This paper presents a new method to estimate the height of the atmospheric boundary layer(ABL) by using COSMIC radio occultation bending angle(BA) data. Using the numerical differentiation method combined with the reg...This paper presents a new method to estimate the height of the atmospheric boundary layer(ABL) by using COSMIC radio occultation bending angle(BA) data. Using the numerical differentiation method combined with the regularization technique, the first derivative of BA profiles is retrieved, and the height at which the first derivative of BA has the global minimum is defined to be the ABL height. To reflect the reliability of estimated ABL heights, the sharpness parameter is introduced, according to the relative minimum of the BA derivative. Then, it is applied to four months of COSMIC BA data(January, April, July, and October in 2008), and the ABL heights estimated are compared with two kinds of ABL heights from COSMIC products and with the heights determined by the finite difference method upon the refractivity data. For sharp ABL tops(large sharpness parameters), there is little difference between the ABL heights determined by different methods, i.e.,the uncertainties are small; whereas, for non-sharp ABL tops(small sharpness parameters), big differences exist in the ABL heights obtained by different methods, which means large uncertainties for different methods. In addition, the new method can detect thin ABLs and provide a reference ABL height in the cases eliminated by other methods. Thus, the application of the numerical differentiation method combined with the regularization technique to COSMIC BA data is an appropriate choice and has further application value.展开更多
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.
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
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
We propose an ？~1 regularized method for numerical differentiation using empirical eigenfunctions. Compared with traditional methods for numerical differentiation, the output of our method can be considered directly ...We propose an ？~1 regularized method for numerical differentiation using empirical eigenfunctions. Compared with traditional methods for numerical differentiation, the output of our method can be considered directly as the derivative of the underlying function. Moreover,our method could produce sparse representations with respect to empirical eigenfunctions.Numerical results show that our method is quite effective.展开更多
We consider the problem of estimating the derivative of a function f from its noisy version fδby using the derivatives of the partial sums of Fourier-Legendre series of f. Instead of the observation Lspace, we perfor...We consider the problem of estimating the derivative of a function f from its noisy version fδby using the derivatives of the partial sums of Fourier-Legendre series of f. Instead of the observation Lspace, we perform the reconstruction of the derivative in a weighted Lspace. This takes full advantage of the properties of Legendre polynomials and results in a slight improvement on the convergence order. Finally, we provide several numerical examples to demonstrate the efficiency of the proposed method.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
Many effective optimization algorithms require partial derivatives of objective functions,while some optimization problems'objective functions have no derivatives.According to former research studies,some search d...Many effective optimization algorithms require partial derivatives of objective functions,while some optimization problems'objective functions have no derivatives.According to former research studies,some search directions are obtained using the quadratic hypothesis of objective functions.Based on derivatives,quadratic function assumptions,and directional derivatives,the computational formulas of numerical first-order partial derivatives,second-order partial derivatives,and numerical second-order mixed partial derivatives were constructed.Based on the coordinate transformation relation,a set of orthogonal vectors in the fixed coordinate system was established according to the optimization direction.A numerical algorithm was proposed,taking the second order approximation direction as an example.A large stepsize numerical algorithm based on coordinate transformation was proposed.Several algorithms were validated by an unconstrained optimization of the two-dimensional Rosenbrock objective function.The numerical second order approximation direction with the numerical mixed partial derivatives showed good results.Its calculated amount is 0.2843%of that of without second-order mixed partial derivative.In the process of rotating the local coordinate system 360°,because the objective function is more complex than the quadratic function,if the numerical direction derivative is used instead of the analytic partial derivative,the optimization direction varies with a range of 103.05°.Because theoretical error is in the numerical negative gradient direction,the calculation with the coordinate transformation is 94.71%less than the calculation without coordinate transformation.If there is no theoretical error in the numerical negative gradient direction or in the large-stepsize numerical optimization algorithm based on the coordinate transformation,the sawtooth phenomenon occurs.When each numerical mixed partial derivative takes more than one point,the optimization results cannot be improved.The numerical direction based on the quadratic hypothesis only requires the objective function to be obtained,but does not require derivability and does not take into account truncation error and rounding error.Thus,the application scopes of many optimization methods are extended.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 presents the application of a novel AI-based approach,Neural Physics,to produce high-fidelity simulations of train aerodynamics.Neural Physics is built upon convolutional neural networks(CNNs),where the wei...This paper presents the application of a novel AI-based approach,Neural Physics,to produce high-fidelity simulations of train aerodynamics.Neural Physics is built upon convolutional neural networks(CNNs),where the weights are explicitly determined by classical numerical discretisation schemes rather than by training.By leveraging the power of AI technology,this recent approach results in code that can run easily on GPUs and AI processors,achieving high computational speed without sacrificing accuracy.The approach uses an implicit large eddy simulation method based on a non-linear Petrov-Galerkin method to model the unresolved turbulence.Furthermore,for higher-order finite elements,the convolutional finite element method(ConvFEM)is used,which greatly simplifies the implementation of higher-order elements within the NN 4 DPEs approach.We demonstrate the capability of Neural Physics by simulating a freight Locomotive Class 66 and a partially loaded freight train operating in an open field environment with and without cross wind.This is the first time that ConvFEM has been applied to high-speed fluid flow problems in complex geometries.The results are validated against existing numerical results and experimental measurements,and show good agreement in terms of pressure and velocity distributions around the train body.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
A novel adaptive fault-tolerant control scheme in the differential algebraic framework was proposed for attitude control of a heavy lift launch vehicle （HLLV）. By using purely mathematical transformations, the decou...A novel adaptive fault-tolerant control scheme in the differential algebraic framework was proposed for attitude control of a heavy lift launch vehicle （HLLV）. By using purely mathematical transformations, the decoupled input-output representations of HLLV were derived, rendering three decoupled second-order systems, i.e., pitch, yaw and roll channels. Based on a new type of numerical differentiator, a differential algebraic observer （DAO） was proposed for estimating the system states and the generalized disturbances, including various disturbances and additive fault torques. Driven by DAOs, three improved proportional-integral- differential （PID） controllers with disturbance compensation were designed for pitch, yaw and roll control. All signals in the closed-loop system were guaranteed to be ultimately uniformly bounded by utilization of Lyapunov＇s indirect method. The convincing numerical simulations indicate that the proposed control scheme is successful in achieving high performance in the presence of parametric perturbations, external disturbances, noisy corruptions, and actuator faults.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
The high-accuracy, wide-range frequency estimation algorithm for multi-component signals presented in this paper, is based on a numerical differentiation and central Lagrange interpolation. With the sample sequences, ...The high-accuracy, wide-range frequency estimation algorithm for multi-component signals presented in this paper, is based on a numerical differentiation and central Lagrange interpolation. With the sample sequences, which need at most 7 points and are sampled at a sample frequency of 25600 Hz, and computation sequences, using employed a formulation proposed in this paper, the frequencies of each component of the signal are all estimated at an accuracy of 0.001% over 1 Hz to 800 kHz with the amplitudes of each component of the signal varying from 1 V to 200 V and the phase angle of each component of the signal varying from 0° to 360°. The proposed algorithm needs at most a half cycle for the frequencies of each component of the signal under noisy or non-noisy conditions. A testing example is given to illustrate the proposed algorithm in Matlab environment.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
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
Accurately approximating higher order derivatives is an inherently difficult problem. It is shown that a random variable shape parameter strategy can improve the accuracy of approximating higher order derivatives with...Accurately approximating higher order derivatives is an inherently difficult problem. It is shown that a random variable shape parameter strategy can improve the accuracy of approximating higher order derivatives with Radial Basis Function methods. The method is used to solve fourth order boundary value problems. The use and location of ghost points are examined in order to enforce the extra boundary conditions that are necessary to make a fourth-order problem well posed. The use of ghost points versus solving an overdetermined linear system via least squares is studied. For a general fourth-order boundary value problem, the recommended approach is to either use one of two novel sets of ghost centers introduced here or else to use a least squares approach. When using either ghost centers or least squares, the random variable shape parameter strategy results in significantly better accuracy than when a constant shape parameter is used.展开更多
Numerical simulation of the high order derivatives based on the sampling data is an important and basic problem in numerical approximation,especially for solving the differential equations numerically.The classical me...Numerical simulation of the high order derivatives based on the sampling data is an important and basic problem in numerical approximation,especially for solving the differential equations numerically.The classical method is the divided difference method.However,it has been shown strongly unstable in practice.Actually,it can only be used to simulate the lower order derivatives in applications.To simulate the high order derivatives,this paper suggests a new method using multiquadric quasi-interpolation.The stability of the multiquadric quasi-interpolation method is compared with the classical divided difference method.Moreover,some numerical examples are presented to confirm the theoretical results.Both theoretical results and numerical examples show that the multiquadric quasi-interpolation method is much stabler than the divided difference method.This property shows that multiquadric quasi-interpolation method is an efficient tool to construct an approximation of high order derivatives based on scattered sampling data even with noise.展开更多
基金supported by the National Natural Science Foundation of China (Grant No. 41475021)
文摘This paper presents a new method to estimate the height of the atmospheric boundary layer(ABL) by using COSMIC radio occultation bending angle(BA) data. Using the numerical differentiation method combined with the regularization technique, the first derivative of BA profiles is retrieved, and the height at which the first derivative of BA has the global minimum is defined to be the ABL height. To reflect the reliability of estimated ABL heights, the sharpness parameter is introduced, according to the relative minimum of the BA derivative. Then, it is applied to four months of COSMIC BA data(January, April, July, and October in 2008), and the ABL heights estimated are compared with two kinds of ABL heights from COSMIC products and with the heights determined by the finite difference method upon the refractivity data. For sharp ABL tops(large sharpness parameters), there is little difference between the ABL heights determined by different methods, i.e.,the uncertainties are small; whereas, for non-sharp ABL tops(small sharpness parameters), big differences exist in the ABL heights obtained by different methods, which means large uncertainties for different methods. In addition, the new method can detect thin ABLs and provide a reference ABL height in the cases eliminated by other methods. Thus, the application of the numerical differentiation method combined with the regularization technique to COSMIC BA data is an appropriate choice and has further application value.
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 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 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 by the National Nature Science Foundation of China(Grant Nos.11301052,11301045,11271060,11601064,11671068)the Fundamental Research Funds for the Central Universities(Grant No.DUT16LK33)the Fundamental Research of Civil Aircraft(Grant No.MJ-F-2012-04)
文摘We propose an ？~1 regularized method for numerical differentiation using empirical eigenfunctions. Compared with traditional methods for numerical differentiation, the output of our method can be considered directly as the derivative of the underlying function. Moreover,our method could produce sparse representations with respect to empirical eigenfunctions.Numerical results show that our method is quite effective.
基金Supported by the National Nature Science Foundation of China(Grant Nos.1130105211301045+4 种基金114010771127106011290143)the Fundamental Research Funds for the Central Universities(Grant No.DUT15RC(3)058)the Fundamental Research of Civil Aircraft(Grant No.MJ-F-2012-04)
文摘We consider the problem of estimating the derivative of a function f from its noisy version fδby using the derivatives of the partial sums of Fourier-Legendre series of f. Instead of the observation Lspace, we perform the reconstruction of the derivative in a weighted Lspace. This takes full advantage of the properties of Legendre polynomials and results in a slight improvement on the convergence order. Finally, we provide several numerical examples to demonstrate the efficiency of the proposed method.
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 in part by the Teaching Reform Research Foundation of Shengli College in China University of Petroleum(East China)(JG201725)the Natural Science Foundation Shandong Province of China(ZR2018PEE009)the Project of Science and Technology of Shandong Universities in China(J17KA044,J17KB061)。
文摘Many effective optimization algorithms require partial derivatives of objective functions,while some optimization problems'objective functions have no derivatives.According to former research studies,some search directions are obtained using the quadratic hypothesis of objective functions.Based on derivatives,quadratic function assumptions,and directional derivatives,the computational formulas of numerical first-order partial derivatives,second-order partial derivatives,and numerical second-order mixed partial derivatives were constructed.Based on the coordinate transformation relation,a set of orthogonal vectors in the fixed coordinate system was established according to the optimization direction.A numerical algorithm was proposed,taking the second order approximation direction as an example.A large stepsize numerical algorithm based on coordinate transformation was proposed.Several algorithms were validated by an unconstrained optimization of the two-dimensional Rosenbrock objective function.The numerical second order approximation direction with the numerical mixed partial derivatives showed good results.Its calculated amount is 0.2843%of that of without second-order mixed partial derivative.In the process of rotating the local coordinate system 360°,because the objective function is more complex than the quadratic function,if the numerical direction derivative is used instead of the analytic partial derivative,the optimization direction varies with a range of 103.05°.Because theoretical error is in the numerical negative gradient direction,the calculation with the coordinate transformation is 94.71%less than the calculation without coordinate transformation.If there is no theoretical error in the numerical negative gradient direction or in the large-stepsize numerical optimization algorithm based on the coordinate transformation,the sawtooth phenomenon occurs.When each numerical mixed partial derivative takes more than one point,the optimization results cannot be improved.The numerical direction based on the quadratic hypothesis only requires the objective function to be obtained,but does not require derivability and does not take into account truncation error and rounding error.Thus,the application scopes of many optimization methods are extended.
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 .
基金Projects(EPSRC EP/Y005732/1,EP/Y018680/1,EP/T003189/1,EP/V040235/1,EP/Y024257/1 and EP/T000414/1)supported by UK Research and Innovation(UKRI),UKProject(APP44894/UKRI 1281)supported by UKRI councils(NERC,AHRC,ESRC,MRC and DEFRA),UK。
文摘This paper presents the application of a novel AI-based approach,Neural Physics,to produce high-fidelity simulations of train aerodynamics.Neural Physics is built upon convolutional neural networks(CNNs),where the weights are explicitly determined by classical numerical discretisation schemes rather than by training.By leveraging the power of AI technology,this recent approach results in code that can run easily on GPUs and AI processors,achieving high computational speed without sacrificing accuracy.The approach uses an implicit large eddy simulation method based on a non-linear Petrov-Galerkin method to model the unresolved turbulence.Furthermore,for higher-order finite elements,the convolutional finite element method(ConvFEM)is used,which greatly simplifies the implementation of higher-order elements within the NN 4 DPEs approach.We demonstrate the capability of Neural Physics by simulating a freight Locomotive Class 66 and a partially loaded freight train operating in an open field environment with and without cross wind.This is the first time that ConvFEM has been applied to high-speed fluid flow problems in complex geometries.The results are validated against existing numerical results and experimental measurements,and show good agreement in terms of pressure and velocity distributions around the train body.
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Foundation item： Project（2012M521538） supported by China Postdoctoral Science Foundation Project suppolted by Postdoctoral Science Foundation of Central South University
文摘A novel adaptive fault-tolerant control scheme in the differential algebraic framework was proposed for attitude control of a heavy lift launch vehicle （HLLV）. By using purely mathematical transformations, the decoupled input-output representations of HLLV were derived, rendering three decoupled second-order systems, i.e., pitch, yaw and roll channels. Based on a new type of numerical differentiator, a differential algebraic observer （DAO） was proposed for estimating the system states and the generalized disturbances, including various disturbances and additive fault torques. Driven by DAOs, three improved proportional-integral- differential （PID） controllers with disturbance compensation were designed for pitch, yaw and roll control. All signals in the closed-loop system were guaranteed to be ultimately uniformly bounded by utilization of Lyapunov＇s indirect method. The convincing numerical simulations indicate that the proposed control scheme is successful in achieving high performance in the presence of parametric perturbations, external disturbances, noisy corruptions, and actuator faults.
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.
文摘The high-accuracy, wide-range frequency estimation algorithm for multi-component signals presented in this paper, is based on a numerical differentiation and central Lagrange interpolation. With the sample sequences, which need at most 7 points and are sampled at a sample frequency of 25600 Hz, and computation sequences, using employed a formulation proposed in this paper, the frequencies of each component of the signal are all estimated at an accuracy of 0.001% over 1 Hz to 800 kHz with the amplitudes of each component of the signal varying from 1 V to 200 V and the phase angle of each component of the signal varying from 0° to 360°. The proposed algorithm needs at most a half cycle for the frequencies of each component of the signal under noisy or non-noisy conditions. A testing example is given to illustrate the proposed algorithm in Matlab environment.
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 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Accurately approximating higher order derivatives is an inherently difficult problem. It is shown that a random variable shape parameter strategy can improve the accuracy of approximating higher order derivatives with Radial Basis Function methods. The method is used to solve fourth order boundary value problems. The use and location of ghost points are examined in order to enforce the extra boundary conditions that are necessary to make a fourth-order problem well posed. The use of ghost points versus solving an overdetermined linear system via least squares is studied. For a general fourth-order boundary value problem, the recommended approach is to either use one of two novel sets of ghost centers introduced here or else to use a least squares approach. When using either ghost centers or least squares, the random variable shape parameter strategy results in significantly better accuracy than when a constant shape parameter is used.
基金supported by the Major State Basic Research Development Program of China(973 Program)(Grant No.2006CB303102)the Science and Technology Commission of Shanghai Municipality(Grant No.09DZ2272900)
文摘Numerical simulation of the high order derivatives based on the sampling data is an important and basic problem in numerical approximation,especially for solving the differential equations numerically.The classical method is the divided difference method.However,it has been shown strongly unstable in practice.Actually,it can only be used to simulate the lower order derivatives in applications.To simulate the high order derivatives,this paper suggests a new method using multiquadric quasi-interpolation.The stability of the multiquadric quasi-interpolation method is compared with the classical divided difference method.Moreover,some numerical examples are presented to confirm the theoretical results.Both theoretical results and numerical examples show that the multiquadric quasi-interpolation method is much stabler than the divided difference method.This property shows that multiquadric quasi-interpolation method is an efficient tool to construct an approximation of high order derivatives based on scattered sampling data even with noise.
