By utilizing symmetric functions,this paper presents explicit representations for Hermite interpolation and its numerical differentiation formula.And the corresponding error estimates are also provided.
A new numerical differentiation method with local opti- mum by data segmentation is proposed. The segmentation of data is based on the second derivatives computed by a Fourier devel- opment method. A filtering process...A new numerical differentiation method with local opti- mum by data segmentation is proposed. The segmentation of data is based on the second derivatives computed by a Fourier devel- opment method. A filtering process is used to achieve acceptable segmentation. Numerical results are presented by using the data segmentation method, compared with the regularization method. For further investigation, the proposed algorithm is applied to the resistance capacitance （RC） networks identification problem, and improvements of the result are obtained by using this algorithm.展开更多
It is shown that in Lagrangian numerical differentiation formulas, the coefficients are explicitly expressed by means of cycle indicator polynomials of symmetric group. Moreover, asymptotic expansions of the remainder...It is shown that in Lagrangian numerical differentiation formulas, the coefficients are explicitly expressed by means of cycle indicator polynomials of symmetric group. Moreover, asymptotic expansions of the remainders are also explicitly represented as a fixed number of interpolation nodes approaching infinitely to the point at which the derivative is evaluated. This implies that complete explicit formulas for local Lagrangian numerical differentiation can be obtained.展开更多
We investigate a novel adaptive choice rule of the Tikhonov regularization parameter in numerical differentiation which is a classic ill-posed problem. By assuming a general unknown Holder type error estimate derived ...We investigate a novel adaptive choice rule of the Tikhonov regularization parameter in numerical differentiation which is a classic ill-posed problem. By assuming a general unknown Holder type error estimate derived for numerical differentiation, we choose a regularization parameter in a geometric set providing a nearly optimal convergence rate with very limited a-priori information. Numerical simulation in image edge detection verifies reliability and efficiency of the new adaptive approach.展开更多
A framework to obtain numerical solution of the fractional partial differential equation using Bernstein polynomials is presented. The main characteristic behind this approach is that a fractional order operational ma...A framework to obtain numerical solution of the fractional partial differential equation using Bernstein polynomials is presented. The main characteristic behind this approach is that a fractional order operational matrix of Bernstein polynomials is derived. With the operational matrix, the equation is transformed into the products of several dependent matrixes which can also be regarded as the system of linear equations after dispersing the variable. By solving the linear equations, the numerical solutions are acquired. Only a small number of Bernstein polynomials are needed to obtain a satisfactory result. Numerical examples are provided to show that the method is computationally efficient.展开更多
In this paper we consider a quasilinear second order ordinary diferential equation with a small parameter Firstly an approximate problem is constructed. Then an iterative procedure is developed. Finally we give an alg...In this paper we consider a quasilinear second order ordinary diferential equation with a small parameter Firstly an approximate problem is constructed. Then an iterative procedure is developed. Finally we give an algorithm whose accuracy is good for arbitrary e>0 .展开更多
This paper focuses on the synchronisation between fractional-order and integer-order chaotic systems. Based on Lyapunov stability theory and numerical differentiation, a nonlinear feedback controller is obtained to ac...This paper focuses on the synchronisation between fractional-order and integer-order chaotic systems. Based on Lyapunov stability theory and numerical differentiation, a nonlinear feedback controller is obtained to achieve the synchronisation between fractional-order and integer-order chaotic systems. Numerical simulation results are presented to illustrate the effectiveness of this method.展开更多
The aim of this survey paper is to propose a new concept ＂generator＂. In fact, generator is a single function that can generate the basis as well as the whole function space. It is a more fundamental concept than ba...The aim of this survey paper is to propose a new concept ＂generator＂. In fact, generator is a single function that can generate the basis as well as the whole function space. It is a more fundamental concept than basis. Various properties of generator are also discussed. Moreover, a special generator named multiquadric function is introduced. Based on the multiquadric generator, the multiquadric quasi-interpolation scheme is constructed, and furthermore, the properties of this kind of quasi-interpolation are discussed to show its better capacity and stability in approximating the high order derivatives.展开更多
In this paper, the problem of reconstructing numerical derivatives from noisy data is considered. A new framework of mollification methods based on the L generalized solution regularization methods is proposed. A spec...In this paper, the problem of reconstructing numerical derivatives from noisy data is considered. A new framework of mollification methods based on the L generalized solution regularization methods is proposed. A specific algorithm for the first three derivatives is presented in the paper, in which a modification of TSVD, termed cTSVD is chosen as the regularization technique. Numerical examples given in the paper verify the theoretical results and show efficiency of the new method.展开更多
A new algorithm for reconstructing the three-dimensional flow field of the oceanic mesoscale eddies is proposed in this paper,based on variational method.Firstly,with the numerical differentiation Tikhonov regularizer...A new algorithm for reconstructing the three-dimensional flow field of the oceanic mesoscale eddies is proposed in this paper,based on variational method.Firstly,with the numerical differentiation Tikhonov regularizer,we reconstruct the continuous horizontal flow field on discrete grid points at each layer in the oceanic region,in terms of the horizontal flow field observations.Secondly,benefitting from the variational optimization analysis and its improvement,we reconstruct a three-dimensional flow field under the constraint of the horizontal flow and the vertical flow.The results of simulation experiments validate that the relative error of the new algorithm is lower than that of the finite difference method in the case of high grid resolution,which still holds in the case of unknown observational errors or in the absence of vertical velocity boundary conditions.Finally,using the reanalysis horizontal data sourcing from SODA and the proposed algorithm,we reconstruct three-dimensional flow field structure for the real oceanic mesoscale eddy.展开更多
Some new conclusions on asymptotic properties and inverse problems of numerical differentiation formulae have been drawn in this paper.In the first place,several asymptotic properties of intermediate points of numeric...Some new conclusions on asymptotic properties and inverse problems of numerical differentiation formulae have been drawn in this paper.In the first place,several asymptotic properties of intermediate points of numerical differentiation formulae are presented by using Taylor's formula.And then,based on the ideas of algebraic accuracy,several inverse problems of numerical differentiation formulae are given.展开更多
For solving higher dimensional diffusion equations with an inhomogeneous diffusion coefficient,Monte Carlo（MC） techniques are considered to be more effective than other algorithms, such as finite element method or f...For solving higher dimensional diffusion equations with an inhomogeneous diffusion coefficient,Monte Carlo（MC） techniques are considered to be more effective than other algorithms, such as finite element method or finite difference method. The inhomogeneity of diffusion coefficient strongly limits the use of different numerical techniques. For better convergence, methods with higher orders have been kept forward to allow MC codes with large step size. The main focus of this work is to look for operators that can produce converging results for large step sizes. As a first step, our comparative analysis has been applied to a general stochastic problem.Subsequently, our formulization is applied to the problem of pitch angle scattering resulting from Coulomb collisions of charge particles in the toroidal devices.展开更多
We propose a clustering-based approach for identifying coherent flow structuresin continuous dynamical systems. We first treat a particle trajectory over a finitetime interval as a high-dimensional data point and then...We propose a clustering-based approach for identifying coherent flow structuresin continuous dynamical systems. We first treat a particle trajectory over a finitetime interval as a high-dimensional data point and then cluster these data from differentinitial locations into groups. The method then uses the normalized standarddeviation or mean absolute deviation to quantify the deformation. Unlike the usualfinite-time Lyapunov exponent (FTLE), the proposed algorithm considers the completetraveling history of the particles. We also suggest two extensions of the method. To improvethe computational efficiency, we develop an adaptive approach that constructsdifferent subsamples of the whole particle trajectory based on a finite time interval. Tostart the computation in parallel to the flow trajectory data collection, we also developan on-the-fly approach to improve the solution as we continue to provide more measurementsfor the algorithm. The method can efficiently compute the WCVE over adifferent time interval by modifying the available data points.展开更多
In this paper, a robust model-free controller for a grid-connected photovoltaic (PV) system is designed. The system consists of a PV generator connected to a three-phase grid by a DC/AC converter. The control objectiv...In this paper, a robust model-free controller for a grid-connected photovoltaic (PV) system is designed. The system consists of a PV generator connected to a three-phase grid by a DC/AC converter. The control objectives of the overall system are to extract maximum power from the PV source, to control reactive power exchange and to improve the quality of the current injected into the grid. The model-free control technique is based on the use of an ultra-local model instead of the dynamic model of the overall system. The local model is continuously updated based on a numerical differentiator using only the input–output behavior of the controlled system. The model-free controller consists of a classical feedback controller and a compensator for the effects of internal parameter changes and external disturbances. Simulation results illustrate the efficiency of the controller for grid-connected PV systems.展开更多
基金Supported by the Education Department of Zhejiang Province (Y200806015)
文摘By utilizing symmetric functions,this paper presents explicit representations for Hermite interpolation and its numerical differentiation formula.And the corresponding error estimates are also provided.
基金supported by the National Basic Research Program of China(2011CB013103)
文摘A new numerical differentiation method with local opti- mum by data segmentation is proposed. The segmentation of data is based on the second derivatives computed by a Fourier devel- opment method. A filtering process is used to achieve acceptable segmentation. Numerical results are presented by using the data segmentation method, compared with the regularization method. For further investigation, the proposed algorithm is applied to the resistance capacitance （RC） networks identification problem, and improvements of the result are obtained by using this algorithm.
基金supported in part by the National Natural Science Foundation of China(Grant No.10471128)
文摘It is shown that in Lagrangian numerical differentiation formulas, the coefficients are explicitly expressed by means of cycle indicator polynomials of symmetric group. Moreover, asymptotic expansions of the remainders are also explicitly represented as a fixed number of interpolation nodes approaching infinitely to the point at which the derivative is evaluated. This implies that complete explicit formulas for local Lagrangian numerical differentiation can be obtained.
文摘We investigate a novel adaptive choice rule of the Tikhonov regularization parameter in numerical differentiation which is a classic ill-posed problem. By assuming a general unknown Holder type error estimate derived for numerical differentiation, we choose a regularization parameter in a geometric set providing a nearly optimal convergence rate with very limited a-priori information. Numerical simulation in image edge detection verifies reliability and efficiency of the new adaptive approach.
基金supported by the Natural Science Foundation of Hebei Province under Grant No.A2012203407
文摘A framework to obtain numerical solution of the fractional partial differential equation using Bernstein polynomials is presented. The main characteristic behind this approach is that a fractional order operational matrix of Bernstein polynomials is derived. With the operational matrix, the equation is transformed into the products of several dependent matrixes which can also be regarded as the system of linear equations after dispersing the variable. By solving the linear equations, the numerical solutions are acquired. Only a small number of Bernstein polynomials are needed to obtain a satisfactory result. Numerical examples are provided to show that the method is computationally efficient.
文摘In this paper we consider a quasilinear second order ordinary diferential equation with a small parameter Firstly an approximate problem is constructed. Then an iterative procedure is developed. Finally we give an algorithm whose accuracy is good for arbitrary e>0 .
文摘This paper focuses on the synchronisation between fractional-order and integer-order chaotic systems. Based on Lyapunov stability theory and numerical differentiation, a nonlinear feedback controller is obtained to achieve the synchronisation between fractional-order and integer-order chaotic systems. Numerical simulation results are presented to illustrate the effectiveness of this method.
基金Supported by the 973program-2006CB303102SGST 09DZ 2272900NSFC No.11026089
文摘The aim of this survey paper is to propose a new concept ＂generator＂. In fact, generator is a single function that can generate the basis as well as the whole function space. It is a more fundamental concept than basis. Various properties of generator are also discussed. Moreover, a special generator named multiquadric function is introduced. Based on the multiquadric generator, the multiquadric quasi-interpolation scheme is constructed, and furthermore, the properties of this kind of quasi-interpolation are discussed to show its better capacity and stability in approximating the high order derivatives.
文摘In this paper, the problem of reconstructing numerical derivatives from noisy data is considered. A new framework of mollification methods based on the L generalized solution regularization methods is proposed. A specific algorithm for the first three derivatives is presented in the paper, in which a modification of TSVD, termed cTSVD is chosen as the regularization technique. Numerical examples given in the paper verify the theoretical results and show efficiency of the new method.
基金Project sported by the National Natural Science Foundation of China(Grant Nos.41875045 and 61371119)the Blue Project of Jiangsu Province,China。
文摘A new algorithm for reconstructing the three-dimensional flow field of the oceanic mesoscale eddies is proposed in this paper,based on variational method.Firstly,with the numerical differentiation Tikhonov regularizer,we reconstruct the continuous horizontal flow field on discrete grid points at each layer in the oceanic region,in terms of the horizontal flow field observations.Secondly,benefitting from the variational optimization analysis and its improvement,we reconstruct a three-dimensional flow field under the constraint of the horizontal flow and the vertical flow.The results of simulation experiments validate that the relative error of the new algorithm is lower than that of the finite difference method in the case of high grid resolution,which still holds in the case of unknown observational errors or in the absence of vertical velocity boundary conditions.Finally,using the reanalysis horizontal data sourcing from SODA and the proposed algorithm,we reconstruct three-dimensional flow field structure for the real oceanic mesoscale eddy.
基金Supported by the Science and Technology Project of the Education Department of Jiangxi Province(GJJ08224 )Supported by the Transformation of Education Project of the Education Department of Jiangxi Province(JxJG-09-7-28)
文摘Some new conclusions on asymptotic properties and inverse problems of numerical differentiation formulae have been drawn in this paper.In the first place,several asymptotic properties of intermediate points of numerical differentiation formulae are presented by using Taylor's formula.And then,based on the ideas of algebraic accuracy,several inverse problems of numerical differentiation formulae are given.
基金supported in part by the Higher Education Commission of Pakistan under PPCR programsupported by the National Magnetic Confinement Fusion Program under Grant No.2013GB104004Fundamental Research Fund for Chinese Central Universities
文摘For solving higher dimensional diffusion equations with an inhomogeneous diffusion coefficient,Monte Carlo（MC） techniques are considered to be more effective than other algorithms, such as finite element method or finite difference method. The inhomogeneity of diffusion coefficient strongly limits the use of different numerical techniques. For better convergence, methods with higher orders have been kept forward to allow MC codes with large step size. The main focus of this work is to look for operators that can produce converging results for large step sizes. As a first step, our comparative analysis has been applied to a general stochastic problem.Subsequently, our formulization is applied to the problem of pitch angle scattering resulting from Coulomb collisions of charge particles in the toroidal devices.
文摘We propose a clustering-based approach for identifying coherent flow structuresin continuous dynamical systems. We first treat a particle trajectory over a finitetime interval as a high-dimensional data point and then cluster these data from differentinitial locations into groups. The method then uses the normalized standarddeviation or mean absolute deviation to quantify the deformation. Unlike the usualfinite-time Lyapunov exponent (FTLE), the proposed algorithm considers the completetraveling history of the particles. We also suggest two extensions of the method. To improvethe computational efficiency, we develop an adaptive approach that constructsdifferent subsamples of the whole particle trajectory based on a finite time interval. Tostart the computation in parallel to the flow trajectory data collection, we also developan on-the-fly approach to improve the solution as we continue to provide more measurementsfor the algorithm. The method can efficiently compute the WCVE over adifferent time interval by modifying the available data points.
文摘In this paper, a robust model-free controller for a grid-connected photovoltaic (PV) system is designed. The system consists of a PV generator connected to a three-phase grid by a DC/AC converter. The control objectives of the overall system are to extract maximum power from the PV source, to control reactive power exchange and to improve the quality of the current injected into the grid. The model-free control technique is based on the use of an ultra-local model instead of the dynamic model of the overall system. The local model is continuously updated based on a numerical differentiator using only the input–output behavior of the controlled system. The model-free controller consists of a classical feedback controller and a compensator for the effects of internal parameter changes and external disturbances. Simulation results illustrate the efficiency of the controller for grid-connected PV systems.
