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Articles 1 - 14 of 14
Full-Text Articles in Partial Differential Equations
Dynamics Of Microscale Droplets In Respiratory Airways, Md Shamser Ali Javed
Dynamics Of Microscale Droplets In Respiratory Airways, Md Shamser Ali Javed
Mathematics Theses and Dissertations
We investigate trajectories of microscale evaporating droplets in a stagnation point flow near a wall of a respiratory airway. The configuration is motivated by the problem of advection and deposition of microscale droplets of respiratory fluids in human airways during transmission of infectious diseases such as tuberculosis and COVID-19. Laminar boundary layer equations are solved to describe the air flow while the equations of motion of the droplet include contributions from gravity, aerodynamic drag, and Saffman force. Evaporation is accounted for at both the droplet surface and the wall of the respiratory airway and is shown to delay droplet deposition …
Cdt-1d Cnn Integration With Simpson-Sobolev Regularization For High-Frequency Options Trading: With Fem-Based Heston Option Pricing, Daniel M. Margolis, Johannes Tausch, Arthur K. Selender
Cdt-1d Cnn Integration With Simpson-Sobolev Regularization For High-Frequency Options Trading: With Fem-Based Heston Option Pricing, Daniel M. Margolis, Johannes Tausch, Arthur K. Selender
Mathematics Theses and Dissertations
This dissertation presents a computational framework for high-frequency options trading that combines Cross-Data-Type 1-D Convolutional Neural Networks (CDT-1D CNN) with Simpson-Sobolev regularization for directional prediction, and finite element methods (FEM) for realistic option pricing during backtesting. The core innovation lies in developing a mathematically rigorous regularization approach that maintains the adaptability of modern deep learning while enabling accurate evaluation through stochastic volatility models. The primary contribution is the Simpson-Sobolev regularization scheme, which extends traditional Sobolev regularization by incorporating Simpson’s rule for numerical integration. This approach achieves higher-order accuracy in approximating the Sobolev norms that control function smoothness. Simpson’s rule attains …
Stochastic Derivative-Free Deep Learning Methods For Solving High Dimensional Partial Differential Equations, Qing He Mr.
Stochastic Derivative-Free Deep Learning Methods For Solving High Dimensional Partial Differential Equations, Qing He Mr.
Mathematics Theses and Dissertations
Solving high-dimensional partial differential equations (PDEs) is a fundamental challenge in scientific computing, with applications ranging from quantum chemistry and computational finance to statistical physics and stochastic optimal control. Classical numerical methods such as finite element or finite difference schemes suffer from the curse of dimensionality, rendering them computationally infeasible when the dimension $d$ exceeds a handful. Physics-informed neural network (PINN) methods alleviate this by embedding the PDE residual directly into a loss function, but they require computing derivatives of the network with respect to its spatial inputs---an operation that scales poorly in high dimensions and demands that the approximate …
Predicting Biomolecular Properties And Interactions Using Numerical, Statistical And Machine Learning Methods, Elyssa Sliheet
Predicting Biomolecular Properties And Interactions Using Numerical, Statistical And Machine Learning Methods, Elyssa Sliheet
Mathematics Theses and Dissertations
We investigate machine learning and electrostatic methods to predict biophysical properties of proteins, such as solvation energy and protein ligand binding affinity, for the purpose of drug discovery/development. We focus on the Poisson-Boltzmann model and various high performance computing considerations such as parallelization schemes.
Tools For Biomolecular Modeling And Simulation, Xin Yang
Tools For Biomolecular Modeling And Simulation, Xin Yang
Mathematics Theses and Dissertations
Electrostatic interactions play a pivotal role in understanding biomolecular systems, influencing their structural stability and functional dynamics. The Poisson-Boltzmann (PB) equation, a prevalent implicit solvent model that treats the solvent as a continuum while describes the mobile ions using the Boltzmann distribution, has become a standard tool for detailed investigations into biomolecular electrostatics. There are two primary methodologies: grid-based finite difference or finite element methods and body-fitted boundary element methods. This dissertation focuses on developing fast and accurate PB solvers, leveraging both methodologies, to meet diverse scientific needs and overcome various obstacles in the field.
Generation, Dynamics, And Interaction Of Quartic Solitary Waves In Nonlinear Laser Systems, Sabrina Hetzel
Generation, Dynamics, And Interaction Of Quartic Solitary Waves In Nonlinear Laser Systems, Sabrina Hetzel
Mathematics Theses and Dissertations
Solitons are self-reinforcing localized wave packets that have remarkable stability features that arise from the balanced competition of nonlinear and dispersive effects in the medium. Traditionally, the dominant order of dispersion has been the lowest (second), however in recent years, experimental and theoretical research has shown that high, even order dispersion may lead to novel applications. Here, the focus is on investigating the interplay of dominant quartic (fourth-order) dispersion and the self-phase modulation due to the nonlinear Kerr effect in laser systems. One big factor to consider for experimentalists working in laser systems is the effect of noise on the …
Neural Network Learning For Pdes With Oscillatory Solutions And Causal Operators, Lizuo Liu
Neural Network Learning For Pdes With Oscillatory Solutions And Causal Operators, Lizuo Liu
Mathematics Theses and Dissertations
In this thesis, we focus on developing neural networks algorithms for scientific computing. First, we proposed a phase shift deep neural network (PhaseDNN), which provides a uniform wideband convergence in approximating high frequency functions and solutions of wave equations. Several linearized learning schemes have been proposed for neural networks solving nonlinear Navier-Stokes equations. We also proposed a causality deep neural network (Causality-DeepONet) to learn the causal response of a physical system. An extension of the Causality-DeepONet to time-dependent PDE systems is also proposed. The PhaseDNN makes use of the fact that common DNNs often achieve convergence in the low frequency …
Finite Element Approximation Of Solutions Of The Equations Of Electroporoelasticity, Yu Hu
Finite Element Approximation Of Solutions Of The Equations Of Electroporoelasticity, Yu Hu
Mathematics Theses and Dissertations
In this thesis we consider the solution of the equations of electroporoelasticity, which are a combination of Maxwell's equations and the poroelasticity equations. Included is a description of suitable initial and boundary conditions, weak formulation of the equations, and the error estimate for a general numerical method.
Advection-Reaction-Diffusion Model Of Drug Concentration In A Lymph Node, Ting Yan
Advection-Reaction-Diffusion Model Of Drug Concentration In A Lymph Node, Ting Yan
Mathematics Theses and Dissertations
It is recognized that there exist reservoirs of HIV located outside the bloodstream, and that these reservoirs hinder the efficacy of antiretroviral medication regimens in combating the virus. The prevailing theories regarding these reservoirs point to the lymphatic system. In this work, we discuss a novel computational model of viral dynamics in the lymph node, to allow numerical studies of viral “reservoirs” causing reinfection. Our model consists of a system of advection-reaction-diffusion partial differential equations (PDEs), where the diffusion coefficients vary between species (virus, drugs, lymphocytes) and include discontinuous jumps to capture differing properties of internal lymph node structures. We …
A New Class Of Discontinuous Galerkin Methods For Wave Equations In Second-Order Form, Lu Zhang
A New Class Of Discontinuous Galerkin Methods For Wave Equations In Second-Order Form, Lu Zhang
Mathematics Theses and Dissertations
Discontinuous Galerkin methods are widely used in many practical fields. In this thesis, we focus on a new class of discontinuous Galerkin methods for second-order wave equations. This thesis is constructed by three main parts. In the first part, we study the convergence properties of the energy-based discontinuous Galerkin proposed in [3] for wave equations. We improve the existing suboptimal error estimates to an optimal convergence rate in the energy norm. In the second part, we generalize the energy-based discontinuous Galerkin method proposed in [3] to the advective wave equation and semilinear wave equation in second-order form. Energy-conserving or energy-dissipating …
Parallel Multipole Expansion Algorithms And Their Biology Applications, Jiahui Chen
Parallel Multipole Expansion Algorithms And Their Biology Applications, Jiahui Chen
Mathematics Theses and Dissertations
N-body pairwise interactions are ubiquitous in scientific areas such as astrophysics, fluids mechanics, electrical engineering, molecular biology, etc. Computing these interactions using direct sum of an O(N) cost is expensive, whereas multipole expansion methods, such as the fast multipole method (FMM) or treecode, can reduce the cost to O(N) or O(N log N). This thesis focuses on developing numerical algorithms of Cartesian FMM and treecode, as well as using these algorithms to directly or implicitly solve biological problems involving pairwise interactions. This thesis consists of the following topics. 1) A cyclic parallel scheme is developed to handle the load balancing …
Swelling As A Stabilizing Mechanism During Ion Bombardment Of Thin Films: An Analytical And Numerical Study, Jennifer M. Swenson
Swelling As A Stabilizing Mechanism During Ion Bombardment Of Thin Films: An Analytical And Numerical Study, Jennifer M. Swenson
Mathematics Theses and Dissertations
Irradiation of semiconductor surfaces often leads to the spontaneous formation of rippled structures at certain irradiation angles. However, at high enough energies, these structures are observed to vanish for all angles, despite the absence of any identified, universally-stabilizing physical mechanisms in operation. Here, we examine the effect on pattern formation of radiation-induced swelling, which has been excluded from prior treatments of stress in irradiated films. After developing a suitable continuum model, we perform a linear stability analysis to determine its effect on stability. Under appropriate simplifying assumptions, we find swelling indeed to be stabilizing at wavenumbers typical of experimental observations. …
Heterogeneous Multiscale Modeling Of Advection-Diffusion Problems, David J. Gardner, Daniel R. Reynolds
Heterogeneous Multiscale Modeling Of Advection-Diffusion Problems, David J. Gardner, Daniel R. Reynolds
Mathematics Research
No abstract provided.
Effects Of Stochastic Freshwater Flux On The Atlantic Thermohaline Circulation, Alyssa Pampell, Alejandra Aceves
Effects Of Stochastic Freshwater Flux On The Atlantic Thermohaline Circulation, Alyssa Pampell, Alejandra Aceves
Mathematics Research
No abstract provided.