A Robust Solver for a Second Order Mixed Finite Element Method for the Cahn-Hilliard Equation

We develop a robust solver for a second order mixed finite element splitting scheme for the Cahn-Hilliard equation. This work is an extension of our previous work in which we developed a robust solver for a first order mixed finite element splitting scheme for the Cahn-Hilliard equaion. The key ingredient of the solver is a preconditioned minimal residual algorithm (with a multigrid preconditioner) whose performance is independent of the spacial mesh size and the time step size for a given interfacial width parameter. The dependence on the interfacial width parameter is also mild.Comment: 17 pages, 3 figures, 4 tables. arXiv admin note: substantial text overlap with arXiv:1709.0400

Convergence Analysis and Error Estimates for a Second Order Accurate Finite Element Method for the Cahn-Hilliard-Navier-Stokes System

In this paper, we present a novel second order in time mixed finite element scheme for the Cahn-Hilliard-Navier-Stokes equations with matched densities. The scheme combines a standard second order Crank-Nicholson method for the Navier-Stokes equations and a modification to the Crank-Nicholson method for the Cahn-Hilliard equation. In particular, a second order Adams-Bashforth extrapolation and a trapezoidal rule are included to help preserve the energy stability natural to the Cahn-Hilliard equation. We show that our scheme is unconditionally energy stable with respect to a modification of the continuous free energy of the PDE system. Specifically, the discrete phase variable is shown to be bounded in $\ell^\infty \left(0,T;L^\infty\right)$ and the discrete chemical potential bounded in $\ell^\infty \left(0,T;L^2\right)$, for any time and space step sizes, in two and three dimensions, and for any finite final time $T$. We subsequently prove that these variables along with the fluid velocity converge with optimal rates in the appropriate energy norms in both two and three dimensions.Comment: 33 pages. arXiv admin note: text overlap with arXiv:1411.524

Quantifying Uncertainty in Ensemble Deep Learning

Neural networks are an emerging topic in the data science industry due to their high versatility and efficiency with large data sets. The purpose of this modern machine learning technique is to recognize relationships and patterns in vast amounts of data that would not be explored otherwise. Past research has utilized machine learning on experimental data in the material sciences and chemistry field to predict properties of metal oxides. Neural networks can determine underlying optical properties in complex images of metal oxides and capture essential features which are unrecognizable by observation. However, neural networks are often referred to as a “black box algorithm” due to the underlying process during the training of the model. The explanation for a prediction is unable to be traced, therefore poses a concern on how robust and reliable the prediction model actually is. Building ensemble neural networks allows for the analysis of the error bars of the prediction model. The project’s objective is to determine the comparative differences between the predictive ability of each individual convolutional neural network versus the ensemble neural network. Additionally, the paper explores how to use the ensemble model as a method of uncertainty quantification. Overall, ensemble neural networks outperform singular networks and demonstrate areas of uncertainty and robustness in the model

A robust solver for a second order mixed finite element method for the Cahn–Hilliard equation

We develop a robust solver for a second order mixed finite element splitting scheme for the Cahn–Hilliard equation. This work is an extension of our previous work in which we developed a robust solver for a first order mixed finite element splitting scheme for the Cahn–Hilliard equation. The key ingredient of the solver is a preconditioned minimal residual algorithm (with a multigrid preconditioner) whose performance is independent of the spatial mesh size and the time step size for a given interfacial width parameter. The dependence on the interfacial width parameter is also mild

Ensemble Deep Learning

Machine learning has become a common tool within the tech industry due to its high versatility and efficiency with large datasets. Partnering with the Nevada National Security Site, our goal is to improve accuracy of machine predictions by utilizing deep learning, which will enable the power and accuracy of a prediction to grow from the model. To build a deep learning model, multiple neural network architectures were developed and combined to create an ensemble neural network. The project’s objective is to determine the comparative differences between the efficiency of the ensemble neural network versus each individual neural network. The data set used to test, validate, and train the networks is 1D regressive. After testing architecture and determining accuracy of certain networks, the model will be updated and tested again to compare accuracies. Accuracy is the number of correct predictions over the total number of predictions. As model precision is a key aspect of machine learning, emphasis is placed on the efficiency of ensemble neural networks

Developing Predictive Algorithm for Possible Fuel Stops for Private Aviation

Machine learning algorithms\u27 capacity to improve over time is one of their main advantages. When more and more data is handled, machine learning technology often becomes more effective and accurate. Machine learning can be used to address problems in industry. OneSky Flight is an aviation company under an umbrella of companies offering technology services for other private jet companies. One problem they face as a business is predicting when a flight will need a fuel stop upon a booking request. Given a data set of approximately 230,000 flights from OneSky, dating back to 2019, a prediction model will be made in order to achieve the overall objective of the project: a fuel stop predictor. Therefore, the customer is aware and can upgrade the aircraft or plan for the necessary fuel stop, saving the customer money and time

Integrating ecology and technology to create innovative pest control devices

Blackie, H., MacMorran, D., Shapiro, L., Woodhead, I., Diegel, O., Murphy, E., Eason, C.T