In this paper, we propose a globally convergent Polak-Ribiere-Polyak (PRP) conjugate gradient method for nonconvex minimization of differentiable functions by employing an Armijo-type line search which is simpler and ...In this paper, we propose a globally convergent Polak-Ribiere-Polyak (PRP) conjugate gradient method for nonconvex minimization of differentiable functions by employing an Armijo-type line search which is simpler and less demanding than those defined in [4,10]. A favorite property of this method is that we can choose the initial stepsize as the one-dimensional minimizer of a quadratic modelΦ(t):= f(xk)+tgkTdk+(1/2) t2dkTQkdk, where Qk is a positive definite matrix that carries some second order information of the objective function f. So, this line search may make the stepsize tk more easily accepted. Preliminary numerical results show that this method is efficient.展开更多
We introduce multilevel augmentation methods for solving operator equations based on direct sum decompositions of the range space of the operator and the solution space of the operator equation and a matrix splitting ...We introduce multilevel augmentation methods for solving operator equations based on direct sum decompositions of the range space of the operator and the solution space of the operator equation and a matrix splitting scheme. We establish a general setting for the analysis of these methods, showing that the methods yield approximate solutions of the same convergence order as the best approximation from the subspace. These augmentation methods allow us to develop fast, accurate and stable nonconventional numerical algorithms for solving operator equations. In particular, for second kind equations, special splitting techniques are proposed to develop such algorithms. These algorithms are then applied to solve the linear systems resulting from matrix compression schemes using wavelet-like functions for solving Fredholm integral equations of the second kind. For this special case, a complete analysis for computational complexity and convergence order is presented. Numerical examples are included to demonstrate the efficiency and accuracy of the methods. In these examples we use the proposed augmentation method to solve large scale linear systems resulting from the recently developed wavelet Galerkin methods and fast collocation methods applied to integral equations of the secondkind. Our numerical results confirm that this augmentation method is particularly efficient for solving large scale linear systems induced from wavelet compression schemes.展开更多
In this paper we develop multilevel iteration methods for solving linear systems resulting from the Galerkin method and Tikhonov regularization for ill-posed problems. The algorithm and its convergence analysis are pr...In this paper we develop multilevel iteration methods for solving linear systems resulting from the Galerkin method and Tikhonov regularization for ill-posed problems. The algorithm and its convergence analysis are presented in an abstract framework.展开更多
We consider solving linear ill-posed operator equations. Based on a multi-scale decomposition for the solution space, we propose a multi-parameter regularization for solving the equations. We establish weak and strong...We consider solving linear ill-posed operator equations. Based on a multi-scale decomposition for the solution space, we propose a multi-parameter regularization for solving the equations. We establish weak and strong convergence theorems for the multi-parameter regularization solution. In particular, based on the eigenfunction decomposition, we develop a posteriori choice strategy for multi-parameters which gives a regularization solution with the optimal error bound. Several practical choices of multi-parameters are proposed. We also present numerical experiments to demonstrate the outperformance of the multiparameter regularization over the single parameter regularization.展开更多
In this paper we establish a high order finite volume method for the fourth order singular perturbation problems.In conjunction with the optimal meshes,the numerical solutions resulting from the method have optimal co...In this paper we establish a high order finite volume method for the fourth order singular perturbation problems.In conjunction with the optimal meshes,the numerical solutions resulting from the method have optimal convergence order.Numerical experiments are presented to verify our theoretical estimates.展开更多
The focus of this paper is on the relationship between accuracy of multivariate refinable vector and vector cascade algorithm. We show that, if the vector cascade algorithm (1.5) with isotropic dilation converges to a...The focus of this paper is on the relationship between accuracy of multivariate refinable vector and vector cascade algorithm. We show that, if the vector cascade algorithm (1.5) with isotropic dilation converges to a vector-valued function with regularity, then the initial function must satisfy the Strang-Fix conditions.展开更多
基金This work is supported by the Chinese NSF grants 60475042 Guangxi NSF grants 0542043the Foundation of Advanced Research Center of Zhongshan University and Hong Kong
文摘In this paper, we propose a globally convergent Polak-Ribiere-Polyak (PRP) conjugate gradient method for nonconvex minimization of differentiable functions by employing an Armijo-type line search which is simpler and less demanding than those defined in [4,10]. A favorite property of this method is that we can choose the initial stepsize as the one-dimensional minimizer of a quadratic modelΦ(t):= f(xk)+tgkTdk+(1/2) t2dkTQkdk, where Qk is a positive definite matrix that carries some second order information of the objective function f. So, this line search may make the stepsize tk more easily accepted. Preliminary numerical results show that this method is efficient.
基金Supported in part by the Natural Science Foundation of China under grants 10371137and 10201034Foundation of Doctoral Program of National Higher Education of China under under grant 20030558008Guangdong Provincial Natural Science Foundation of China u
文摘We introduce multilevel augmentation methods for solving operator equations based on direct sum decompositions of the range space of the operator and the solution space of the operator equation and a matrix splitting scheme. We establish a general setting for the analysis of these methods, showing that the methods yield approximate solutions of the same convergence order as the best approximation from the subspace. These augmentation methods allow us to develop fast, accurate and stable nonconventional numerical algorithms for solving operator equations. In particular, for second kind equations, special splitting techniques are proposed to develop such algorithms. These algorithms are then applied to solve the linear systems resulting from matrix compression schemes using wavelet-like functions for solving Fredholm integral equations of the second kind. For this special case, a complete analysis for computational complexity and convergence order is presented. Numerical examples are included to demonstrate the efficiency and accuracy of the methods. In these examples we use the proposed augmentation method to solve large scale linear systems resulting from the recently developed wavelet Galerkin methods and fast collocation methods applied to integral equations of the secondkind. Our numerical results confirm that this augmentation method is particularly efficient for solving large scale linear systems induced from wavelet compression schemes.
基金Natural Science Foundation of China under grants 10371137 and 10201034 the Foundation of Doctoral Program of National Higher Education of China under grant 20030558008 Guangdong Provincial Natural Science Foundation of China under grant 1011170 the Foundation of Zhongshan University Advanced Research Center.
文摘In this paper we develop multilevel iteration methods for solving linear systems resulting from the Galerkin method and Tikhonov regularization for ill-posed problems. The algorithm and its convergence analysis are presented in an abstract framework.
基金supported in part by the Natural Science Foundation of China under grants 10371137the Foundation of Doctoral Program of National Higher Education of China under grant 20030558008+5 种基金Guangdong Provincial Natural Science Foundation of China under grant 05003308the Foundation of Zhongshan University Advanced Research Centersupported in part by the US National Science Foundation under grant CCR-0407476National Aeronautics and Space Administration under Cooperative Agreement NNX07AC37Athe Natural Science Foundation of China under grants 10371122 and 10631080the Education Ministry of the People's Republic of China under the Changjiang Scholar Chair Professorship Program through Zhongshan University
文摘We consider solving linear ill-posed operator equations. Based on a multi-scale decomposition for the solution space, we propose a multi-parameter regularization for solving the equations. We establish weak and strong convergence theorems for the multi-parameter regularization solution. In particular, based on the eigenfunction decomposition, we develop a posteriori choice strategy for multi-parameters which gives a regularization solution with the optimal error bound. Several practical choices of multi-parameters are proposed. We also present numerical experiments to demonstrate the outperformance of the multiparameter regularization over the single parameter regularization.
基金the National Natural Science Foundation of China(Grant Nos.10771224,10601070)the Guangdong Provincial Natural Science Foundation of China(Grant No.05003308)+1 种基金MOE Project of Key Research Institute of Humanities and Social Sciences at UniversitiesChina-France-Russia Mathematics Collaboration(Grant No.34000-3275100)
文摘In this paper we establish a high order finite volume method for the fourth order singular perturbation problems.In conjunction with the optimal meshes,the numerical solutions resulting from the method have optimal convergence order.Numerical experiments are presented to verify our theoretical estimates.
基金The paper is supported financially by State Nature Foundation (No.40004003) Major State Basic Research Program of People's Republic of China (G1999032803).
文摘The focus of this paper is on the relationship between accuracy of multivariate refinable vector and vector cascade algorithm. We show that, if the vector cascade algorithm (1.5) with isotropic dilation converges to a vector-valued function with regularity, then the initial function must satisfy the Strang-Fix conditions.
