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Analysis of a two-grid method for semiconductor device problem

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Applied Mathematics and Mechanics(English Edition) 2021年第1期42卷 143-158页

作者： Ying LIU Yanping CHEN Yunqing HUANG Qingfeng LI School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and EngineeringXiangtan UniversityXiangtan 411105Hunan ProvinceChina College of Information and Intelligence Hunan Agricultural UniversityChangsha 410128China School of Mathematical Sciences South China Normal UniversityGuangzhou 510631China

The mathematical model of a semiconductor device is governed by a system of quasi-linear partial differential *** electric potential equation is approximated by a mixed finite element method,and the concentration equations are approximated by a standard Galerkin *** estimate the error of the numerical solutions in the sense of the *** linearize the full discrete scheme of the problem,we present an efficient two-grid method based on the idea of Newton *** main procedures are to solve the small scaled nonlinear equations on the coarse grid and then deal with the linear equations on the fine *** estimation for the two-grid solutions is analyzed in *** is shown that this method still achieves asymptotically optimal approximations as long as a mesh size satisfies H=O(h^1/2).Numerical experiments are given to illustrate the efficiency of the two-grid method.

关键词： two-grid method semiconductor device mixed finite element method Galerkin method L^q error estimate

New Upper Solution Bounds of a Class of the Continuous Algebraic Riccati Equations and Their Application in Redundant Control Inputs Systems

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SSRN

SSRN 2023年

作者： Liu, Jianzhou Tang, Fuying School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan Xiangtan411105 China

In the redundant control inputs problems, the norm of the feedback matrix is smaller, the controlled system is better for control. Therefore, it is the key to solve these problems for studying upper and lower bounds of the symmetric positive definite solutions of the continuous algebra Riccati equations(CAREs) corresponding to the controlled systems. In this paper, we obtain a convergent matrix series as upper bound of the symmetric positive definite solution for a special class of the CAREs. It is the limit of monotonically decreasing upper bound sequence of the symmetric positive definite solution. We use numerous numerical experiments to illustrate that upper bounds obtained are better than the relevant results in the existing literature when the terms are sufficiently large. Moreover, we apply the main theoretical results to the related redundant control inputs systems. Under a wider range of conditions, a class of redundant control inputs problems is solved, and the recent relevant results are improved. © 2023, The Authors. All rights reserved.

关键词： Riccati equations

Low-rank alternating direction doubling algorithm for solving large-scale continuous time algebraic Riccati equations

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arXiv 2024年

作者： Zhang, Juan Xun, Wenlu Key Laboratory of Intelligent Computing and Information Processing Ministry of Education Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan China School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan China

This paper proposes an effective low-rank alternating direction doubling algorithm (R-ADDA) for computing numerical low-rank solutions to large-scale sparse continuous-time algebraic Riccati matrix equations. The method is based on the alternating direction doubling algorithm (ADDA), utilizing the low-rank property of matrices and employing Cholesky factorization for solving. The advantage of the new algorithm lies in computing only the 2k-th approximation during the iterative process, instead of every approximation. Its efficient low-rank formula saves storage space and is highly effective from a computational perspective. Finally, the effectiveness of the new algorithm is demonstrated through theoretical analysis and numerical experiments. Copyright © 2024, The Authors. All rights reserved.

关键词： Matrix algebra

Low-rank generalized alternating direction implicit iteration method for solving matrix equations

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arXiv 2024年

作者： Zhang, Juan Xun, Wenlu Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan China School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan China

This paper presents an effective low-rank generalized alternating direction implicit iteration (R-GADI) method for solving large-scale sparse and stable Lyapunov matrix equations and continuous-time algebraic Riccati matrix equations. The method is based on generalized alternating direction implicit iteration (GADI), which exploits the low-rank property of matrices and utilizes the Cholesky factorization approach for solving. The advantage of the new algorithm lies in its direct and efficient low-rank formulation, which is a variant of the Cholesky decomposition in the Lyapunov GADI method, saving storage space and making it computationally effective. When solving the continuous-time algebraic Riccati matrix equation, the Riccati equation is first simplified to a Lyapunov equation using the Newton method, and then the R-GADI method is employed for computation. Additionally, we analyze the convergence of the R-GADI method and prove its consistency with the convergence of the GADI method. Finally, the effectiveness of the new algorithm is demonstrated through corresponding numerical experiments. Copyright © 2024, The Authors. All rights reserved.

关键词： Riccati equations

Optimization methods for solving matrix equations

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arXiv 2024年

作者： Zhang, Juan Luo, Xiao Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan China School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan China

In this paper, we focus on using optimization methods to solve matrix equations by transforming the problem of solving the Sylvester matrix equation or continuous algebraic Riccati equation into an optimization problem. Initially, we use a constrained convex optimization method (CCOM) to solve the Sylvester matrix equation with 2,1-norm, where we provide a convergence analysis and numerical examples of CCOM;however, the results show that the algorithm is not efficient. To address this issue, we employ classical quasi-Newton methods such as DFP and BFGS algorithms to solve the Sylvester matrix equation and present the convergence and numerical results of the algorithm. Additionally, we compare these algorithms with the CG algorithm and AR algorithm, and our results demonstrate that the presented algorithms are effective. Furthermore, we propose a unified framework of the alternating direction multiplier method (ADMM) for directly solving the continuous algebraic Riccati equation (CARE), and we provide the convergence and numerical results of ADMM. Our experimental results indicate that ADMM is an effective optimization algorithm for solving CARE. Finally, to improve the effectiveness of the optimization method for solving Riccati equation, we propose the Newton-ADMM algorithm framework, where the outer iteration of this method is the classical Newton method, and the inner iteration involves using ADMM to solve Lyapunov matrix equations inexactly. We also provide the convergence and numerical results of this algorithm, which our results demonstrate are more efficient than ADMM for solving CARE. Copyright © 2024, The Authors. All rights reserved.

关键词： Newton-Raphson method

A general alternating direction implicit iteration method for solving complex symmetric linear systems

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arXiv 2024年

We have introduced the generalized alternating direction implicit iteration (GADI) method for solving large sparse complex symmetric linear systems and proved its convergence properties. Additionally, some numerical results have demonstrated the effectiveness of this algorithm. Furthermore, as an application of the GADI method in solving complex symmetric linear systems, we utilized the flattening operator and Kronecker product properties to solve Lyapunov and Riccati equations with complex coefficients using the GADI method. In solving the Riccati equation, we combined inner and outer iterations, first simplifying the Riccati equation into a Lyapunov equation using the Newton method, and then applying the GADI method for solution. Finally, we provided convergence analysis of the method and corresponding numerical results. Copyright © 2024, The Authors. All rights reserved.

关键词： Riccati equations

The grad-div conforming virtual element method for the quad-div problem in three dimensions

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arXiv 2024年

作者： Dong, Xiaojing Han, Yibing Huang, Yunqing Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing & Information Processing of Ministry of Education School of Mathematics and Computational Science Xiangtan University Er huan Road Hunan Xiangtan411105 China

We propose a new stable variational formulation for the quad-div problem in three dimensions and prove its well-posedness. Using this weak form, we develop and analyze the H(grad-div)-conforming virtual element method of arbitrary approximation orders on polyhedral meshes. Three families of H(grad-div)conforming virtual elements are constructed based on the structure of a de Rham sub-complex with enhanced smoothness, resulting in an exact discrete virtual element complex. In the lowest-order case, the simplest element has only one degree of freedom at each vertex and face, respectively. We rigorously prove the interpolation error estimates, the stability of discrete bilinear forms, the well-posedness of discrete formulation and the optimal error estimates. Some numerical examples are shown to verify the theoretical results. © 2024, CC BY-NC-ND.

关键词： Mesh generation

THREE-PRECISION ITERATIVE REFINEMENT WITH PARAMETER REGULARIZATION AND PREDICTION FOR SOLVING LARGE SPARSE LINEAR SYSTEMS

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arXiv 2025年

作者： Ge, Jifeng Zhang, Juan Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Hunan KeyLaboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China

This study presents a novel mixed-precision iterative refinement algorithm, GADI-IR, within the general alternating-direction implicit (GADI) framework, designed for efficiently solving large-scale sparse linear systems. By employing low-precision arithmetic, particularly half-precision (FP16), for computationally intensive inner iterations, the method achieves substantial acceleration while maintaining high numerical accuracy. key challenges such as overflow in half-precision and convergence issues for low precision are addressed through careful backward error analysis and the application of a regularization parameter α. Furthermore, the integration of low-precision arithmetic into the parameter prediction process, using Gaussian Process Regression (GPR), significantly reduces computational time without degrading performance. The method is particularly effective for large-scale linear systems arising from discretized partial differential equations and other high-dimensional problems, where both accuracy and efficiency are critical. Numerical experiments demonstrate that the use of mixed-precision strategies not only accelerates computation but also ensures robust convergence, making the approach advantageous for various applications. The results highlight the potential of leveraging lower-precision arithmetic to achieve superior computational efficiency in high-performance computing. © 2025, CC BY.

关键词： Convergence of numerical methods

Micro-nanobubble–assisted As(III) removal from water by Ni-doped MOF materials

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Environmental science and Pollution Research 2024年第31期31卷 43913-43926页

作者： Ye, Jian Zuo, Yize Chen, Qiang Yang, Zhiming Liu, Shaobo Yang, Chunping Tan, Xiaofei Hunan University Changsha410082 China State Key Joint Laboratory of Environment Simulation and Pollution Control Research Center for Eco-Environmental Sciences Chinese Academy of Sciences Beijing100085 China Shenzhen Research Institute of Hunan University Shenzhen518055 China School of Architecture and Art Central South University Changsha410083 China Academy of Environmental and Resource Sciences School of Environmental Science and Engineering Guangdong University of Petrochemical Technology Guangdong Maoming525000 China Hunan Chuke Taiyan New Materials Co. Ltd. Jishou416000 China

Micro-nanobubbles (MNBs) can form reactive oxygen species (ROS) with high oxidizing potential. In this study, nickel-doped metal–organic framework materials (MOFs) capable of activating molecular oxygen were synthesized using trivalent arsenic (As(III)) as a target pollutant and combined with peroxymonosulfate (PMS) to construct a MOF/MNB/PMS system. The results included the rapid oxidation of As(III), the successful absorption of oxidized As(V), and finally the efficient removal of As. The effects of pH, amount of PMS used, and preparation time of MNBs on the As removal performance of the MOF/MNB/PMS system were investigated experimentally. The changes in the properties of the materials before and after the reaction were analyzed by XPS, and it was found that the main active sites on the surface of the MOFs were the metal elements and the pyridine nitrogen near the carbon atom. The regular morphology and elemental composition of the MOFs were determined by TEM scanning and EDS test, which indicated the presence of nickel. XRD tests before and after the reaction showed that the MOFs were structurally stable. The results of the free radical burst experiments show that the single linear oxygen (1O2) is the main active substance in the system, and that the MNBs are key factors by which the system achieves efficient oxidation performance. In addition to providing a sustainable supply of molecular oxygen to the MOFs during the reaction process, coupling the MNBs with PMS was found to improve the oxidation capacity of the system. The results of this study thus provide a new concept for As removal and advanced oxidation in water bodies. © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2024.

关键词： Molecular oxygen