This paper presents mechanical quadrature methods for solving first-kind boundary integral equations on polygonal regions, which possesses high accuracy O(h0^3)and low computing complexities. Moreover, the multivariat...This paper presents mechanical quadrature methods for solving first-kind boundary integral equations on polygonal regions, which possesses high accuracy O(h0^3)and low computing complexities. Moreover, the multivariate asymptotic expansion of the error with hi^3(i = 1,…,d) power is shown. Using the multi-parameter asymptotic expansion, we not only get a high precisioin approximation solution by means of the splitting extrapolation, but also derive a posteriori estimation.展开更多
The finite element solutions of elliptic eigenvalue equations are shown to have a multi-parameter asymptotic error expansion. Based on this expansion and a splitting extrapolation technique, a parallel algorithm for s...The finite element solutions of elliptic eigenvalue equations are shown to have a multi-parameter asymptotic error expansion. Based on this expansion and a splitting extrapolation technique, a parallel algorithm for solving multi-dimensional equations with high order accuracy is developed.展开更多
Splitting extrapolation based on domain decomposition for finite element approximations is a new technique for solving large scale scientific and engineering problems in parallel. By means of domain decomposition, a l...Splitting extrapolation based on domain decomposition for finite element approximations is a new technique for solving large scale scientific and engineering problems in parallel. By means of domain decomposition, a large scale multidimensional problem is turned to many discrete problems involving several grid parameters The multi-variate asymptotic expansions of finite element errors on independent grid parameters are proved for linear and nonlin ear second order elliptic equations as well as eigenvalue problems. Therefore after solving smaller problems with similar sizes in parallel, a global fine grid approximation with higher accuracy is computed by the splitting extrapolation method.展开更多
This paper deals with the splitting extrapolation for mixed finite element used in the approximation of the steady Stokes equation. Applying the multi variate asymptotic expansion of the error on independent grid p...This paper deals with the splitting extrapolation for mixed finite element used in the approximation of the steady Stokes equation. Applying the multi variate asymptotic expansion of the error on independent grid parameters, we can get a parallel algorithm and a global fine grid approximations with high accuracy.展开更多
A new splitting extrapolation based on multivariate asymptotic expansionsof finite elemellt eraes for differellt mesh parameters is described. By means of splittingextrapolation, a far 3 scale problem is decomposed in...A new splitting extrapolation based on multivariate asymptotic expansionsof finite elemellt eraes for differellt mesh parameters is described. By means of splittingextrapolation, a far 3 scale problem is decomposed into many subproblems, which can besolved in parallel. 111 this paper) we prove that the splitting extrapolation algorithm possesses a high order accuracy and the computation is almost independent of the dimension ofthe problem. Moreover, an extrapolation algorithm at global fine grid points is presented,several numerical examples including interface problems are discussed.展开更多
Taking hm as the mesh width of a curved edge Гm （m = 1, ..., d ） of polygons and using quadrature rules for weakly singular integrals, this paper presents mechanical quadrature methods for solving BIES of the first...Taking hm as the mesh width of a curved edge Гm （m = 1, ..., d ） of polygons and using quadrature rules for weakly singular integrals, this paper presents mechanical quadrature methods for solving BIES of the first kind of plane elasticity Dirichlet problems on curved polygons, which possess high accuracy O（h0^3） and low computing complexities. Since multivariate asymptotic expansions of approximate errors with power hi^3 （i = 1, 2, ..., d） are shown, by means of the splitting extrapolations high precision approximations and a posteriori estimate are obtained.展开更多
In this paper, multidimensional weakly singular integrals are solved by using rectangular quadrature rules which base on quadrature rules of one dimensional weakly singular and multidimensional regular integrals with ...In this paper, multidimensional weakly singular integrals are solved by using rectangular quadrature rules which base on quadrature rules of one dimensional weakly singular and multidimensional regular integrals with their Euler-Maclaurin asymptotic expansions of the errors. The presented method is suit for solving multidimensional and singular integrals by comparing with Gauss quadrature rule. The error asymptotic expansions show that the convergence order of the initial quadrature rules is , where . The order of accuracy can reach to by using extrapolation and splitting extrapolation, where h0 is the maximum mesh width. Some numerical examples are constructed to show the efficiency of the method.展开更多
In this paper,the collocation methods are used to solve the boundary integral equations of the first kind on the polygon.By means of Sidi’s periodic transformation and domain decomposition,the errors are proved to po...In this paper,the collocation methods are used to solve the boundary integral equations of the first kind on the polygon.By means of Sidi’s periodic transformation and domain decomposition,the errors are proved to possess the multi-parameter asymptotic expansion at the interior point with the powers h^(3)/_(i)(i=1,...,d),which means that the approximations of higher accuracy and a posteriori estimation of the errors can be obtained by splitting extrapolations.Numerical experiments are carried out to show that the methods are very efficient.展开更多
The finite element solutions of elliptic equations are shown to have a multiparameter asymptotic error expansion. Based on this expansion and a multi-parametersplitting extrapolation technique, a parallel algorithm fo...The finite element solutions of elliptic equations are shown to have a multiparameter asymptotic error expansion. Based on this expansion and a multi-parametersplitting extrapolation technique, a parallel algorithm for solving multi-dimensional equations with high order accuracy is introduced.展开更多
X-ray computed tomography （CT） is one of widely used diagnostic tools for medical and dental tomographic imaging of the human body. However, the standard filtered back- projection reconstruction method requires the ...X-ray computed tomography （CT） is one of widely used diagnostic tools for medical and dental tomographic imaging of the human body. However, the standard filtered back- projection reconstruction method requires the complete knowledge of the projection data. In the case of limited data, the inverse problem of CT becomes more ill-posed, which makes the reconstructed image deteriorated by the artifacts. In this paper, we consider two dimensional CT reconstruction using the projections truncated along the spatial direc- tion in the Radon domain. Over the decades, the numerous results including the sparsity model based approach has enabled the reconstruction of the image inside the region of interest （ROI） from the limited knowledge of the data. However, unlike these existing methods, we try to reconstruct the entire CT image from the limited knowledge of the sinogram via the tight frame regularization and the simultaneous sinogram extrapolation. Our proposed model shows more promising numerical simulation results compared with the existing sparsity model based approach.展开更多
The splitting extrapolation is an important technique for solving multidimensionalproblems.In the case that error u<sup>h</sup>-u has an asymptotic expansion of form Σc<sub>α</sub>h<sup&g...The splitting extrapolation is an important technique for solving multidimensionalproblems.In the case that error u<sup>h</sup>-u has an asymptotic expansion of form Σc<sub>α</sub>h<sup>2α</sup>,whereα=（α<sub>1</sub>,…,α<sub>s</sub>） and h<sup>α</sup>=h<sub>1</sub><sup>α<sub>1</sub></sup>,…h<sub>s</sub><sup>α<sub>s</sub></sup>,the method gives an approximation involving less computerstorage and less computational work in comparison with the classical Richardson extrapolation.In this paper we present a recurrence rule of the splitting extrapolation and discuss itsapplications in the fields of multiple integrals,multidimensional integral equations,partialdifferential equations and singular perturbation problems.展开更多
文摘This paper presents mechanical quadrature methods for solving first-kind boundary integral equations on polygonal regions, which possesses high accuracy O(h0^3)and low computing complexities. Moreover, the multivariate asymptotic expansion of the error with hi^3(i = 1,…,d) power is shown. Using the multi-parameter asymptotic expansion, we not only get a high precisioin approximation solution by means of the splitting extrapolation, but also derive a posteriori estimation.
基金This research is supported by National Science Foundation of China
文摘The finite element solutions of elliptic eigenvalue equations are shown to have a multi-parameter asymptotic error expansion. Based on this expansion and a splitting extrapolation technique, a parallel algorithm for solving multi-dimensional equations with high order accuracy is developed.
文摘Splitting extrapolation based on domain decomposition for finite element approximations is a new technique for solving large scale scientific and engineering problems in parallel. By means of domain decomposition, a large scale multidimensional problem is turned to many discrete problems involving several grid parameters The multi-variate asymptotic expansions of finite element errors on independent grid parameters are proved for linear and nonlin ear second order elliptic equations as well as eigenvalue problems. Therefore after solving smaller problems with similar sizes in parallel, a global fine grid approximation with higher accuracy is computed by the splitting extrapolation method.
文摘This paper deals with the splitting extrapolation for mixed finite element used in the approximation of the steady Stokes equation. Applying the multi variate asymptotic expansion of the error on independent grid parameters, we can get a parallel algorithm and a global fine grid approximations with high accuracy.
文摘A new splitting extrapolation based on multivariate asymptotic expansionsof finite elemellt eraes for differellt mesh parameters is described. By means of splittingextrapolation, a far 3 scale problem is decomposed into many subproblems, which can besolved in parallel. 111 this paper) we prove that the splitting extrapolation algorithm possesses a high order accuracy and the computation is almost independent of the dimension ofthe problem. Moreover, an extrapolation algorithm at global fine grid points is presented,several numerical examples including interface problems are discussed.
基金Supported by Natioal Science Foundation of China （10171073）.
文摘Taking hm as the mesh width of a curved edge Гm （m = 1, ..., d ） of polygons and using quadrature rules for weakly singular integrals, this paper presents mechanical quadrature methods for solving BIES of the first kind of plane elasticity Dirichlet problems on curved polygons, which possess high accuracy O（h0^3） and low computing complexities. Since multivariate asymptotic expansions of approximate errors with power hi^3 （i = 1, 2, ..., d） are shown, by means of the splitting extrapolations high precision approximations and a posteriori estimate are obtained.
文摘In this paper, multidimensional weakly singular integrals are solved by using rectangular quadrature rules which base on quadrature rules of one dimensional weakly singular and multidimensional regular integrals with their Euler-Maclaurin asymptotic expansions of the errors. The presented method is suit for solving multidimensional and singular integrals by comparing with Gauss quadrature rule. The error asymptotic expansions show that the convergence order of the initial quadrature rules is , where . The order of accuracy can reach to by using extrapolation and splitting extrapolation, where h0 is the maximum mesh width. Some numerical examples are constructed to show the efficiency of the method.
文摘In this paper,the collocation methods are used to solve the boundary integral equations of the first kind on the polygon.By means of Sidi’s periodic transformation and domain decomposition,the errors are proved to possess the multi-parameter asymptotic expansion at the interior point with the powers h^(3)/_(i)(i=1,...,d),which means that the approximations of higher accuracy and a posteriori estimation of the errors can be obtained by splitting extrapolations.Numerical experiments are carried out to show that the methods are very efficient.
文摘The finite element solutions of elliptic equations are shown to have a multiparameter asymptotic error expansion. Based on this expansion and a multi-parametersplitting extrapolation technique, a parallel algorithm for solving multi-dimensional equations with high order accuracy is introduced.
文摘X-ray computed tomography （CT） is one of widely used diagnostic tools for medical and dental tomographic imaging of the human body. However, the standard filtered back- projection reconstruction method requires the complete knowledge of the projection data. In the case of limited data, the inverse problem of CT becomes more ill-posed, which makes the reconstructed image deteriorated by the artifacts. In this paper, we consider two dimensional CT reconstruction using the projections truncated along the spatial direc- tion in the Radon domain. Over the decades, the numerous results including the sparsity model based approach has enabled the reconstruction of the image inside the region of interest （ROI） from the limited knowledge of the data. However, unlike these existing methods, we try to reconstruct the entire CT image from the limited knowledge of the sinogram via the tight frame regularization and the simultaneous sinogram extrapolation. Our proposed model shows more promising numerical simulation results compared with the existing sparsity model based approach.
文摘The splitting extrapolation is an important technique for solving multidimensionalproblems.In the case that error u<sup>h</sup>-u has an asymptotic expansion of form Σc<sub>α</sub>h<sup>2α</sup>,whereα=（α<sub>1</sub>,…,α<sub>s</sub>） and h<sup>α</sup>=h<sub>1</sub><sup>α<sub>1</sub></sup>,…h<sub>s</sub><sup>α<sub>s</sub></sup>,the method gives an approximation involving less computerstorage and less computational work in comparison with the classical Richardson extrapolation.In this paper we present a recurrence rule of the splitting extrapolation and discuss itsapplications in the fields of multiple integrals,multidimensional integral equations,partialdifferential equations and singular perturbation problems.
