In this paper, standard and economical cascadic multigrid methods are considered for solving the algebraic systems resulting from the mortar finite element methods. Both cascadic multigrid methods do not need full ell...In this paper, standard and economical cascadic multigrid methods are considered for solving the algebraic systems resulting from the mortar finite element methods. Both cascadic multigrid methods do not need full elliptic regularity, so they can be used to tackle more general elliptic problems. Numerical experiments are reported to suonort our theorv.展开更多
Based on an asymptotic expansion of finite element, an extrapolation cascadic multigrid method （EXCMG） is proposed, in which the new extrapolation and quadratic interpolation are used to provide a better initial val...Based on an asymptotic expansion of finite element, an extrapolation cascadic multigrid method （EXCMG） is proposed, in which the new extrapolation and quadratic interpolation are used to provide a better initial value on refined grid. In the case of multiple grids, both superconvergence error in H^1-norm and the optimal error in l2-norm are analyzed. The numerical experiment shows the advantage of EXCMG in comparison with CMG.展开更多
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
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, especiall...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
Based on an asymptotic expansion of finite element,a new extrapolation formula and extrapolation cascadic multigrid method(EXCMG)are proposed,in which the new extrapolation and quadratic interpolation are used to prov...Based on an asymptotic expansion of finite element,a new extrapolation formula and extrapolation cascadic multigrid method(EXCMG)are proposed,in which the new extrapolation and quadratic interpolation are used to provide a better initial value on refined grid.In the case of triple grids,the error of the new initial value is analyzed in detail.A larger scale computation is completed in PC.展开更多
Based on an asymptotic expansion of （bi）linear finite elements, a new extrapolation formula and extrapolation cascadic multigrid method （EXCMG） are proposed. The key ingredients of the proposed methods are some ne...Based on an asymptotic expansion of （bi）linear finite elements, a new extrapolation formula and extrapolation cascadic multigrid method （EXCMG） are proposed. The key ingredients of the proposed methods are some new extrapolations and quadratic interpolations, which are used to provide better initial values on the refined grid. In the case of triple grids, the errors of the new initial values are analyzed in detail. The numerical experiments show that EXCMG has higher accuracy and efficiency.展开更多
In this paper, we develop the cascadic multigrid method for parabolic problems. The optimal convergence accuracy and computation complexity are obtained.
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
A cascadic multigrid method is proposed for eigenvalue problems based on the multilevel correction scheme. With this new scheme, an eigenvalue problem on the finest space can be solved by linear smoothing steps on a s...A cascadic multigrid method is proposed for eigenvalue problems based on the multilevel correction scheme. With this new scheme, an eigenvalue problem on the finest space can be solved by linear smoothing steps on a series of multilevel finite element spaces and nonlinear correcting steps on special coarsest spaces. Once the sequence of finite element spaces and the number of smoothing steps are appropriately chosen, the optimal convergence rate with the optimal computational work can be obtained. Some numerical experiments are presented to validate our theoretical analysis.展开更多
In this paper, we discuss the finite volume element method of P1-nonconforming quadrilateral element for elliptic problems and obtain optimal error estimates for general quadrilateral partition. An optimal cascadic mu...In this paper, we discuss the finite volume element method of P1-nonconforming quadrilateral element for elliptic problems and obtain optimal error estimates for general quadrilateral partition. An optimal cascadic multigrid algorithm is proposed to solve the non-symmetric large-scale system resulting from such discretization. Numerical experiments are reported to support our theoretical results.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
In this paper, we develop a cascadic multigrid algorithm for fast computation of the Fiedler vector of a graph Laplacian, namely, the eigenvector corresponding to the second smallest eigenvalne. This vector has been f...In this paper, we develop a cascadic multigrid algorithm for fast computation of the Fiedler vector of a graph Laplacian, namely, the eigenvector corresponding to the second smallest eigenvalne. This vector has been found to have applications in fields such as graph partitioning and graph drawing. The algorithm is a purely algebraic approach based on a heavy edge coarsening scheme and pointwise smoothing for refinement. To gain theoretical insight, we also consider the related cascadic multigrid method in the geometric setting for elliptic eigenvalue problems and show its uniform convergence under certain assumptions. Numerical tests are presented for computing the Fiedler vector of several practical graphs, and numerical results show the efficiency and optimality of our proposed cascadic multigrid algorithm.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
This paper introduce a cascadic multigrid method for solving semilinear elliptic equations based on a multilevel correction method.Instead of the common costly way of directly solving semilinear elliptic equation on a...This paper introduce a cascadic multigrid method for solving semilinear elliptic equations based on a multilevel correction method.Instead of the common costly way of directly solving semilinear elliptic equation on a very fine space,the new method contains some smoothing steps on a series of multilevel finite element spaces and some solving steps to semilinear elliptic equations on a very coarse space.To prove the efficiency of the new method,we derive two results,one of the optimal convergence rate by choosing the appro- priate sequence of finite element spaces and the number of smoothing steps,and the other of the optimal computational work by applying the parallel computing technique.Moreover,the requirement of bounded second order derivatives of nonlinear term in the existing multigrid methods is reduced to a bounded first order derivative in the new method.Some numerical experiments are presented to validate our theoretical analysis.展开更多
Amyloid-β(Aβ)and tau,the two hallmark proteins associated with Alzheimer’s disease(AD),exhibit distinct toxic effects but also interact synergistically within the disease pathology.The prevailing theory in AD patho...Amyloid-β(Aβ)and tau,the two hallmark proteins associated with Alzheimer’s disease(AD),exhibit distinct toxic effects but also interact synergistically within the disease pathology.The prevailing theory in AD pathology-the amyloid cascade hypothesis-highlights the pivotal role of increased processing of the amyloid precursor protein(APP).Initially cleaved by the majorβ-secretase(β-amyloid cleaving enzyme-1,BACE1)in the brain,then undergoes further cleavage by theγ-secretase complex,resulting in the production of Aβ_(40-42)and a set of intracellular C-terminal peptides known as Aβand APP intracellular domain(β-AICDs)and soluble amyloid precursor proteinβ(sAPPβ)(Orobets and Karamyshev,2023).展开更多
Currently,our understanding of the pathogenesis of major neurodegenerative disorders,such as Alzheimer's,Parkinson's,and Huntington's diseases,is largely shaped by the amyloid cascade hypothesis.Pa rticula...Currently,our understanding of the pathogenesis of major neurodegenerative disorders,such as Alzheimer's,Parkinson's,and Huntington's diseases,is largely shaped by the amyloid cascade hypothesis.Pa rticularly,this hypothesis posits that in Alzheimer's disease,the aggregation of amyloid-beta peptide initiates a series of pathological processes leading to neuronal dysfunction and death(Zhang et al.,2024).展开更多
Deep learning-based methods have become alternatives to traditional numerical weather prediction systems,offering faster computation and the ability to utilize large historical datasets.However,the application of deep...Deep learning-based methods have become alternatives to traditional numerical weather prediction systems,offering faster computation and the ability to utilize large historical datasets.However,the application of deep learning to medium-range regional weather forecasting with limited data remains a significant challenge.In this work,three key solutions are proposed:(1)motivated by the need to improve model performance in data-scarce regional forecasting scenarios,the authors innovatively apply semantic segmentation models,to better capture spatiotemporal features and improve prediction accuracy;(2)recognizing the challenge of overfitting and the inability of traditional noise-based data augmentation methods to effectively enhance model robustness,a novel learnable Gaussian noise mechanism is introduced that allows the model to adaptively optimize perturbations for different locations,ensuring more effective learning;and(3)to address the issue of error accumulation in autoregressive prediction,as well as the challenge of learning difficulty and the lack of intermediate data utilization in one-shot prediction,the authors propose a cascade prediction approach that effectively resolves these problems while significantly improving model forecasting performance.The method achieves a competitive result in The East China Regional AI Medium Range Weather Forecasting Competition.Ablation experiments further validate the effectiveness of each component,highlighting their contributions to enhancing prediction performance.展开更多
基金supported by the National Basic Research Program of China under the grant 2005CB321701the National Science Foundation(NSF) of China(10731060)111 project(B08018)
文摘In this paper, standard and economical cascadic multigrid methods are considered for solving the algebraic systems resulting from the mortar finite element methods. Both cascadic multigrid methods do not need full elliptic regularity, so they can be used to tackle more general elliptic problems. Numerical experiments are reported to suonort our theorv.
基金Supported by National Natural Science Foundation of China (10771063)the Doctor Programme of the National Education Committee (20050542006)
文摘Based on an asymptotic expansion of finite element, an extrapolation cascadic multigrid method （EXCMG） is proposed, in which the new extrapolation and quadratic interpolation are used to provide a better initial value on refined grid. In the case of multiple grids, both superconvergence error in H^1-norm and the optimal error in l2-norm are analyzed. The numerical experiment shows the advantage of EXCMG in comparison with CMG.
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 Natural Science Foundation of China(Grant Nos.10771063,10571053)Doctoral Programme of National Education Ministry of China(Grant No.20050542006)Programme for New Century Excellent Talents in University(Grant No.NCET-060712)
文摘Based on an asymptotic expansion of finite element,a new extrapolation formula and extrapolation cascadic multigrid method(EXCMG)are proposed,in which the new extrapolation and quadratic interpolation are used to provide a better initial value on refined grid.In the case of triple grids,the error of the new initial value is analyzed in detail.A larger scale computation is completed in PC.
基金The research is supported by the National Natural Science Foundation of China （No. 11071067） and the Key Laboratory of Education Ministry.
文摘Based on an asymptotic expansion of （bi）linear finite elements, a new extrapolation formula and extrapolation cascadic multigrid method （EXCMG） are proposed. The key ingredients of the proposed methods are some new extrapolations and quadratic interpolations, which are used to provide better initial values on the refined grid. In the case of triple grids, the errors of the new initial values are analyzed in detail. The numerical experiments show that EXCMG has higher accuracy and efficiency.
基金Subeidized by the Special Funds for Major State Basic Research Projects.
文摘In this paper, we develop the cascadic multigrid method for parabolic problems. The optimal convergence accuracy and computation complexity are obtained.
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Acknowledgments. This work is supported in part by the National Natural Science Foundation of China （NSFC 91330202, 11371026, 11001259, 11031006, 2011CB309703） and the National Center for Mathematics and Interdisciplinary Science, CAS.
文摘A cascadic multigrid method is proposed for eigenvalue problems based on the multilevel correction scheme. With this new scheme, an eigenvalue problem on the finest space can be solved by linear smoothing steps on a series of multilevel finite element spaces and nonlinear correcting steps on special coarsest spaces. Once the sequence of finite element spaces and the number of smoothing steps are appropriately chosen, the optimal convergence rate with the optimal computational work can be obtained. Some numerical experiments are presented to validate our theoretical analysis.
文摘In this paper, we discuss the finite volume element method of P1-nonconforming quadrilateral element for elliptic problems and obtain optimal error estimates for general quadrilateral partition. An optimal cascadic multigrid algorithm is proposed to solve the non-symmetric large-scale system resulting from such discretization. Numerical experiments are reported to support our theoretical results.
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.
文摘In this paper, we develop a cascadic multigrid algorithm for fast computation of the Fiedler vector of a graph Laplacian, namely, the eigenvector corresponding to the second smallest eigenvalne. This vector has been found to have applications in fields such as graph partitioning and graph drawing. The algorithm is a purely algebraic approach based on a heavy edge coarsening scheme and pointwise smoothing for refinement. To gain theoretical insight, we also consider the related cascadic multigrid method in the geometric setting for elliptic eigenvalue problems and show its uniform convergence under certain assumptions. Numerical tests are presented for computing the Fiedler vector of several practical graphs, and numerical results show the efficiency and optimality of our proposed cascadic multigrid algorithm.
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the National Science Foundation of China (NSFC 11401129).
文摘This paper introduce a cascadic multigrid method for solving semilinear elliptic equations based on a multilevel correction method.Instead of the common costly way of directly solving semilinear elliptic equation on a very fine space,the new method contains some smoothing steps on a series of multilevel finite element spaces and some solving steps to semilinear elliptic equations on a very coarse space.To prove the efficiency of the new method,we derive two results,one of the optimal convergence rate by choosing the appro- priate sequence of finite element spaces and the number of smoothing steps,and the other of the optimal computational work by applying the parallel computing technique.Moreover,the requirement of bounded second order derivatives of nonlinear term in the existing multigrid methods is reduced to a bounded first order derivative in the new method.Some numerical experiments are presented to validate our theoretical analysis.
文摘Amyloid-β(Aβ)and tau,the two hallmark proteins associated with Alzheimer’s disease(AD),exhibit distinct toxic effects but also interact synergistically within the disease pathology.The prevailing theory in AD pathology-the amyloid cascade hypothesis-highlights the pivotal role of increased processing of the amyloid precursor protein(APP).Initially cleaved by the majorβ-secretase(β-amyloid cleaving enzyme-1,BACE1)in the brain,then undergoes further cleavage by theγ-secretase complex,resulting in the production of Aβ_(40-42)and a set of intracellular C-terminal peptides known as Aβand APP intracellular domain(β-AICDs)and soluble amyloid precursor proteinβ(sAPPβ)(Orobets and Karamyshev,2023).
基金funded by the Russian Science Foundation(grant No.23-74-10092)(to AIS)。
文摘Currently,our understanding of the pathogenesis of major neurodegenerative disorders,such as Alzheimer's,Parkinson's,and Huntington's diseases,is largely shaped by the amyloid cascade hypothesis.Pa rticularly,this hypothesis posits that in Alzheimer's disease,the aggregation of amyloid-beta peptide initiates a series of pathological processes leading to neuronal dysfunction and death(Zhang et al.,2024).
基金supported by the National Natural Science Foundation of China[grant number 62376217]the Young Elite Scientists Sponsorship Program by CAST[grant number 2023QNRC001]the Joint Research Project for Meteorological Capacity Improvement[grant number 24NLTSZ003]。
文摘Deep learning-based methods have become alternatives to traditional numerical weather prediction systems,offering faster computation and the ability to utilize large historical datasets.However,the application of deep learning to medium-range regional weather forecasting with limited data remains a significant challenge.In this work,three key solutions are proposed:(1)motivated by the need to improve model performance in data-scarce regional forecasting scenarios,the authors innovatively apply semantic segmentation models,to better capture spatiotemporal features and improve prediction accuracy;(2)recognizing the challenge of overfitting and the inability of traditional noise-based data augmentation methods to effectively enhance model robustness,a novel learnable Gaussian noise mechanism is introduced that allows the model to adaptively optimize perturbations for different locations,ensuring more effective learning;and(3)to address the issue of error accumulation in autoregressive prediction,as well as the challenge of learning difficulty and the lack of intermediate data utilization in one-shot prediction,the authors propose a cascade prediction approach that effectively resolves these problems while significantly improving model forecasting performance.The method achieves a competitive result in The East China Regional AI Medium Range Weather Forecasting Competition.Ablation experiments further validate the effectiveness of each component,highlighting their contributions to enhancing prediction performance.
