An improved learning algorithm for hyperball CMAC was presented. Only one parameter is needed to determine the learning rate, and the parameter can be obtained by a self-optimizing method. The convergence of the impro...An improved learning algorithm for hyperball CMAC was presented. Only one parameter is needed to determine the learning rate, and the parameter can be obtained by a self-optimizing method. The convergence of the improved learning algorithm was proved. The simulation research shows that the learning speed and the learning accuracy are both improved.展开更多
A moving collocation method has been shown to be very effcient for the adaptive solution of second- and fourth-order time-dependent partial differential equations and forms the basis for the two robust codes MOVCOL an...A moving collocation method has been shown to be very effcient for the adaptive solution of second- and fourth-order time-dependent partial differential equations and forms the basis for the two robust codes MOVCOL and MOVCOL4. In this paper, the relations between the method and the traditional collocation and finite volume methods are investigated. It is shown that the moving collocation method inherits desirable properties of both methods： the ease of implementation and high-order convergence of the traditional collocation method and the mass conservation of the finite volume method. Convergence of the method in the maximum norm is proven for general linear two-point boundary value problems. Numerical results are given to demonstrate the convergence order of the method.展开更多
This paper deals with the solvability and the convergence of a class of unsymmetric Meshless Local Petrov-Galerkin(MLPG)method with radial basis function(RBF)kernels generated trial spaces.Local weak-form testings are...This paper deals with the solvability and the convergence of a class of unsymmetric Meshless Local Petrov-Galerkin(MLPG)method with radial basis function(RBF)kernels generated trial spaces.Local weak-form testings are done with stepfunctions.It is proved that subject to sufficiently many appropriate testings,solvability of the unsymmetric RBF-MLPG resultant systems can be guaranteed.Moreover,an error analysis shows that this numerical approximation converges at the same rate as found in RBF interpolation.Numerical results(in double precision)give good agreement with the provided theory.展开更多
We analyze a least-squares asymmetric radial basis function collocation method for solving the modified Helmholtz equations.In the theoretical part,we proved the convergence of the proposed method providing that the c...We analyze a least-squares asymmetric radial basis function collocation method for solving the modified Helmholtz equations.In the theoretical part,we proved the convergence of the proposed method providing that the collocation points are sufficiently dense.For numerical verification,direct solver and a subspace selection process for the trial space(the so-called adaptive greedy algorithm)is employed,respectively,for small and large scale problems.展开更多
We propose a multiscale multilevel Monte Carlo(MsMLMC)method to solve multiscale elliptic PDEs with random coefficients in the multi-query setting.Our method consists of offline and online stages.In the offline stage,...We propose a multiscale multilevel Monte Carlo(MsMLMC)method to solve multiscale elliptic PDEs with random coefficients in the multi-query setting.Our method consists of offline and online stages.In the offline stage,we construct a small number of reduced basis functions within each coarse grid block,which can then be used to approximate the multiscale finite element basis functions.In the online stage,we can obtain the multiscale finite element basis very efficiently on a coarse grid by using the pre-computed multiscale basis.The MsMLMC method can be applied to multiscale RPDE starting with a relatively coarse grid,without requiring the coarsest grid to resolve the smallestscale of the solution.We have performed complexity analysis and shown that the MsMLMC offers considerable savings in solving multiscale elliptic PDEs with random coefficients.Moreover,we provide convergence analysis of the proposed method.Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several multiscale stochastic problems without scale separation.展开更多
Pointwise convergence and uniform convergence for wavelet frame series is a new topic. With the help of band-limited dual wavelet frames, this topic is first researched.
基金National High Technology Research andDevelopment Program of China ( Project 863 ,G2 0 0 1AA413 13 0 )
文摘An improved learning algorithm for hyperball CMAC was presented. Only one parameter is needed to determine the learning rate, and the parameter can be obtained by a self-optimizing method. The convergence of the improved learning algorithm was proved. The simulation research shows that the learning speed and the learning accuracy are both improved.
基金supported by Natural Sciences and Engineering Research Council of Canada (Grant No. A8781)National Natural Science Foundation of China (Grant No. 11171274)National Science Foundation of USA (Grant No. DMS-0712935)
文摘A moving collocation method has been shown to be very effcient for the adaptive solution of second- and fourth-order time-dependent partial differential equations and forms the basis for the two robust codes MOVCOL and MOVCOL4. In this paper, the relations between the method and the traditional collocation and finite volume methods are investigated. It is shown that the moving collocation method inherits desirable properties of both methods： the ease of implementation and high-order convergence of the traditional collocation method and the mass conservation of the finite volume method. Convergence of the method in the maximum norm is proven for general linear two-point boundary value problems. Numerical results are given to demonstrate the convergence order of the method.
基金supported by the CERG Grant of the Hong Kong Research Grant Council and the FRG Grant of the Hong Kong Baptist University.
文摘This paper deals with the solvability and the convergence of a class of unsymmetric Meshless Local Petrov-Galerkin(MLPG)method with radial basis function(RBF)kernels generated trial spaces.Local weak-form testings are done with stepfunctions.It is proved that subject to sufficiently many appropriate testings,solvability of the unsymmetric RBF-MLPG resultant systems can be guaranteed.Moreover,an error analysis shows that this numerical approximation converges at the same rate as found in RBF interpolation.Numerical results(in double precision)give good agreement with the provided theory.
基金supported by CERG Grants of Hong Kong Research Grant CouncilFRG grants of Hong Kong Baptist University.
文摘We analyze a least-squares asymmetric radial basis function collocation method for solving the modified Helmholtz equations.In the theoretical part,we proved the convergence of the proposed method providing that the collocation points are sufficiently dense.For numerical verification,direct solver and a subspace selection process for the trial space(the so-called adaptive greedy algorithm)is employed,respectively,for small and large scale problems.
基金partially supported by the Hong Kong Ph D Fellowship Schemesupported by the Hong Kong RGC General Research Funds(Projects 27300616,17300817,and 17300318)+2 种基金National Natural Science Foundation of China(Project 11601457)Seed Funding Programme for Basic Research(HKU)Basic Research Programme(JCYJ20180307151603959)of the Science,Technology and Innovation Commission of Shenzhen Municipality。
文摘We propose a multiscale multilevel Monte Carlo(MsMLMC)method to solve multiscale elliptic PDEs with random coefficients in the multi-query setting.Our method consists of offline and online stages.In the offline stage,we construct a small number of reduced basis functions within each coarse grid block,which can then be used to approximate the multiscale finite element basis functions.In the online stage,we can obtain the multiscale finite element basis very efficiently on a coarse grid by using the pre-computed multiscale basis.The MsMLMC method can be applied to multiscale RPDE starting with a relatively coarse grid,without requiring the coarsest grid to resolve the smallestscale of the solution.We have performed complexity analysis and shown that the MsMLMC offers considerable savings in solving multiscale elliptic PDEs with random coefficients.Moreover,we provide convergence analysis of the proposed method.Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several multiscale stochastic problems without scale separation.
文摘Pointwise convergence and uniform convergence for wavelet frame series is a new topic. With the help of band-limited dual wavelet frames, this topic is first researched.
