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Articles 121 - 150 of 434
Full-Text Articles in Statistics and Probability
Energy Stable Numerical Schemes For Ternary Cahn-Hilliard System, Wenbin Chen, Cheng Wang, Shufen Wang, Xiaoming Wang, Steven M. Wise
Energy Stable Numerical Schemes For Ternary Cahn-Hilliard System, Wenbin Chen, Cheng Wang, Shufen Wang, Xiaoming Wang, Steven M. Wise
Mathematics and Statistics Faculty Research & Creative Works
We present and analyze a uniquely solvable and unconditionally energy stable numerical scheme for the ternary Cahn-Hilliard system, with a polynomial pattern nonlinear free energy expansion. One key difficulty is associated with presence of the three mass components, though a total mass constraint reduces this to two components. Another numerical challenge is to ensure the energy stability for the nonlinear energy functional in the mixed product form, which turns out to be non-convex, non-concave in the three-phase space. to overcome this subtle difficulty, we add a few auxiliary terms to make the combined energy functional convex in the three-phase space, …
On The Noisy Gradient Descent That Generalizes As Sgd, Jingfeng Wu, Wenqing Hu, Haoyi Xiong, Jun Huan, Vladimir Braverman, Zhanxing Zhu
On The Noisy Gradient Descent That Generalizes As Sgd, Jingfeng Wu, Wenqing Hu, Haoyi Xiong, Jun Huan, Vladimir Braverman, Zhanxing Zhu
Mathematics and Statistics Faculty Research & Creative Works
The gradient noise of SGD is considered to play a central role in the observed strong generalization abilities of deep learning. While past studies confirm that the magnitude and covariance structure of gradient noise are critical for regularization, it remains unclear whether or not the class of noise distributions is important. In this work we provide negative results by showing that noises in classes different from the SGD noise can also effectively regularize gradient descent. Our finding is based on a novel observation on the structure of the SGD noise: it is the multiplication of the gradient matrix and a …
Vanishing Porosity Limit Of The Coupled Stokes-Brinkman System, Mingwen Fei, Dongjuan Niu, Xiaoming Wang
Vanishing Porosity Limit Of The Coupled Stokes-Brinkman System, Mingwen Fei, Dongjuan Niu, Xiaoming Wang
Mathematics and Statistics Faculty Research & Creative Works
We investigate the small porosity asymptotic behavior of the coupled Stokes-Brinkman system in the presence of a curved interface between the Stokes region and the Brinkman region. in particular, we derive a set of approximate solutions, validated via rigorous analysis, to the coupled Stokes-Brinkman system. of particular interest is that the approximate solution satisfies a generalized Beavers-Joseph-Saffman-Jones interface condition (1.9) with the constant of proportionality independent of the curvature of the interface.
Handling Missing Data For Unsupervised Learning With An Application On A Fitbir Traumatic Brain Injury (Tbi) Dataset, Louis Steinmeister, Dacosta Yeboah, Gayla Olbricht, Tayo Obafemi-Ajayi, Bassam Hadi, Daniel Hier, Donald C. Wunsch
Handling Missing Data For Unsupervised Learning With An Application On A Fitbir Traumatic Brain Injury (Tbi) Dataset, Louis Steinmeister, Dacosta Yeboah, Gayla Olbricht, Tayo Obafemi-Ajayi, Bassam Hadi, Daniel Hier, Donald C. Wunsch
Mathematics and Statistics Faculty Research & Creative Works
"The problem of missing data and imputation have been widely discussed amongst specialists. However, many data scientists and applied statisticians fail to appropriately consider this issue. Often, it seems intuitive to discard observations containing missing data or simply to substitute means. This can lead to disastrous consequences, particularly in an era of exponentially increasing data volumes. In the following, we show how inappropriate handling of missing data and an insufficient analysis of the censoring mechanism can lead to a bias, overconfidence in the estimation of parameters, could challenge the reproducibility of obtained results, and may distort the structure of the …
Rejoinder On ‘A Selective Overview Of Sparse Sufficient Dimension Reduction’, Lu Li, Xuewong Meggie Wen, Zhou Yu
Rejoinder On ‘A Selective Overview Of Sparse Sufficient Dimension Reduction’, Lu Li, Xuewong Meggie Wen, Zhou Yu
Mathematics and Statistics Faculty Research & Creative Works
No abstract provided.
Existence And Multiplicity Of Weak Solutions For A Neumann Elliptic Problem With -Laplacian, Martin Bohner, Giuseppe Caristi, Fariba Gharehgazlouei, Shapour Heidarkhani
Existence And Multiplicity Of Weak Solutions For A Neumann Elliptic Problem With -Laplacian, Martin Bohner, Giuseppe Caristi, Fariba Gharehgazlouei, Shapour Heidarkhani
Mathematics and Statistics Faculty Research & Creative Works
We are interested in the existence of multiple weak solutions for the Neumann elliptic problem involving the anisotropic -Laplacian operator, on a bounded domain with smooth boundary. We work on the anisotropic variable exponent Sobolev space, and by using a consequence of the local minimum theorem due to Bonanno, we establish existence of at least one weak solution under algebraic conditions on the nonlinear term. Also, we discuss existence of at least two weak solutions for the problem, under algebraic conditions including the classical Ambrosetti-Rabinowitz condition on the nonlinear term. Furthermore, by employing a three critical point theorem due to …
Efficient Smooth Non-Convex Stochastic Compositional Optimization Via Stochastic Recursive Gradient Descent, Wenqing Hu, Chris Junchi Li, Xiangru Lian, Ji Liu, Huizhuo Yuan
Efficient Smooth Non-Convex Stochastic Compositional Optimization Via Stochastic Recursive Gradient Descent, Wenqing Hu, Chris Junchi Li, Xiangru Lian, Ji Liu, Huizhuo Yuan
Mathematics and Statistics Faculty Research & Creative Works
Stochastic compositional optimization arises in many important machine learning applications. The objective function is the composition of two expectations of stochastic functions, and is more challenging to optimize than vanilla stochastic optimization problems. In this paper, we investigate the stochastic compositional optimization in the general smooth non-convex setting. We employ a recently developed idea of Stochastic Recursive Gradient Descent to design a novel algorithm named SARAH-Compositional, and prove a sharp Incremental First-order Oracle (IFO) complexity upper bound for stochastic compositional optimization: 𝒪((n + m)1/2ε-2) in the finite-sum case and 𝒪(ε-3) in the online case. …
Approximation Dynamics, Ethan Akin
Approximation Dynamics, Ethan Akin
Mathematics and Statistics Faculty Research & Creative Works
We describe the approximation of a continuous dynamical system on a p.l. manifold or Cantor set by a tractable system. A system is tractable when it has a finite number of chain components and, with respect to a given full background measure, almost every point is generic for one of a finite number of ergodic invariant measures. The approximations we use are non-degenerate simplicial dynamical systems for p.l. manifolds and shift-like dynamical systems for Cantor Sets.
A Multi-Step Approach To Modeling The 24-Hour Daily Profiles Of Electricity Load Using Daily Splines, Abdelmonaem Jornaz, V. A. Samaranayake
A Multi-Step Approach To Modeling The 24-Hour Daily Profiles Of Electricity Load Using Daily Splines, Abdelmonaem Jornaz, V. A. Samaranayake
Mathematics and Statistics Faculty Research & Creative Works
Forecasting of real-time electricity load has been an important research topic over many years. Electricity load is driven by many factors, including economic conditions and weather. Furthermore, the demand for electricity varies with time, with different hours of the day and different days of the week having an effect on the load. This paper proposes a hybrid load-forecasting method that combines classical time series formulations with cubic splines to model electricity load. It is shown that this approach produces a model capable of making short-term forecasts with reasonable accuracy. In contrast to forecasting models that utilize a multitude of regressor …
An Optimal Edg Method For Distributed Control Of Convection Diffusion Pdes, X. Zhang, Y. Zhang, John R. Singler
An Optimal Edg Method For Distributed Control Of Convection Diffusion Pdes, X. Zhang, Y. Zhang, John R. Singler
Mathematics and Statistics Faculty Research & Creative Works
We propose an embedded discontinuous Galerkin (EDG) method to approximate the solution of a distributed control problem governed by convection diffusion PDEs, and obtain optimal a priori error estimates for the state, dual state, their uxes, and the control. Moreover, we prove the optimize-then-discretize (OD) and discrtize-then-optimize (DO) approaches coincide. Numerical results confirm our theoretical results.
Three-Dimensional Rotation Of Paramagnetic And Ferromagnetic Prolate Spheroids In Simple Shear And Uniform Magnetic Field, Christopher A. Sobecki, Yanzhi Zhang, Cheng Wang
Three-Dimensional Rotation Of Paramagnetic And Ferromagnetic Prolate Spheroids In Simple Shear And Uniform Magnetic Field, Christopher A. Sobecki, Yanzhi Zhang, Cheng Wang
Mathematics and Statistics Faculty Research & Creative Works
We examine a time-dependent, three-dimensional rotation of magnetic ellipsoidal particles in a two-dimensional, simple shear flow and a uniform magnetic field. We consider that the particles have paramagnetic and ferromagnetic properties, and we compare their rotational dynamics due to the strengths and directions of the applied uniform magnetic field. We determine the critical magnetic field strength that can pin the particles' rotations. Above the critical field strength, the particles' stable steady angles were determined. In a weak magnetic regime (below the critical field strength), a paramagnetic particle's polar angle will oscillate toward the magnetic field plane while its azimuthal angle …
On Nonoscillatory Solutions Of Three Dimensional Time-Scale Systems, Elvan Akin, Taher Hassan, Ozkan Ozturk, Ismail U. Tiryaki
On Nonoscillatory Solutions Of Three Dimensional Time-Scale Systems, Elvan Akin, Taher Hassan, Ozkan Ozturk, Ismail U. Tiryaki
Mathematics and Statistics Faculty Research & Creative Works
In this article, we classify nonoscillatory solutions of a system of three-dimensional time scale systems. We use the method of considering the sign of components of such solutions. Examples are given to highlight some of our results. Moreover, the existence of such solutions is obtained by Knaster's fixed point theorem.
An Hdg Method For Dirichlet Boundary Control Of Convection Dominated Diffusion Pdes, Gang Chen, John R. Singler, Yangwen Zhang
An Hdg Method For Dirichlet Boundary Control Of Convection Dominated Diffusion Pdes, Gang Chen, John R. Singler, Yangwen Zhang
Mathematics and Statistics Faculty Research & Creative Works
We first propose a hybridizable discontinuous Galerkin (HDG) method to approximate the solution of a convection dominated Dirichlet boundary control problem without constraints. Dirichlet boundary control problems and convection dominated problems are each very challenging numerically due to solutions with low regularity and sharp layers, respectively. Although there are some numerical analysis works in the literature on diffusion dominated convection diffusion Dirichlet boundary control problems, we are not aware of any existing numerical analysis works for convection dominated boundary control problems. Moreover, the existing numerical analysis techniques for convection dominated PDEs are not directly applicable for the Dirichlet boundary control …
On The Instabilities And Transitions Of The Western Boundary Current, Daozhi Han, Marco Hernandez, Quan Wang
On The Instabilities And Transitions Of The Western Boundary Current, Daozhi Han, Marco Hernandez, Quan Wang
Mathematics and Statistics Faculty Research & Creative Works
We study the stability and dynamic transitions of the western boundary currents in a rectangular closed basin. By reducing the infinite dynamical system to a finite dimensional one via center manifold reduction, we derive a non-dimensional transition number that determines the types of dynamical transition. We show by careful numerical evaluation of the transition number that both continuous transitions (supercritical Hopf bifurcation) and catastrophic transitions (subcritical Hopf bifurcation) can happen at the critical Reynolds number, depending on the aspect ratio and stratification. The regions separating the continuous and catastrophic transitions are delineated on the parameter plane.
Positivity-Preserving, Energy Stable Numerical Schemes For The Cahn-Hilliard Equation With Logarithmic Potential, Wenbin Chen, Cheng Wang, Xiaoming Wang, Steven M. Wise
Positivity-Preserving, Energy Stable Numerical Schemes For The Cahn-Hilliard Equation With Logarithmic Potential, Wenbin Chen, Cheng Wang, Xiaoming Wang, Steven M. Wise
Mathematics and Statistics Faculty Research & Creative Works
In this paper we present and analyze finite difference numerical schemes for the Cahn-Hilliard equation with a logarithmic Flory Huggins energy potential. Both first and second order accurate temporal algorithms are considered. in the first order scheme, we treat the nonlinear logarithmic terms and the surface diffusion term implicitly and update the linear expansive term and the mobility explicitly. We provide a theoretical justification that this numerical algorithm has a unique solution, such that the positivity is always preserved for the logarithmic arguments, i.e., the phase variable is always between −1 and 1, at a point-wise level. in particular, our …
Smale Strategies For The $N$-Person Iterated Prisoner's Dilemma, Ethan Akin, Sławomir Plaskacz, Joanna Zwierzchowska
Smale Strategies For The $N$-Person Iterated Prisoner's Dilemma, Ethan Akin, Sławomir Plaskacz, Joanna Zwierzchowska
Mathematics and Statistics Faculty Research & Creative Works
Adapting methods introduced by Steven Smale, we describe good strategies for a symmetric version of the Iterated Prisoner's Dilemma with n players.
Extensions Of Schauder's And Darbo's Fixed Point Theorems, Zhaocai Hao, Martin Bohner, Junjun Wang
Extensions Of Schauder's And Darbo's Fixed Point Theorems, Zhaocai Hao, Martin Bohner, Junjun Wang
Mathematics and Statistics Faculty Research & Creative Works
In this paper, some new extensions of Schauder's and Darbo's fixed point theorems are given. As applications of the main results, the existence of global solutions for first-order nonlinear integro-differential equations of mixed type in a real Banach space is investigated.
A Second Order Bdf Numerical Scheme With Variable Steps For The Cahn-Hilliard Equation, Wenbin Chen, Xiaoming Wang, Yue Yan, Zhuying Zhang
A Second Order Bdf Numerical Scheme With Variable Steps For The Cahn-Hilliard Equation, Wenbin Chen, Xiaoming Wang, Yue Yan, Zhuying Zhang
Mathematics and Statistics Faculty Research & Creative Works
We present and analyze a second order in time variable step BDF2 numerical scheme for the Cahn-Hilliard equation. the construction relies on a second order backward difference, convex-splitting technique and viscous regularizing at the discrete level. We show that the scheme is unconditionally stable and uniquely solvable. in addition, under mild restriction on the ratio of adjacent time-steps, an optimal second order in time convergence rate is established. the proof involves a novel generalized discrete Gronwall-type inequality. as far as we know, this is the first rigorous proof of second order convergence for a variable step BDF2 scheme, even in …
Almost Periodic Functions In Quantum Calculus, Martin Bohner, Jaqueline Godoy Mesquita
Almost Periodic Functions In Quantum Calculus, Martin Bohner, Jaqueline Godoy Mesquita
Mathematics and Statistics Faculty Research & Creative Works
In this article, we introduce the concepts of Bochner and Bohr almost periodic functions in quantum calculus and show that both concepts are equivalent. Also, we present a correspondence between almost periodic functions defined in quantum calculus and N0, proving several important properties for this class of functions. We investigate the existence of almost periodic solutions of linear and nonlinear q-difference equations. Finally, we provide some examples of almost periodic functions in quantum calculus.
A Further Extension Of The Extended Riemann-Liouville Fractional Derivative Operator, Martin Bohner, Gauhar Rahman, Shahid Mubeen, Kottakkaran Sooppy Nisar
A Further Extension Of The Extended Riemann-Liouville Fractional Derivative Operator, Martin Bohner, Gauhar Rahman, Shahid Mubeen, Kottakkaran Sooppy Nisar
Mathematics and Statistics Faculty Research & Creative Works
The main objective of this paper is to establish the extension of an extended fractional derivative operator by using an extended beta function recently defined by Parmar et al. by considering the Bessel functions in its kernel. We also give some results related to the newly defined fractional operator, such as Mellin transform and relations to extended hypergeometric and Appell's function via generating functions.
Dynamics Of Paramagnetic And Ferromagnetic Ellipsoidal Particles In Shear Flow Under A Uniform Magnetic Field, Christopher A. Sobecki, Jie Zhang, Yanzhi Zhang, Cheng Wang
Dynamics Of Paramagnetic And Ferromagnetic Ellipsoidal Particles In Shear Flow Under A Uniform Magnetic Field, Christopher A. Sobecki, Jie Zhang, Yanzhi Zhang, Cheng Wang
Mathematics and Statistics Faculty Research & Creative Works
We investigate the two-dimensional dynamic motion of magnetic particles of ellipsoidal shapes in shear flow under the influence of a uniform magnetic field. In the first part, we present a theoretical analysis of the rotational dynamics of the particles in simple shear flow. By considering paramagnetic and ferromagnetic particles, we study the effects of the direction and strength of the magnetic field on the particle rotation. The critical magnetic-field strength, at which particle rotation is impeded, is determined. In a weak-field regime (i.e., below the critical strength) where the particles execute complete rotations, the symmetry property of the rotational velocity …
A Multi-Step Nonlinear Dimension-Reduction Approach With Applications To Bigdata, R. Krishnan, V. A. Samaranayake, Jagannathan Sarangapani
A Multi-Step Nonlinear Dimension-Reduction Approach With Applications To Bigdata, R. Krishnan, V. A. Samaranayake, Jagannathan Sarangapani
Mathematics and Statistics Faculty Research & Creative Works
In this paper, a multi-step dimension-reduction approach is proposed for addressing nonlinear relationships within attributes. In this work, the attributes in the data are first organized into groups. In each group, the dimensions are reduced via a parametric mapping that takes into account nonlinear relationships. Mapping parameters are estimated using a low rank singular value decomposition (SVD) of distance covariance. Subsequently, the attributes are reorganized into groups based on the magnitude of their respective singular values. The group-wise organization and the subsequent reduction process is performed for multiple steps until a singular value-based user-defined criterion is satisfied. Simulation analysis is …
Approximation Degree Of Durrmeyer-Bézier Type Operators, Purshottam N. Agrawal, Serkan Araci, Martin Bohner, Kumari Lipi
Approximation Degree Of Durrmeyer-Bézier Type Operators, Purshottam N. Agrawal, Serkan Araci, Martin Bohner, Kumari Lipi
Mathematics and Statistics Faculty Research & Creative Works
Recently, a mixed hybrid operator, generalizing the well-known Phillips operators and Baskakov-Szász type operators, was introduced. In this paper, we study Bézier variant of these new operators. We investigate the degree of approximation of these operators by means of the Lipschitz class function, the modulus of continuity, and a weighted space. We study a direct approximation theorem by means of the unified Ditzian-Totik modulus of smoothness. Furthermore, the rate of convergence for functions having derivatives of bounded variation is discussed.
Phytoforensics: Trees As Bioindicators Of Potential Indoor Exposure Via Vapor Intrusion, Jordan L. Wilson, V. A. Samaranayake, Matt A. Limmer, Joel Gerard Burken
Phytoforensics: Trees As Bioindicators Of Potential Indoor Exposure Via Vapor Intrusion, Jordan L. Wilson, V. A. Samaranayake, Matt A. Limmer, Joel Gerard Burken
Mathematics and Statistics Faculty Research & Creative Works
Human exposure to volatile organic compounds (VOCs) via vapor intrusion (VI) is an emerging public health concern with notable detrimental impacts on public health. Phytoforensics, plant sampling to semi-quantitatively delineate subsurface contamination, provides a potential non-invasive screening approach to detect VI potential, and plant sampling is effective and also time- and cost-efficient. Existing VI assessment methods are time- and resource-intensive, invasive, and require access into residential and commercial buildings to drill holes through basement slabs to install sampling ports or require substantial equipment to install groundwater or soil vapor sampling outside the home. Tree-core samples collected in 2 days at …
Mini Review: A Note On Nonoscillatory Solutions For Higher Dimensional Time Scale Systems, Elvan Akin, Ozkan Ozturk, Ismail Ugur Tiryaki, Gulsah Yeni
Mini Review: A Note On Nonoscillatory Solutions For Higher Dimensional Time Scale Systems, Elvan Akin, Ozkan Ozturk, Ismail Ugur Tiryaki, Gulsah Yeni
Mathematics and Statistics Faculty Research & Creative Works
In this paper, we focus on nonoscillatory solutions of two (2D) and three (3D) dimensional time scale systems and discuss nonexistence of such solutions.
The Generalized Hypergeometric Difference Equation, Martin Bohner, Tom Cuchta
The Generalized Hypergeometric Difference Equation, Martin Bohner, Tom Cuchta
Mathematics and Statistics Faculty Research & Creative Works
A difference equation analogue of the generalized hypergeometric differential equation is defined, its contiguous relations are developed, and its relation to numerous well-known classical special functions are demonstrated.
A Higher-Order Ensemble/Proper Orthogonal Decomposition Method For The Nonstationary Navier-Stokes Equations, Max Gunzburger, Nan Jiang, Michael Schneier
A Higher-Order Ensemble/Proper Orthogonal Decomposition Method For The Nonstationary Navier-Stokes Equations, Max Gunzburger, Nan Jiang, Michael Schneier
Mathematics and Statistics Faculty Research & Creative Works
Partial differential equations (PDE) often involve parameters, such as viscosity or density. An analysis of the PDE may involve considering a large range of parameter values, as occurs in uncertainty quantification, control and optimization, inference, and several statistical techniques. The solution for even a single case may be quite expensive; whereas parallel computing may be applied, this reduces the total elapsed time but not the total computational effort. In the case of flows governed by the Navier-Stokes equations, a method has been devised for computing an ensemble of solutions. Recently, a reduced-order model derived from a proper orthogonal decomposition (POD) …
Parametrization Of Scale-Invariant Self-Adjoint Extensions Of Scale-Invariant Symmetric Operators, Miron B. Bekker, Martin Bohner, Alexander P. Ugol'nikov, Hristo Voulov
Parametrization Of Scale-Invariant Self-Adjoint Extensions Of Scale-Invariant Symmetric Operators, Miron B. Bekker, Martin Bohner, Alexander P. Ugol'nikov, Hristo Voulov
Mathematics and Statistics Faculty Research & Creative Works
On a Hilbert space H, we consider a symmetric scale-invariant operator with equal defect numbers. It is assumed that the operator has at least one scale invariant self-adjoint extension in H. We prove that there is a one-to-one correspondence between (generalized) resolvents of scale-invariant extensions and solutions of some functional equation. Two examples of Dirac-type operators are considered.
Decoupled, Linear, And Energy Stable Finite Element Method For The Cahn-Hilliard-Navier-Stokes-Darcy Phase Field Model, Yali Gao, Xiaoming He, Liquan Mei, Xiaofeng Yang
Decoupled, Linear, And Energy Stable Finite Element Method For The Cahn-Hilliard-Navier-Stokes-Darcy Phase Field Model, Yali Gao, Xiaoming He, Liquan Mei, Xiaofeng Yang
Mathematics and Statistics Faculty Research & Creative Works
In this paper, we consider the numerical approximation for a phase field model of the coupled two-phase free flow and two-phase porous media flow. This model consists of Cahn—Hilliard—Navier—Stokes equations in the free flow region and Cahn—Hilliard—Darcy equations in the porous media region that are coupled by seven interface conditions. The coupled system is decoupled based on the interface conditions and the solution values on the interface from the previous time step. A fully discretized scheme with finite elements for the spatial discretization is developed to solve the decoupled system. In order to deal with the difficulties arising from the …
Convergence Analysis And Error Estimates For A Second Order Accurate Finite Element Method For The Cahn–Hilliard–Navier–Stokes System, Amanda E. Diegel, Cheng Wang, Xiaoming Wang, Steven M. Wise
Convergence Analysis And Error Estimates For A Second Order Accurate Finite Element Method For The Cahn–Hilliard–Navier–Stokes System, Amanda E. Diegel, Cheng Wang, Xiaoming Wang, Steven M. Wise
Mathematics and Statistics Faculty Research & Creative Works
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–Nicolson method for the Navier–Stokes equations and a modification to the Crank–Nicolson 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 …