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operator splitting and Local Time-Stepping Methods for Transport Problems in Fractured Porous Media

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Communications in Computational Physics 2023年第10期34卷 1215-1246页

作者： Phuoc-Toan Huynh Yanzhao Cao Thi-Thao-Phuong Hoang Department of Mathematics and Statistics Auburn UniversityAuburn 36849AlabamaUSA

This paper is concerned with efficient numerical methods for the advectiondiffusion equation in a heterogeneous porous medium containing fractures.A dimensionally reduced fracture model is considered,in which the fracture is represented as an interface between subdomains and is assumed to have larger permeability than the surrounding *** develop three global-in-time domain decomposition methods coupled with operator splitting for the reduced fracture model,where the advection and the diffusion are treated separately by different numerical schemes and with different time ***,smaller time steps can be used in the fracture-interface than in the *** first two methods are based on the physical transmission conditions,while the third one is based on the optimized Schwarz waveform relaxation approach with Ventcel-Robin transmission conditions.A discrete space-time interface system is formulated for each method and is solved iteratively and globally in *** results for two-dimensional problems with various P′eclet numbers and different types of fracture are presented to illustrate and compare the convergence and accuracy in time of the proposed methods with local time stepping.

关键词： Domain decomposition reduced fracture model operator splitting nonconforming time grids time-dependent Steklov-Poincare operator optimized Schwarz waveform relaxation

Differentially weighted operator splitting Monte Carlo method for simulating complex aerosol dynamic processes

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Particuology 2018年第1期16卷 114-126页

作者： Hongmei Liu Tat Leung Chan Department of Mechanical Engineering The Hong Kong Polytechnic University Kowloon Hong Kong China

A differentially weighted operator splitting Monte Carlo （DWOSMC） method is developed to solve com- plex aerosol dynamic processes by coupling the differentially weighted Monte Carlo method and the operator splitting technique. This method is validated by analytical solutions and a sectional method in different aerosol dynamic processes. It is first validated in coagulation and condensation processes, and nucleation and coagulation processes, and then validated through simultaneous nucleation, coagulation, and condensation processes. The results show that the DWOSMC method is a computationally efficient and quantitatively accurate method for simulating complex aerosol dynamic processes.

关键词： Differentially weighted Monte Carlo operator splitting Aerosol dynamics Particle size distribution

A Penalty-Regularization-operator splitting Method for the Numerical Solution of a Scalar Eikonal Equation

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Chinese Annals of Mathematics,Series B 2015年第5期36卷 659-688页

作者： Alexandre CABOUSSAT Roland GLOWINSKI Haute Ecole de Gestion de Genève HES-SO//University of Applied Sciences Department of Mathematics University of Houston

In this article, we discuss a numerical method for the computation of the minimal and maximal solutions of a steady scalar Eikonal equation. This method relies on a penalty treatment of the nonlinearity, a biharmonic regularization of the resulting variational problem, and the time discretization by operator-splitting of an initial value problem associated with the Euler-Lagrange equations of the regularized variational problem. A low-order finite element discretization is advocated since it is well-suited to the low regularity of the solutions. Numerical experiments show that the method sketched above can capture efficiently the extremal solutions of various two-dimensional test problems and that it has also the ability of handling easily domains with curved boundaries.

关键词： Eikonal equation Minimal and maximal solutions Regularization methods Penalization of equality constraints Dynamical flow operator splitting Finite element methods

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A Class of Preconditioners Based on Positive-Definite operator splitting Iteration Methods for Variable-Coefficient Space-Fractional Diffusion Equations

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Communications on Applied Mathematics and Computation 2021年第1期3卷 157-176页

作者： Jun-Feng Yin Yi-Shu Du School of Mathematical Sciences Tongji UniversityShanghai 200092China

After discretization by the finite volume method,the numerical solution of fractional diffusion equations leads to a linear system with the Toeplitz-like *** theoretical analysis gives sufficient conditions to guarantee the positive-definite property of the discretized ***,we develop a class of positive-definite operator splitting iteration methods for the numerical solution of fractional diffusion equations,which is unconditionally convergent for any positive ***,the iteration methods introduce a new preconditioner for Krylov subspace *** experiments verify the convergence of the positive-definite operator splitting iteration methods and show the efficiency of the proposed preconditioner,compared with the existing approaches.

关键词： Fractional diffusion equations Finite volume method operator splitting Positive-definite

Connections between operator-splitting Methods and Deep Neural Networks with Applications in Image Segmentation

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Annals of Applied Mathematics 2023年第4期39卷 406-428页

作者： Hao Liu Xue-Cheng Tai Raymond Chan Department of Mathematics Hong Kong Baptist UniversityKowloon TongHong Kong Norwegian Research Centre(NORCE) Nyg˚ardsgaten 1125008 BergenNorway Department of Mathematics City University of Hong Kong and Hong Kong Center for Cerebro-Cardiovascular Health EngineeringHong Kong

Deep neural network is a powerful tool for many *** why it is so successful and providing a mathematical explanation is an important problem and has been one popular research direction in past *** the literature of mathematical analysis of deep neural networks,a lot of works is dedicated to establishing representation *** to make connections between deep neural networks and mathematical algorithms is still under *** this paper,we give an algorithmic explanation for deep neural networks,especially in their connections with operator *** show that with certain splitting strategies,operator-splitting methods have the same structure as *** this connection and the Potts model for image segmentation,two networks inspired by operator-splitting methods are *** two networks are essentially two operator-splitting algorithms solving the Potts *** experiments are presented to demonstrate the effectiveness of the proposed networks.

关键词： Potts model operator splitting deep neural network image segmentation

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Propagation and Pinning of Travelling Wave for Nagumo Type Equation

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Journal of Applied Mathematics and Physics 2024年第3期12卷 861-869页

作者： Sharon-Yasotha Veerayah-Mcgregor Valipuram Manoranjan Department of Mathematics Washington State University Pullman Washington USA

In this paper, we study the propagation and its failure to propagate (pinning) of a travelling wave in a Nagumo type equation, an equation that describes impulse propagation in nerve axons that also models population growth with Allee effect. An analytical solution is derived for the traveling wave and the work is extended to a discrete formulation with a piecewise linear reaction function. We propose an operator splitting numerical scheme to solve the equation and demonstrate that the wave either propagates or gets pinned based on how the spatial mesh is chosen.

关键词： operator splitting Travelling Wave Piecewise Reaction Nagumo Equation Pinning Finite Differences

An　operator　Spliting　Method　with　FDM　and　FEM　for　the　Convection－Difusion　Equation　Using　Boundary－fited　Coordinate　System

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Communications in Nonlinear Science and Numerical Simulation 1996年第4期 57-62页

作者： Jie-Min ZHAN & Yao-Song CHEN  (Dept. of Applied Mechanics and Engineering, Zhongshan Univ., Guangzhou, China)  (Dept. of Mech., Peking Univ., Beijing 100871, China) E-mail: rd185079@hkpucc.polyu.edu.hk

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关键词： operator splitting Combined method of FDM and FEM Boundary-fitted coordinate system Convection-diffusion

An Inertial Semi-forward-reflected-backward splitting and Its Application

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Acta Mathematica Sinica,English Series 2022年第2期38卷 443-464页

作者： Chun Xiang ZONG Yu Chao TANG Guo Feng ZHANG School of Mathematics and Statistics Lanzhou UniversityLanzhou 730000.P.R.China Department of Mathematics Nanchang UniversityNanchang 330031P.R.China

Inertial methods play a vital role in accelerating the convergence speed of optimization *** work is concerned with an inertial semi-forward-reflected-backward splitting algorithm of approaching the solution of sum of a maximally monotone operator,a cocoercive operator and a monotone-Lipschitz continuous *** theoretical convergence properties of the proposed iterative algorithm are also presented under mild *** importantly,we use an adaptive stepsize rule in our algorithm to avoid calculating Lipschitz constant,which is generally unknown or difficult to estimate in practical *** addition,a large class of composite monotone inclusion problem involving mixtures of linearly composed and parallel-sum type monotone operators is solved by combining the primal-dual approach and our proposed *** a direct application,the obtained inertial algorithm is exploited to composite convex optimization problem and some numerical experiments on image deblurring problem are also investigated to demonstrate the efficiency of the proposed algorithm.

关键词： operator splitting inertial scheme composite monotone inclusions composite convex optimization total variation

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Finite Difference Method for Inhomogeneous Fractional Dirichlet Problem

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Numerical Mathematics(Theory,Methods and Applications) 2022年第3期15卷 744-767页

作者： Jing Sun Weihua Deng Daxin Nie School of Mathematics and Statistics Gansu Key Laboratory of Applied Mathematics and Complex SystemsLanzhou UniversityLanzhou 730000P.R.China

We make the split of the integral fractional Laplacian as(−△)^(s)u=(−△)(−△)^(s−1)u,where s∈(0,1/2)∪(1/2,1).Based on this splitting,we respectively discretize the oneand two-dimensional integral fractional Laplacian with the inhomogeneous Dirichlet boundary condition and give the corresponding truncation errors with the help of the interpolation ***,the suitable corrections are proposed to guarantee the convergence in solving the inhomogeneous fractional Dirichlet problem and an O(h^(1+α)2s))convergence rate is obtained when the solution u∈C^(1,α)(Ω_(n)^(δ)),where n is the dimension of the space,∈(max(0,2s−1),1],δis a fixed positive constant,and h denotes mesh ***,the performed numerical experiments confirm the theoretical results.

关键词： One-and two-dimensional integral fractional Laplacian Lagrange interpolation operator splitting finite difference the inhomogeneous fractional Dirichlet problem error estimates