Article
Mathematics, Applied
X. I. A. O. C. H. U. N. Chen, C. H. E. N. G. Wang, Steven M. Wise
Summary: This paper provides a detailed analysis of the preconditioned steepest descent (PSD) iteration solver for a convex splitting numerical scheme to the Cahn-Hilliard equation with variable mobility function. The paper presents a method to handle the energy functional, compute the mobility function, and address challenges in the implementation of the numerical scheme. The PSD iteration solver improves the efficiency and stability of the numerical solution for the Cahn-Hilliard equation.
INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING
(2022)
Article
Automation & Control Systems
Alfonso Landeros, Oscar Hernan Madrid Padilla, Hua Zhou, Kenneth Lange
Summary: This paper studies the problem of minimizing a loss function subject to constraints and proposes a method that combines the Beltrami-Courant penalty method with the proximal distance principle. The algorithm is driven by a large tuning constant rho and the distance between Dx and S, and convergence and convergence rates are proven. Additionally, a steepest descent variant is constructed to improve the efficiency of the algorithm.
JOURNAL OF MACHINE LEARNING RESEARCH
(2022)
Article
Mathematics, Applied
Min Wang, Qiumei Huang, Cheng Wang
Summary: In this paper, a second order accurate numerical scheme for the square phase field crystal equation is proposed and analyzed, with the introduction of a 4-Laplacian term leading to higher nonlinearity. The scheme is made linear while preserving non-linear energy stability through the use of the scalar auxiliary variable (SAV) approach, with energy stability achieved by introducing an auxiliary variable and constant-coefficient diffusion terms with positive eigenvalues. The proposed method demonstrates efficiency and accuracy through numerical experiments.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Engineering, Electrical & Electronic
Sultan Sial, Aly R. Seadawy, Nauman Raza, Adnan Khan, Ahmad Javid
Summary: Mahavier and Montgomery constructed a Sobolev space for the approximate solution of linear initial value problems using a single-iteration descent method, demonstrating the existence of a best Sobolev gradient for finite difference approximation. They then explored the potential application of single-iteration convergence in an appropriate Sobolev space for a broader class of problems.
OPTICAL AND QUANTUM ELECTRONICS
(2021)
Article
Mathematics, Applied
Min-Li Zeng, Zhong Zheng
Summary: This paper studies efficient algorithms for solving nonlinear saddle point problems and proposes an improved algorithmic framework. The effectiveness of the algorithm is demonstrated through theoretical analysis and numerical experiments.
NUMERICAL ALGORITHMS
(2023)
Article
Computer Science, Information Systems
Jose de Jesus Rubio, Marco Antonio Islas, Genaro Ochoa, David Ricardo Cruz, Enrique Garcia, Jaime Pacheco
Summary: The article proposes a convergent Newton method that combines the Newton method and convergent gradient steepest descent for neural network adaptation, incorporating second-order partial derivatives into time-varying adaptation rates. The method ensures error convergence and minimum finding, with satisfactory results shown in electric energy usage data prediction.
INFORMATION SCIENCES
(2022)
Article
Mathematics, Applied
Min-Li Zeng
Summary: This paper introduces a new accelerated GSOR iteration method for solving large and sparse block two-by-two linear systems of generalized saddle-point structure. Theoretical results on convergence properties and eigenvalues distribution of the preconditioning matrix are studied in detail. Implementations in image restoration and PDE-constraint optimization problems validate the feasibility and efficiency of the new method.
APPLIED NUMERICAL MATHEMATICS
(2021)
Article
Chemistry, Analytical
Chunhua Ren, Dongning Guo, Lu Zhang, Tianhe Wang
Summary: This paper presents an adaptive Fourier series compensation method (AFCM) to improve the output accuracy of MEMS gyroscope under tiny angular velocities. The method improves Fourier series fitting and corrects fitting residuals to reduce nonlinearity. Experiments verify the effectiveness and superiority of the proposed method in gyroscope compensation.
Article
Thermodynamics
Bo Wu, Xing-Bao Gao
Summary: This paper develops a block diagonal preconditioned Uzawa splitting (BDP-US) method for solving saddle point problems and provides a sufficient condition for its convergence. It also proposes a preconditioner based on the BDP-US method, analyzes the spectral properties of the preconditioned matrix, and discusses the choice of parameters for the matrix splitting iteration method. Numerical results support the obtained results and demonstrate the effectiveness of the BDP-US method and the corresponding preconditioner.
ADVANCES IN MECHANICAL ENGINEERING
(2023)
Article
Mathematics
Yu-Jiang Wu, Wei-Hong Zhang, Ai-Li Yang
Summary: By formulating the large sparse linear complementarity problem as implicit fixed-point equations, a modulus-based iteration method is established with the assistance of an inexact non-alternating preconditioned matrix splitting iteration method. The convergence properties of this method are carefully demonstrated under certain conditions, and numerical results validate its superiority over other iteration methods in terms of iteration steps and computing times.
LINEAR & MULTILINEAR ALGEBRA
(2022)
Article
Mathematics, Applied
Zheng Zhou, Bing Tan, Songxiao Li
Summary: In this paper, the split monotone variational inclusion problem is discussed and two new inertial algorithms in infinite-dimensional Hilbert spaces are proposed. The algorithms use a hybrid steepest descent method for convergence and an adaptive step size criterion to avoid the difficulty of calculating the operator norm. The results are also applied to other types of split problems and numerical experiments show the algorithms are realistic and summarize known results.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Mathematics, Applied
Malik Zaka Ullah, Sultan Muaysh Alaslani
Summary: This work investigates an iterative scheme to simultaneously calculate the matrix square root and its inversion, using the concept of matrix sign function. Convergence properties are discussed under certain conditions, and an attempt is made to propose an iterative method with higher convergence order and stability. The extension of the proposed scheme to the pth root of a matrix is also provided. Several tests, including the application of the proposed iterative method to solve matrix differential equations, are presented.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Mathematics, Applied
Kelong Cheng, Cheng Wang, Steven M. Wise
Summary: We propose and analyze a second order accurate in time, energy stable numerical scheme for the strongly anisotropic Cahn-Hilliard system, in which a biharmonic regularization has to be introduced to make the equation well-posed. Convexity analysis on the anisotropic interfacial energy is necessary to overcome the difficulty associated with its highly nonlinear and singular nature. The scheme combines second order backward differentiation formula temporal approximation, Fourier pseudo-spectral spatial discretization, explicit extrapolation formula for updating the nonlinear surface energy, and energy stability enforced by global constant bounding of the second order functional derivatives.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
J. K. Djoko, J. Koko, B. D. Reddy
Summary: This paper focuses on an abstract problem in the form of a variational inequality or a non-differential functional minimization problem. The problem is inspired by the elastoplasticity initial-boundary value problem formulation. The objective is to revisit predictor-corrector algorithms commonly used in computational applications and establish conditions for their convergence or at least for generating decreasing sequences of the functional. Attention is given to various methods such as the tangent predictor, line search approach, steepest descent, and Newton-like method, all of which are shown to result in decreasing sequences.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2023)
Article
Mathematics, Applied
Mei Feng, Xiang Wang, Teng Wang
Summary: In this paper, we propose the Square-Newton method for solving the linear complementary problem (LCP) efficiently. We provide theoretical analysis and numerical experiments to validate the effectiveness of this method.
Article
Mathematics, Applied
Jea-Hyun Park, Abner J. Salgado, Steven M. Wise
Summary: This paper establishes a theoretical foundation for applying Nesterov's accelerated gradient descent method to approximate solutions of a wide class of partial differential equations, showing convergence and existence when the preconditioned version is used. The method is shown to be an explicit time-discretization of a second-order ordinary differential equation, with energy stability requirements. The global convergence and accelerated rate of the PAGD method is demonstrated through numerical experiments.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Physics, Mathematical
Wenbin Chen, Jianyu Jing, Cheng Wang, Xiaoming Wang, Steven M. Wise
Summary: In this paper, a second order accurate numerical scheme for the Cahn-Hilliard equation with logarithmic Flory Huggins energy potential is proposed and analyzed. The scheme includes a modified Crank-Nicolson approximation, explicit second order Adams-Bashforth extrapolation, and a nonlinear artificial regularization term to ensure positivity-preserving property. The numerical scheme demonstrates unconditional energy stability and optimal rate convergence estimate, validated through numerical results.
COMMUNICATIONS IN COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Chun Liu, Cheng Wang, Steven M. Wise, Xingye Yue, Shenggao Zhou
Summary: In this paper, a theoretical analysis is provided for the implementation of a finite difference numerical scheme for the Poisson-Nernst-Planck (PNP) system based on the Energetic Variational Approach (EnVarA). The analysis overcomes the difficulties arising from the nonlinear and singular nature of the logarithmic energy potentials and proposes a modified Newton iteration. A numerical test demonstrates the linear convergence rate of the proposed iteration solver.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Kelong Cheng, Cheng Wang, Steven M. Wise, Yanmei Wu
Summary: In this paper, a backward differentiation formula (BDF) type numerical scheme with third order temporal accuracy for the Cahn-Hilliard equation is proposed and analyzed. The scheme uses the Fourier pseudo-spectral method for space discretization and treats the surface diffusion and nonlinear chemical potential terms implicitly. The expansive term is approximated using a third order explicit extrapolation formula. A third order accurate Douglas-Dupont regularization term is also added in the numerical scheme. Energy stability is derived in a modified version and a uniform bound for the original energy functional is obtained. The coefficient A is theoretically justified. The numerical solution is shown to have a uniform-in-time L-N(6) bound and the optimal convergence rate and error estimate are provided.
NUMERICAL MATHEMATICS-THEORY METHODS AND APPLICATIONS
(2022)
Article
Mathematics, Applied
C. H. U. N. LIU, C. H. E. N. G. WANG, Y. I. W. E. I. WANG, S. T. E. V. E. N. M. WISE
Summary: In this paper, we present a detailed convergence analysis for an operator splitting scheme applied to a reaction-diffusion system with detailed balance. The scheme is based on an energetic variational formulation and ensures energy stability and positivity preservation. By utilizing convexity of nonlinear logarithmic terms and combining rough and refined error estimates, we obtain the convergence estimate of the numerical scheme. The analysis technique can be extended to a more general class of dissipative reaction mechanisms.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2022)
Article
Materials Science, Multidisciplinary
Maik Punke, Steven M. Wise, Axel Voigt, Marco Salvalaglio
Summary: We propose a phase-field crystal model that incorporates thermal transport and a temperature-dependent lattice parameter, and characterizes elasticity effects through the continuous elastic field computed from the microscopic density field. We demonstrate the capabilities of our model through numerical investigations focusing on the growth of two-dimensional crystals from the melt, resulting in faceted shapes and dendrites.
MODELLING AND SIMULATION IN MATERIALS SCIENCE AND ENGINEERING
(2022)
Article
Mathematics, Applied
Lixiu Dong, Cheng Wang, Steven M. Wise, Zhengru Zhang
Summary: This paper rigorously proves the first order convergence in time and second order convergence in space for a fully discrete finite difference scheme for the three-component Macromolecular Microsphere Composite (MMC) hydrogels system. Many non-standard estimates are involved due to the nonlinear and singular nature of the surface diffusion coefficients. This work is the first to provide an optimal rate convergence estimate for a ternary phase field system with singular energy coefficients.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Biology
Xiaoxia Tang, Shuwang Li, John S. S. Lowengrub, Steven M. M. Wise
Summary: We propose a phase field model to describe vesicle growth or shrinkage induced by osmotic pressure. The model consists of an Allen-Cahn equation for the evolution of the phase field parameter and a Cahn-Hilliard-type equation for the evolution of the ionic fluid. We establish conditions for growth or shrinkage using free energy curves and enforce mass conservation and surface area constraint during membrane deformation. Our numerical scheme and multigrid solver demonstrate accuracy and efficiency in evolving the fields to near equilibrium for 2D vesicles. Numerical results confirm the capability of the diffuse interface model in capturing cell shape dynamics.
JOURNAL OF MATHEMATICAL BIOLOGY
(2023)
Article
Thermodynamics
Chang-Ling Lu, Zhao-Fei Hu, Xiao-Rong Kang, Ke-Long Zheng
Summary: Based on the traveling wave reduction method and the F-expansion method, this article obtains a class of explicit exact solutions of the (2+1)-dimensional LGKS equation through symbolic computation. Furthermore, it discusses the interaction behavior between parameters, the perturbation degree of periodic wave and Gauss wave to rational pulse wave, and the correlation of parameters to the superposition degree of the interaction energy between solitary wave and rational pulse wave. Finally, numerical simulations are used to demonstrate the mechanism of the obtained solutions.
Article
Mathematics
Cheng Wang, Steven M. Wise
Summary: This paper introduces a new solidification model with heat flux using the phase field crystal (PFC) framework. The model includes a heat-like equation and a mass-conservation equation, which properly captures the variation in the free energy landscape as the temperature changes. A procedure for constructing a temperature-atom-density phase diagram using this energy landscape is described, along with a simple demonstration of solidification using the model.
JOURNAL OF MATHEMATICAL STUDY
(2022)
Article
Mathematics, Applied
X. I. A. O. C. H. U. N. Chen, C. H. E. N. G. Wang, Steven M. Wise
Summary: This paper provides a detailed analysis of the preconditioned steepest descent (PSD) iteration solver for a convex splitting numerical scheme to the Cahn-Hilliard equation with variable mobility function. The paper presents a method to handle the energy functional, compute the mobility function, and address challenges in the implementation of the numerical scheme. The PSD iteration solver improves the efficiency and stability of the numerical solution for the Cahn-Hilliard equation.
INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING
(2022)
Article
Mathematics, Applied
Maoqin Yuan, Wenbin Chen, Cheng Wang, Steven M. Wise, Zhengru Zhang
Summary: In this paper, a mixed finite element scheme is proposed for the Cahn-Hilliard equation, and the unique solvability and unconditional energy stability of the scheme are proved. The theoretical properties are verified through numerical experiments.
ADVANCES IN APPLIED MATHEMATICS AND MECHANICS
(2022)
Article
Thermodynamics
Jun Zhou, Ke-Long Cheng
Summary: A variant of second order accurate backward differentiation formula schemes for the Cahn-Hilliard equation is proposed in this paper, with a Fourier collocation spectral approximation in space. The introduction of an additional Douglas-Dupont regularization term ensures energy stability with a mild requirement, and various numerical simulations validate the efficiency and robustness of the proposed schemes.
Article
Thermodynamics
Xiao-Rong Kang, Yan-Mei Wu, Ke-Long Cheng
Summary: This paper presents a second order numerical scheme for the Cahn-Hilliard equation with a Fourier pseudo-spectral approximation in space. An additional Douglas-Dupont regularization term is introduced for energy stability. A linear iteration algorithm is proposed to solve the non-linear system, and its efficiency is verified through numerical simulations.