In this paper a cascadic multigrid algorithm for the mortar finite element approximation of the semilinear elliptic problem is proposed,and corresponding theorems aregiven,which display the error estimate and the comp...In this paper a cascadic multigrid algorithm for the mortar finite element approximation of the semilinear elliptic problem is proposed,and corresponding theorems aregiven,which display the error estimate and the computational complexity of the method.展开更多
In this paper, a new extrapolation economy cascadic multigrid method is proposed to solve the image restoration model. The new method combines the new extrapolation formula and quadratic interpolation to design a nonl...In this paper, a new extrapolation economy cascadic multigrid method is proposed to solve the image restoration model. The new method combines the new extrapolation formula and quadratic interpolation to design a nonlinear prolongation operator, which provides more accurate initial values for the fine grid level. An edge preserving denoising operator is constructed to remove noise and preserve image edges. The local smoothing operator reduces the influence of staircase effect. The experiment results show that the new method not only improves the computational efficiency but also ensures good recovery quality.展开更多
Aiming at providing theoretical basis for effective protection of biodiversity, the study presents a cascade method which combines both qualitative and quantitative methods, incorporates basic data with RS（remote se...Aiming at providing theoretical basis for effective protection of biodiversity, the study presents a cascade method which combines both qualitative and quantitative methods, incorporates basic data with RS（remote sense） technology, and ranks the ecosystems according to its ability of biodiversity sustainability in Hebei Province. The results indicate that the most important areas for protection in Hebei Province are forest and meadow ecosystems in some highlands around Xiaowutai Mountain, Wuling Mountain, North Hebei, Taihang Mountain and East Hebei; grass ecosystems in part of plateau area and North Hebei; and some scattered wetlands in the plain and inshore areas. This method is suitable for undertaking large-scale investigations especially when the data are not adequate or unevenly distributed spatially.展开更多
We propose the modulus-based cascadic multigrid(MCMG)method and the modulus-based economical cascadic multigrid method for solving the quasi-variational inequalities problem.The modulus-based matrix splitting iterativ...We propose the modulus-based cascadic multigrid(MCMG)method and the modulus-based economical cascadic multigrid method for solving the quasi-variational inequalities problem.The modulus-based matrix splitting iterative method is adopted as a smoother,which can accelerate the convergence of the new methods.We also give the convergence analysis of these methods.Finally,some numerical experiments confirm the theoretical analysis and show that the new methods can achieve high efficiency and lower costs simultaneously.展开更多
In this paper,an economical cascadic multigrid method is proposed.Compared with the usual cascadic multigrid method developed by Bornemann and Deuflhard,the new one requires less iterations on each level,especially on...In this paper,an economical cascadic multigrid method is proposed.Compared with the usual cascadic multigrid method developed by Bornemann and Deuflhard,the new one requires less iterations on each level,especially on the coarser grids.Many operations can be saved in the new cascadic multigrid algorithms.The main ingredient is the control of the iteration numbers on the each level to preserve the accuracy without over iterations.The theoretical justification is based on the observations that the error reduction rate of an iteration scheme in terms of the smoothing property is no longer accurate while the iteration number is big enough.A new formulae of the error reduction rate is employed in our new algorithm.Numerical experiments are reported to support our theory.展开更多
In this paper,we consider the cascadic multigrid method for a parabolic type equation.Backward Euler approximation in time and linear finite element approximation in space are employed.A stability result is establishe...In this paper,we consider the cascadic multigrid method for a parabolic type equation.Backward Euler approximation in time and linear finite element approximation in space are employed.A stability result is established under some conditions on the smoother.Using new and sharper estimates for the smoothers that reflect the precise dependence on the time step and the spatial mesh parameter,these conditions are verified for a number of popular smoothers.Optimal error bound sare derived for both smooth and non-smooth data.Iteration strategies guaranteeing both the optimal accuracy and the optimal complexity are presented.展开更多
In this paper, we consider the cascadic multigrid method for the mortar P1 nonconforming element which is used to solve the Poisson equation and prove that the cascadic conjugate gradient method is accurate with optim...In this paper, we consider the cascadic multigrid method for the mortar P1 nonconforming element which is used to solve the Poisson equation and prove that the cascadic conjugate gradient method is accurate with optimal complexity.展开更多
For the Poisson equation with Robin boundary conditions,by using a few techniques such as orthogonal expansion(M-type),separation of the main part and the finite element projection,we prove for the first time that the...For the Poisson equation with Robin boundary conditions,by using a few techniques such as orthogonal expansion(M-type),separation of the main part and the finite element projection,we prove for the first time that the asymptotic error expansions of bilinear finite element have the accuracy of O(h3)for u∈H3.Based on the obtained asymptotic error expansions for linear finite elements,extrapolation cascadic multigrid method(EXCMG)can be used to solve Robin problems effectively.Furthermore,by virtue of Richardson not only the accuracy of the approximation is improved,but also a posteriori error estimation is obtained.Finally,some numerical experiments that confirm the theoretical analysis are presented.展开更多
The aim of this paper is to develop a fast multigrid solver for interpolation-free finite volume (FV) discretization of anisotropic elliptic interface problems on general bounded domains that can be described as a uni...The aim of this paper is to develop a fast multigrid solver for interpolation-free finite volume (FV) discretization of anisotropic elliptic interface problems on general bounded domains that can be described as a union of blocks. We assume that the curved interface falls exactly on the boundaries of blocks. The transfinite interpolation technique is applied to generate block-wise distorted quadrilateral meshes, which can resolve the interface with fine geometric details. By an extensive study of the harmonic average point method, an interpolation-free nine-point FV scheme is then derived on such multi-block grids for anisotropic elliptic interface problems with non-homogeneous jump conditions. Moreover, for the resulting linear algebraic systems from cell-centered FV discretization, a high-order prolongation operator based fast cascadic multigrid solver is developed and shown to be robust with respect to both the problem size and the jump of the diffusion coefficients. Various non-trivial examples including four interface problems and an elliptic problem in complex domain without interface, all with tens of millions of unknowns, are provided to show that the proposed multigrid solver is dozens of times faster than the classical algebraic multigrid method as implemented in the code AMG1R5 by Stüben.展开更多
In this paper, a type of accurate a posteriori error estimator is proposed for the Steklov eigenvalue problem based on the complementary approach, which provides an asymptotic exact estimate for the approximate eigenp...In this paper, a type of accurate a posteriori error estimator is proposed for the Steklov eigenvalue problem based on the complementary approach, which provides an asymptotic exact estimate for the approximate eigenpair. Besides, we design a type of cascadic adaptive finite element method for the Steklov eigenvalue problem based on the proposed a posteriori error estimator. In this new cascadic adaptive scheme, instead of solving the Steklov eigenvalue problem in each adaptive space directly, we only need to do some smoothing steps for linearized boundary value problems on a series of adaptive spaces and solve some Steklov eigenvalue problems on a low dimensional space. Furthermore, the proposed a posteriori error estimator provides the way to refine meshes and control the number of smoothing steps for the cascadic adaptive method. Some numerical examples are presented to validate the efficiency of the algorithm in this paper.展开更多
In response to the limitedworking capacity and poor energy interaction efficiency of microgrids,it is now common to use multiple microgrids to form microgrid clusters to enhance the reliability of power supply between...In response to the limitedworking capacity and poor energy interaction efficiency of microgrids,it is now common to use multiple microgrids to form microgrid clusters to enhance the reliability of power supply between each other and further improve the penetration rate of distributed power sources.This article focuses on the energy development method of microgrid groups and the problem of scheduling optimization of integrated energy system is discussed for hot and cold electricity within the microgrid group.Firstly,the basic composition ideas of the electric heating interconnection system,electrical interconnection system,and cold and hot electrical interconnection system within the microgrid group were designed layer by layer,and the functional characteristics of the energy equipment within the microgrid group were clarified.Then,in response to the impact of wind and solar uncertainty on system operation,the goal cascading method and robust stochastic optimization method are used to gradually construct a coupled scheduling model for electric heating,a multi-objective scheduling model for electrical interconnection,and a coordinated scheduling optimization model for cold and hot electrical interconnection.Finally,the effectiveness and applicability of the proposed model were verified through case analysis of the relevant models mentioned above,and key factors were selected for sensitivity analysis,providing reliable methodological support for optimizing the operation of an integrated energy system.展开更多
基金supported by Educational Commission of Guangdong Province,China(No.2012LYM-0066)the National Social Science Foundation of China(No.14CJL016)
文摘In this paper a cascadic multigrid algorithm for the mortar finite element approximation of the semilinear elliptic problem is proposed,and corresponding theorems aregiven,which display the error estimate and the computational complexity of the method.
文摘In this paper, a new extrapolation economy cascadic multigrid method is proposed to solve the image restoration model. The new method combines the new extrapolation formula and quadratic interpolation to design a nonlinear prolongation operator, which provides more accurate initial values for the fine grid level. An edge preserving denoising operator is constructed to remove noise and preserve image edges. The local smoothing operator reduces the influence of staircase effect. The experiment results show that the new method not only improves the computational efficiency but also ensures good recovery quality.
基金the National Natural Science Foundation of China (30590381)
文摘Aiming at providing theoretical basis for effective protection of biodiversity, the study presents a cascade method which combines both qualitative and quantitative methods, incorporates basic data with RS（remote sense） technology, and ranks the ecosystems according to its ability of biodiversity sustainability in Hebei Province. The results indicate that the most important areas for protection in Hebei Province are forest and meadow ecosystems in some highlands around Xiaowutai Mountain, Wuling Mountain, North Hebei, Taihang Mountain and East Hebei; grass ecosystems in part of plateau area and North Hebei; and some scattered wetlands in the plain and inshore areas. This method is suitable for undertaking large-scale investigations especially when the data are not adequate or unevenly distributed spatially.
基金National Natural Science Foundation of China(12161027)Guangxi Natural Science Foundation,China(2020GXNSFAA159143)Science and Technology Project of Guangxi,China(AD23023002).
文摘We propose the modulus-based cascadic multigrid(MCMG)method and the modulus-based economical cascadic multigrid method for solving the quasi-variational inequalities problem.The modulus-based matrix splitting iterative method is adopted as a smoother,which can accelerate the convergence of the new methods.We also give the convergence analysis of these methods.Finally,some numerical experiments confirm the theoretical analysis and show that the new methods can achieve high efficiency and lower costs simultaneously.
基金This work was supported by the National Basic Research Program of China(Grant No.2005CB321701)the Research Found for the Doctoral Program of Higher Education
文摘In this paper,an economical cascadic multigrid method is proposed.Compared with the usual cascadic multigrid method developed by Bornemann and Deuflhard,the new one requires less iterations on each level,especially on the coarser grids.Many operations can be saved in the new cascadic multigrid algorithms.The main ingredient is the control of the iteration numbers on the each level to preserve the accuracy without over iterations.The theoretical justification is based on the observations that the error reduction rate of an iteration scheme in terms of the smoothing property is no longer accurate while the iteration number is big enough.A new formulae of the error reduction rate is employed in our new algorithm.Numerical experiments are reported to support our theory.
基金the National Science Foundation(Grant Nos.DMS0409297,DMR0205232,CCF-0430349)US National Institute of Health-National Cancer Institute(Grant No.1R01CA125707-01A1)+2 种基金the National Natural Science Foundation of China(Grant No.10571172)the National Basic Research Program(Grant No.2005CB321704)the Youth's Innovative Program of Chinese Academy of Sciences(Grant Nos.K7290312G9,K7502712F9)
文摘In this paper,we consider the cascadic multigrid method for a parabolic type equation.Backward Euler approximation in time and linear finite element approximation in space are employed.A stability result is established under some conditions on the smoother.Using new and sharper estimates for the smoothers that reflect the precise dependence on the time step and the spatial mesh parameter,these conditions are verified for a number of popular smoothers.Optimal error bound sare derived for both smooth and non-smooth data.Iteration strategies guaranteeing both the optimal accuracy and the optimal complexity are presented.
基金Supported by the National Natural Science Foundation of China under grant 10071015.
文摘In this paper, we consider the cascadic multigrid method for the mortar P1 nonconforming element which is used to solve the Poisson equation and prove that the cascadic conjugate gradient method is accurate with optimal complexity.
基金supported by National Natural Science Foundation of China(Grant Nos.11226332,41204082 and 11071067)the China Postdoctoral Science Foundation(Grant No.2011M501295)+1 种基金the Research Fund for the Doctoral Program of Higher Education of China(Grant No.20120162120036)the Construct Program of the Key Discipline in Hunan Province
文摘For the Poisson equation with Robin boundary conditions,by using a few techniques such as orthogonal expansion(M-type),separation of the main part and the finite element projection,we prove for the first time that the asymptotic error expansions of bilinear finite element have the accuracy of O(h3)for u∈H3.Based on the obtained asymptotic error expansions for linear finite elements,extrapolation cascadic multigrid method(EXCMG)can be used to solve Robin problems effectively.Furthermore,by virtue of Richardson not only the accuracy of the approximation is improved,but also a posteriori error estimation is obtained.Finally,some numerical experiments that confirm the theoretical analysis are presented.
基金supported by the National Natural Science Foundation of China(Grant No.42274101)X.X.Wu was supported by the Fundamental Research Funds for the Central Universities of Central South University(Grant No.2020zzts354)+2 种基金H.L.Hu was supported by the National Natural Science Foundation of China(Grant No.12071128)by the Natural Science Foundation of Hunan Province(Grant No.2021JJ30434)Z.L.Li was supported by a Simons Grant No.633724.
文摘The aim of this paper is to develop a fast multigrid solver for interpolation-free finite volume (FV) discretization of anisotropic elliptic interface problems on general bounded domains that can be described as a union of blocks. We assume that the curved interface falls exactly on the boundaries of blocks. The transfinite interpolation technique is applied to generate block-wise distorted quadrilateral meshes, which can resolve the interface with fine geometric details. By an extensive study of the harmonic average point method, an interpolation-free nine-point FV scheme is then derived on such multi-block grids for anisotropic elliptic interface problems with non-homogeneous jump conditions. Moreover, for the resulting linear algebraic systems from cell-centered FV discretization, a high-order prolongation operator based fast cascadic multigrid solver is developed and shown to be robust with respect to both the problem size and the jump of the diffusion coefficients. Various non-trivial examples including four interface problems and an elliptic problem in complex domain without interface, all with tens of millions of unknowns, are provided to show that the proposed multigrid solver is dozens of times faster than the classical algebraic multigrid method as implemented in the code AMG1R5 by Stüben.
基金supported by National Natural Science Foundation of China(Grant Nos.11801021 and 11571027)Foundation for Fundamental Research of Beijing University of Technology(Grant No.006000546318504)International Research Cooperation Seed Fund of Beijing University of Technology(Grant No.2018B32)。
文摘In this paper, a type of accurate a posteriori error estimator is proposed for the Steklov eigenvalue problem based on the complementary approach, which provides an asymptotic exact estimate for the approximate eigenpair. Besides, we design a type of cascadic adaptive finite element method for the Steklov eigenvalue problem based on the proposed a posteriori error estimator. In this new cascadic adaptive scheme, instead of solving the Steklov eigenvalue problem in each adaptive space directly, we only need to do some smoothing steps for linearized boundary value problems on a series of adaptive spaces and solve some Steklov eigenvalue problems on a low dimensional space. Furthermore, the proposed a posteriori error estimator provides the way to refine meshes and control the number of smoothing steps for the cascadic adaptive method. Some numerical examples are presented to validate the efficiency of the algorithm in this paper.
文摘In response to the limitedworking capacity and poor energy interaction efficiency of microgrids,it is now common to use multiple microgrids to form microgrid clusters to enhance the reliability of power supply between each other and further improve the penetration rate of distributed power sources.This article focuses on the energy development method of microgrid groups and the problem of scheduling optimization of integrated energy system is discussed for hot and cold electricity within the microgrid group.Firstly,the basic composition ideas of the electric heating interconnection system,electrical interconnection system,and cold and hot electrical interconnection system within the microgrid group were designed layer by layer,and the functional characteristics of the energy equipment within the microgrid group were clarified.Then,in response to the impact of wind and solar uncertainty on system operation,the goal cascading method and robust stochastic optimization method are used to gradually construct a coupled scheduling model for electric heating,a multi-objective scheduling model for electrical interconnection,and a coordinated scheduling optimization model for cold and hot electrical interconnection.Finally,the effectiveness and applicability of the proposed model were verified through case analysis of the relevant models mentioned above,and key factors were selected for sensitivity analysis,providing reliable methodological support for optimizing the operation of an integrated energy system.
