modeling |
| Yuwei Geng (University of South Carolina) | |
| Allen-Cahn equation is a stiff nonlinear partial differential equation (PDE) that was introduced in modeling phase field. It has been well studied by numerical methods, such as the finite difference method. As deep learning has achieved tremendous successes over the past decade, it is of great interest in designing deep learning based approaches for numerical solutions to PDEs. In this work, we construct Convolutional Neural Networks to model the discrete operator of Allen-Cahn, linking discrete finite difference solutions at adjacent time steps, associated to the backward Euler scheme with a fixed step size. In particular, the networks are trained based on limited initial data, which can be further used to predict the numerical solutions to Allen-Cahn subject to different initial conditions. |
| 11:30AM | Numerical analysis of continuous data assimilation for semilinear heat equation |
| Elizabeth Hawkins (Clemson University) | |
| We consider a BDF2-Finite element discretization of continuous data assimilation (CDA) applied to a semilinear heat equation. After deriving the fully discretized CDA enhanced semilinear heat system, we prove the system is stable and converges to the solution given by the finite element (FEM) system. Finally, we show results for CDA applied to a Fujita equation and then lab experiment data for vibration induced temperature increase. These results match our analysis in the previous section for CDA. Then, we use \(G\)-stability theory to prove the fully discretized finite element system is stable and converges to the true solution to semilinear heat equation. These results match those found is other papers but $G$-stability allows for simpler proofs. |
| 11:50AM | Space-time domain decomposition methods for flow and transport problems in fractured porous media. |
| Toan Huynh (Auburn University) | |
| This talk is concerned with the numerical solutions to the flow and transport problems in fractured porous media. A dimensionally reduced fracture model written in mixed form is considered, where the fracture is assumed to have larger permeability than the surrounding area and is modeled as an interface between subdomains. We aim to develop fast-convergent and accurate global-in-time domain decomposition (DD) methods for such reduced models. We first focus on the flow problem of a single phase, compressible fluid and derive its reduced fracture models. New global-in-time DD methods together with the existing methods for such models are then discussed. Numerical experiments with different types of fracture are presented to verify and compare the performance of the proposed methods with different time steps in the fracture and in the rock matrix. These methods are then extended to the case of linear advection-diffusion equations where global-in-time domain decomposition is coupled with operator splitting to treat the advection and the diffusion separately by different numerical schemes and with different time steps. |
| 02:00PM | An artificial compressibility CNLF method for the Stokes-Darcy model and application in ensemble simulations |
| Ying Li (University of Florida) | |
| We propose and analyze an efficient, unconditionally stable, second order convergent, artificial compressibility Crank-Nicolson leap-frog (CNLFAC) method for numerically solving the Stokes-Darcy equations. The method decouples the fully coupled Stokes-Darcy system into two smaller subphysics problems, which reduces the size of the linear systems to be solved, at each time step, and allows parallel computing of the two subphysics problems. It also decouples the computation of the velocity and pressure in the free flow region. The pressure only needs to be updated at each time step without solving a Poisson equation, avoiding pressure errors in boundary layers due to imposing artificial boundary conditions. We prove that the method is unconditionally long time stable and second order convergent. We also propose an unconditionally stable ensemble algorithm based on the CNLFAC method. The ensemble algorithm results in a common coefficient matrix for all realizations and consequently allows the use of efficient direct or iterative solvers to reduce the computational cost. Numerical experiments are provided to illustrate the second-order convergence and unconditional stability of the CNLFAC method. Moreover, the CNLFAC ensemble algorithm is demonstrated to reduce the computational time of a CNLF nonensemble algorithm by 95% in our tests. |
| 02:20PM | Flight stability of wedges |
| Pejman Sanaei (Georgia State University) | |
| Recent experiments have shown that cones of intermediate apex angles display orientational stability with apex leading in flight. In this talk we show in experiments and simulations that analogous results hold in the two-dimensional context of solid wedges or triangular prisms in planar flows at Reynolds numbers Re ∼ 100 to 1000. Slender wedges are statically unstable with apex leading and tend to flip over or tumble, and broad wedges oscillate or flutter due to dynamical instabilities, but those of apex half angles between about 40 and 55 degrees maintain stable posture during flight. The existence of ‘‘Goldilocks’’ shapes that possess the ‘‘just right’’ angularity for flight stability is thus robust to dimensionality. We also show that the stability is robust to moderate changes in shape and Reynolds number. |
| 02:40PM | Fourier Transforms for Generalized Sturm-Liouville Differential Equations |
| Steven Redolfi (University of Alabama at Birmingham) | |
| This talk is about the spectral theory associated with the differential equation \(Ju' + qu = wf\) on the real interval \((a, b)\) when \(J\) is an \(n \times n\) constant, invertible, skew-Hermitian matrix and \(q\) and \(w\) are \(n \times n\) matrices whose entries are distributions of order zero (local measures) with \(q\) Hermitian and \(w\) non-negative. Under these hypotheses it may not be possible to uniquely continue a solution from one point to another, thus blunting the standard tools of spectral theory. Despite this fact we are able to show, under certain assumptions on the coefficients of the equation, that any self adjoint restriction \(T\) of the maximal relation for the differential equation has a corresponding Fourier transform \(\mathcal{F}\) and spectral space \(L^2(\tau)\) into which \(\mathcal{F}\) restricted to \(\mathrm{dom}(T)\) is a surjective unitary operator which diagonalizes \(T\). |
| 03:00PM | Learning Green's Functions of Linear Reaction-Diffusion Equations with Application to Fast Numerical Solver |
| Yuankai Teng (University of South Carolina) | |
| Partial differential equations are commonly used to model various physical phenomena, such as heat diffusion, wave propagation, fluid dynamics, elasticity, electrodynamics and so on. Due to their tremendous applications in scientific and engineering research, many numerical methods have been developed in past decades for efficient and accurate solutions of these equations on modern computing systems. Inspired by the rapidly growing impact of deep learning techniques, we propose in this paper a novel neural network method, "GF-Net", for learning the Green's functions of the classic linear reaction-diffusion equation with Dirichlet boundary condition in the unsupervised fashion. The proposed method overcomes the numerical challenges for finding the Green's functions of the equations on general domains by utilizing the physics-informed neural network and the domain decomposition approach. As a consequence, it also leads to a fast numerical solver for the target equation subject to arbitrarily given sources and boundary values without network retraining. We numerically demonstrate the effectiveness of the proposed method by extensive experiments with various domains and operator coefficients. |
| 03:40PM | Spectral ReLu: A Smooth Generalization of ReLu in Functional Space |
| Zezhong Zhang (Florida State University) | |
| In this work, we propose a novel generalization of the classical element-wise ReLu acti-vation neural networks by projecting all neurons in each layer in a function space via basis expansion, and the concept of neuron is replaced by a (continuous) state function. By different choices of the basis functions, the model will have different order of (parametric) continuity. As a result, the ReLu can also yield a smooth model. Our numerical examples show that by choosing particular basis our model can have fast convergence rate like classical ReLu and smooth optimization landscape like Tanh. We also show that the optimization landscape smoothness comes from the continuous activation pattern of the state function. |
| 04:00PM | TBA |
| Pengjun Wang (Auburn University) | |
| TBA |
| 04:20PM | Machine Learning-based Approach to Stellar Blend Image Classification |
| Rafael Bidese (Auburn University) | |
| The Zwicky Transient Facility is an extremely wide-field sky survey designed to image a large area of sky at a time. Its main goal is discovering transients in a timely manner (such as supernova, gamma ray bursts, variable stars, asteroids, and exoplanets). It has a wide field-of-view and gathers large volumes of data every night. Hence, machine learning approaches can help extracting more information out of that data. For instance, one such analysis is the identification of stellar blending, occurs when two or more stars look "blended" together in an image. Information about stellar blends is important to the study of dark matter through gravitational lensing, which is distortion of the observation of the universe that relies on understanding what matter is along the line-of-sight. Stellar blends are typically disambiguated through more "expensive" methods, such as spectroscopy, getting resolved images using space-based telescopes, or getting higher resolution images using ground-based telescopes. However, this project uses data science to create a pipeline to automatically identify stellar blends from low resolution ZTF images. A novel machine learning approach was used to solve the stellar blending problem. Different normalizations methods were evaluated as inputs for computationally efficient gaussian process model (MuyGPs) and other traditional machine learning methods. The best performing model was MuyGPs with an nth-root local min-max normalization that achieved 86% accuracy. Further, the uncertainty classification from the model can be used to determine a threshold where uncertain samples are not misclassified and can be redirected to human specialist to correctly label them. The proposed method is promising for problems that have low dimension complexity data and should be investigated further for applications such as multiclass classification, regression and anomaly detection. |
| 04:40PM | Topologically Protected Edge States |
| Ridvan Ozdemir (Auburn University) | |
| In this talk, we present an approach to topologically protected edge states. Study is based on breaking the inversion symmetry of a periodic media that is constructed via setting unit cells and joining periodic aggregations that are reflections of each other with respect to a unit cell or a group of cells. This construction reveals some topological structures with some dispersion topological invariants, which are Zak phase and Chern numbers in our case. We study both 1D lattice and 2D hexagonal lattice problems in topological photonic structure and existence of interface states when two sides of the system have some certain characteristics. |