While the numerical solution of one-dimensional Volterra integral equations of the second kind with regular kernels is well understood, there exist no systematic studies of asymptotic error expansion for the approxima...While the numerical solution of one-dimensional Volterra integral equations of the second kind with regular kernels is well understood, there exist no systematic studies of asymptotic error expansion for the approximate solution. In this paper,we analyse the Nystrom solution of one-dimensional nonlinear Volterra integral equation of the second kind and show that approkimate solution admits an asymptotic error expansion in even powers of the step-size h, beginning with a term in h2. So that the Richardson's extrapolation can be done. This will increase the accuracy of numerical solution greatly.展开更多
With the reversible data hiding method based on pixel-value-ordering,data are embedded through the modification of the maximum and minimum values of a block.A significant relationship exists between the embedding perf...With the reversible data hiding method based on pixel-value-ordering,data are embedded through the modification of the maximum and minimum values of a block.A significant relationship exists between the embedding performance and the block size.Traditional pixel-value-ordering methods utilize pixel blocks with a fixed size to embed data;the smaller the pixel blocks,greater is the embedding capacity.However,it tends to result in the deterioration of the quality of the marked image.Herein,a novel reversible data hiding method is proposed by incorporating a block merging strategy into Li et al.’s pixel-value-ordering method,which realizes the dynamic control of block size by considering the image texture.First,the cover image is divided into non-overlapping 2×2 pixel blocks.Subsequently,according to their complexity,similarity and thresholds,these blocks are employed for data embedding through the pixel-value-ordering method directly or after being emerged into 2×4,4×2,or 4×4 sized blocks.Hence,smaller blocks can be used in the smooth region to create a high embedding capacity and larger blocks in the texture region to maintain a high peak signal-to-noise ratio.Experimental results prove that the proposed method is superior to the other three advanced methods.It achieves a high embedding capacity while maintaining low distortion and improves the embedding performance of the pixel-value-ordering algorithm.展开更多
The classical composite rectangle (constant) rule for the computation of Cauchy principle value integral with the singular kernel is discussed. We show that the superconvergence rate of the composite midpoint ru...The classical composite rectangle (constant) rule for the computation of Cauchy principle value integral with the singular kernel is discussed. We show that the superconvergence rate of the composite midpoint rule occurs at certain local coodinate of each subinterval and obtain the corresponding superconvergence error estimate. Then collation methods are presented to solve certain kind of Hilbert singular integral equation. At last, some numerical examples are provided to validate the theoretical analysis.展开更多
With the advancement of communication technology,a large number of data are constantly transmitted through the internet for various purposes,which are prone to be illegally accessed by third parties.Therefore,securing...With the advancement of communication technology,a large number of data are constantly transmitted through the internet for various purposes,which are prone to be illegally accessed by third parties.Therefore,securing such data is crucial to protect the transmitted information from falling into the wrong hands.Among data protection schemes,Secret Image Sharing is one of the most popular methods.It protects critical messages or data by embedding them in an image and sharing it with some users.Furthermore,it combines the security concepts in that private data are embedded into a cover image and then secured using the secret-sharing method.Despite its advantages,this method may produce noise,making the resulting stego file much different from its cover.Moreover,the size of private data that can be embedded is limited.This research works on these problems by utilizing prediction-error expansion and histogram-based approaches to embed the data.To recover the cover image,the SS method based on the Chinese remainder theorem is used.The experimental results indicate that this proposed method performs better than similar methods in several cover images and scenarios.展开更多
The composite trapezoidal rule for the computation of Hadamard finite-part integrals in boundary element methods with the hypersingular kernel I/sin2（x- s） is discussed, and the main part of the asymptotic expansion...The composite trapezoidal rule for the computation of Hadamard finite-part integrals in boundary element methods with the hypersingular kernel I/sin2（x- s） is discussed, and the main part of the asymptotic expansion of error function is obtained. Based on the main part of the asymptotic expansion, a series is constructed to approach the singular point. An extrapolation algorithm is presented and the convergence rate is proved. Some numerical results are also presented to confirm the theoretical results and show the efficiency of the algorithms.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
Asymptotic error expansions in H^1-norm for the bilinear finite element approximation to a class of optimal control problems are derived for rectangular meshes. With the rectan- gular meshes, the Richardson extrapolat...Asymptotic error expansions in H^1-norm for the bilinear finite element approximation to a class of optimal control problems are derived for rectangular meshes. With the rectan- gular meshes, the Richardson extrapolation of two different schemes and an interpolation defect correction can be applied. The higher order numerical approximations are used to generate a posteriori error estimators for the finite element approximation.展开更多
In this paper, we consider the solution of the biharmonic equation using Adini nonconforming finite element, and report new results of the asymptotic error expansions of the interpolation error functionals and nonconf...In this paper, we consider the solution of the biharmonic equation using Adini nonconforming finite element, and report new results of the asymptotic error expansions of the interpolation error functionals and nonconforming remainder. These expansions are used to develop two extrapolation formulas and a series of sharp error estimates. Finally, the numerical results have verified the extrapolation theory.展开更多
文摘While the numerical solution of one-dimensional Volterra integral equations of the second kind with regular kernels is well understood, there exist no systematic studies of asymptotic error expansion for the approximate solution. In this paper,we analyse the Nystrom solution of one-dimensional nonlinear Volterra integral equation of the second kind and show that approkimate solution admits an asymptotic error expansion in even powers of the step-size h, beginning with a term in h2. So that the Richardson's extrapolation can be done. This will increase the accuracy of numerical solution greatly.
文摘With the reversible data hiding method based on pixel-value-ordering,data are embedded through the modification of the maximum and minimum values of a block.A significant relationship exists between the embedding performance and the block size.Traditional pixel-value-ordering methods utilize pixel blocks with a fixed size to embed data;the smaller the pixel blocks,greater is the embedding capacity.However,it tends to result in the deterioration of the quality of the marked image.Herein,a novel reversible data hiding method is proposed by incorporating a block merging strategy into Li et al.’s pixel-value-ordering method,which realizes the dynamic control of block size by considering the image texture.First,the cover image is divided into non-overlapping 2×2 pixel blocks.Subsequently,according to their complexity,similarity and thresholds,these blocks are employed for data embedding through the pixel-value-ordering method directly or after being emerged into 2×4,4×2,or 4×4 sized blocks.Hence,smaller blocks can be used in the smooth region to create a high embedding capacity and larger blocks in the texture region to maintain a high peak signal-to-noise ratio.Experimental results prove that the proposed method is superior to the other three advanced methods.It achieves a high embedding capacity while maintaining low distortion and improves the embedding performance of the pixel-value-ordering algorithm.
文摘The classical composite rectangle (constant) rule for the computation of Cauchy principle value integral with the singular kernel is discussed. We show that the superconvergence rate of the composite midpoint rule occurs at certain local coodinate of each subinterval and obtain the corresponding superconvergence error estimate. Then collation methods are presented to solve certain kind of Hilbert singular integral equation. At last, some numerical examples are provided to validate the theoretical analysis.
基金This research was supported by the Ministry of Education,Culture,Research and Technology,The Republic of Indonesia,Institut Teknologi Sepuluh Nopember,and Universitas 17 Agustus 1945 Surabaya.
文摘With the advancement of communication technology,a large number of data are constantly transmitted through the internet for various purposes,which are prone to be illegally accessed by third parties.Therefore,securing such data is crucial to protect the transmitted information from falling into the wrong hands.Among data protection schemes,Secret Image Sharing is one of the most popular methods.It protects critical messages or data by embedding them in an image and sharing it with some users.Furthermore,it combines the security concepts in that private data are embedded into a cover image and then secured using the secret-sharing method.Despite its advantages,this method may produce noise,making the resulting stego file much different from its cover.Moreover,the size of private data that can be embedded is limited.This research works on these problems by utilizing prediction-error expansion and histogram-based approaches to embed the data.To recover the cover image,the SS method based on the Chinese remainder theorem is used.The experimental results indicate that this proposed method performs better than similar methods in several cover images and scenarios.
基金supported by National Natural Science Foundation of China(Grant Nos. 11101247 and 11201209)Shandong Provincial Natural Science Foundation of China (Grant No.ZR2011AQ020)+3 种基金a project of Shandong Province Higher Educational Science and Technology Program (GrantNo. J11LE08)supported by National Natural Science Foundation of China (GrantNo. 11101317)supported by National Basic Research Program of China (Grant No.2005CB321701)the Reward Fund of CAS for National Prize
文摘The composite trapezoidal rule for the computation of Hadamard finite-part integrals in boundary element methods with the hypersingular kernel I/sin2（x- s） is discussed, and the main part of the asymptotic expansion of error function is obtained. Based on the main part of the asymptotic expansion, a series is constructed to approach the singular point. An extrapolation algorithm is presented and the convergence rate is proved. Some numerical results are also presented to confirm the theoretical results and show the efficiency of the algorithms.
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.
基金supported in part by the National Basic Research Program (2007CB814906)the National Natural Science Foundation of China (10471103 and 10771158)+4 种基金Social Science Foundation of the Ministry of Education of China (06JA630047)Tianjin Natural Science Foundation (07JCYBJC14300)Tianjin University of Finance and Economicssupported by the National Basic Research Program under the Grant 2005CB321701the National Natural Science Foundation of China under the Grant 10771211
文摘Asymptotic error expansions in H^1-norm for the bilinear finite element approximation to a class of optimal control problems are derived for rectangular meshes. With the rectan- gular meshes, the Richardson extrapolation of two different schemes and an interpolation defect correction can be applied. The higher order numerical approximations are used to generate a posteriori error estimators for the finite element approximation.
文摘In this paper, we consider the solution of the biharmonic equation using Adini nonconforming finite element, and report new results of the asymptotic error expansions of the interpolation error functionals and nonconforming remainder. These expansions are used to develop two extrapolation formulas and a series of sharp error estimates. Finally, the numerical results have verified the extrapolation theory.
