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Optimal Shape Factor and Fictitious Radius in the MQ-RBF:Solving Ill-Posed Laplacian Problems
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作者 Chein-Shan Liu Chung-Lun Kuo Chih-Wen Chang 《Computer Modeling in Engineering & Sciences》 SCIE EI 2024年第6期3189-3208,共20页
To solve the Laplacian problems,we adopt a meshless method with the multiquadric radial basis function(MQRBF)as a basis whose center is distributed inside a circle with a fictitious radius.A maximal projection techniq... To solve the Laplacian problems,we adopt a meshless method with the multiquadric radial basis function(MQRBF)as a basis whose center is distributed inside a circle with a fictitious radius.A maximal projection technique is developed to identify the optimal shape factor and fictitious radius by minimizing a merit function.A sample function is interpolated by theMQ-RBF to provide a trial coefficient vector to compute the merit function.We can quickly determine the optimal values of the parameters within a preferred rage using the golden section search algorithm.The novel method provides the optimal values of parameters and,hence,an optimal MQ-RBF;the performance of the method is validated in numerical examples.Moreover,nonharmonic problems are transformed to the Poisson equation endowed with a homogeneous boundary condition;this can overcome the problem of these problems being ill-posed.The optimal MQ-RBF is extremely accurate.We further propose a novel optimal polynomial method to solve the nonharmonic problems,which achieves high precision up to an order of 10^(−11). 展开更多
关键词 Laplace equation nonharmonic boundary value problem ill-posed problem maximal projection optimal shape factor and fictitious radius optimal MQ-RBF optimal polynomial method
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A Multi-Baseline PolInSAR Forest Height Inversion Method Taking into Account the Model Ill-posed Problem
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作者 LIN Dongfang ZHU Jianjun +4 位作者 LI Zhiwei FU Haiqiang LIANG Ji ZHOU Fangbin ZHANG Bing 《Journal of Geodesy and Geoinformation Science》 CSCD 2024年第3期42-56,共15页
Affected by the insufficient information of single baseline observation data,the three-stage method assumes the Ground-to-Volume Ratio(GVR)to be zero so as to invert the vegetation height.However,this assumption intro... Affected by the insufficient information of single baseline observation data,the three-stage method assumes the Ground-to-Volume Ratio(GVR)to be zero so as to invert the vegetation height.However,this assumption introduces much biases into the parameter estimates which greatly limits the accuracy of the vegetation height inversion.Multi-baseline observation can provide redundant information and is helpful for the inversion of GVR.Nevertheless,the similar model parameter values in a multi-baseline model often lead to ill-posed problems and reduce the inversion accuracy of conventional algorithm.To this end,we propose a new step-by-step inversion method applied to the multi-baseline observations.Firstly,an adjustment inversion model is constructed by using multi-baseline volume scattering dominant polarization data,and the regularized estimates of model parameters are obtained by regularization method.Then,the reliable estimates of GVR are determined by the MSE(mean square error)analysis of each regularized parameter estimation.Secondly,the estimated GVR is used to extracts the pure volume coherence,and then the vegetation height parameter is inverted from the pure volume coherence by least squares estimation.The experimental results show that the new method can improve the vegetation height inversion result effectively.The inversion accuracy is improved by 26%with respect to the three-stage method and the conventional solution of multi-baseline.All of these have demonstrated the feasibility and effectiveness of the new method. 展开更多
关键词 multi-baseline vegetation height GVR POLINSAR ill-posed problem
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Ridge estimation iterative solution of ill-posed mixed additive and multiplicative random error model with equality constraints 被引量:3
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作者 Leyang Wang Tao Chen 《Geodesy and Geodynamics》 CSCD 2021年第5期336-346,共11页
The reasonable prior information between the parameters in the adjustment processing can significantly improve the precision of the parameter solution. Based on the principle of equality constraints, we establish the ... The reasonable prior information between the parameters in the adjustment processing can significantly improve the precision of the parameter solution. Based on the principle of equality constraints, we establish the mixed additive and multiplicative random error model with equality constraints and derive the weighted least squares iterative solution of the model. In addition, aiming at the ill-posed problem of the coefficient matrix, we also propose the ridge estimation iterative solution of ill-posed mixed additive and multiplicative random error model with equality constraints based on the principle of ridge estimation method and derive the U-curve method to determine the ridge parameter. The experimental results show that the weighted least squares iterative solution can obtain more reasonable parameter estimation and precision information than existing solutions, verifying the feasibility of applying the equality constraints to the mixed additive and multiplicative random error model. Furthermore, the ridge estimation iterative solution can obtain more accurate parameter estimation and precision information than the weighted least squares iterative solution. 展开更多
关键词 ill-posed problem Mixed additive and multiplicative random error model Equality constraints Weighted least squares Ridge estimation method U-curve method
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ILL-POSEDNESS OF MODIFIED KAWAHARA EQUATION AND KAUP-KUPERSHMIDT EQUATION 被引量:2
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作者 闫威 李用声 《Acta Mathematica Scientia》 SCIE CSCD 2012年第2期710-716,共7页
In this article, we consider the Cauchy problems for the modified Kawahara equationδtu+μδx(u3)+αδx5u+βδx3u+γδxu=0 and the Kaup-Kupershmidt equation δtu+μuδx2u+αδx5u+βδx3u+βδx3u+γδxu=0Usi... In this article, we consider the Cauchy problems for the modified Kawahara equationδtu+μδx(u3)+αδx5u+βδx3u+γδxu=0 and the Kaup-Kupershmidt equation δtu+μuδx2u+αδx5u+βδx3u+βδx3u+γδxu=0Using the general well-posedness principle introduced by I. Bejenaru and T. Tao, we prove 1 that the modified Kawahara equation is ill-posed for the initial data in H8 (It) with s 〈 - and that the Kaup-Kupershmidt equation is ill-posed for the initial data in HS(It) with s〈0. 展开更多
关键词 ill-posedNESS modified Kawahaxa equation Kaup-Kupershmidt equation general well-posedness principle
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REGULARIZATION METHODS FOR NONLINEAR ILL-POSED PROBLEMS WITH ACCRETIVE OPERATORS 被引量:2
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作者 王吉安 李景 刘振海 《Acta Mathematica Scientia》 SCIE CSCD 2008年第1期141-150,共10页
This article is devoted to the regularization of nonlinear ill-posed problems with accretive operators in Banach spaces. The data involved are assumed to be known approximately. The authors concentrate their discussio... This article is devoted to the regularization of nonlinear ill-posed problems with accretive operators in Banach spaces. The data involved are assumed to be known approximately. The authors concentrate their discussion on the convergence rates of regular solutions. 展开更多
关键词 REGULARIZATION ill-posed problems nonlinear accretive operator convergence rates
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ITERATIVE REGULARIZATION METHODS FOR NONLINEAR ILL-POSED OPERATOR EQUATIONS WITH M-ACCRETIVE MAPPINGS IN BANACH SPACES 被引量:2
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作者 Ioannis K.ARGYROS Santhosh GEORGE 《Acta Mathematica Scientia》 SCIE CSCD 2015年第6期1318-1324,共7页
In this paper, a modified Newton type iterative method is considered for ap- proximately solving ill-posed nonlinear operator equations involving m-accretive mappings in Banach space. Convergence rate of the method is... In this paper, a modified Newton type iterative method is considered for ap- proximately solving ill-posed nonlinear operator equations involving m-accretive mappings in Banach space. Convergence rate of the method is obtained based on an a priori choice of the regularization parameter. Our analysis is not based on the sequential continuity of the normalized duality mapping. 展开更多
关键词 nonlinear ill-posed equations iterative regularization m-accretive operator Newton type method
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A mixed Newton-Tikhonov method for nonlinear ill-posed problems 被引量:1
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作者 康传刚 贺国强 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2009年第6期741-752,共12页
Newton type methods are one kind of the efficient methods to solve nonlinear ill-posed problems, which have attracted extensive attention. However, computational cost of Newton type methods is high because practical p... Newton type methods are one kind of the efficient methods to solve nonlinear ill-posed problems, which have attracted extensive attention. However, computational cost of Newton type methods is high because practical problems are complicated. We propose a mixed Newton-Tikhonov method, i.e., one step Newton-Tikhonov method with several other steps of simplified Newton-Tikhonov method. Convergence and stability of this method are proved under some conditions. Numerical experiments show that the proposed method has obvious advantages over the classical Newton method in terms of computational costs. 展开更多
关键词 nonlinear ill-posed problem inverse heat conduction problem mixedNewton-Tikhonov method CONVERGENCE stability
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Arnoldi Projection Fractional Tikhonov for Large Scale Ill-Posed Problems 被引量:1
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作者 Wang Zhengsheng Mu Liming +1 位作者 Liu Rongrong Xu Guili 《Transactions of Nanjing University of Aeronautics and Astronautics》 EI CSCD 2018年第3期395-402,共8页
It is well known that Tikhonov regularization in standard form may determine approximate solutions that are too smooth for ill-posed problems,so fractional Tikhonov methods have been introduced to remedy this shortcom... It is well known that Tikhonov regularization in standard form may determine approximate solutions that are too smooth for ill-posed problems,so fractional Tikhonov methods have been introduced to remedy this shortcoming.And Tikhonov regularization for large-scale linear ill-posed problems is commonly implemented by determining apartial Arnoldi decomposition of the given matrix.In this paper,we propose a new method to compute an approximate solution of large scale linear discrete ill-posed problems which applies projection fractional Tikhonov regularization in Krylov subspace via Arnoldi process.The projection fractional Tikhonov regularization combines the fractional matrices and orthogonal projection operators.A suitable value of the regularization parameter is determined by the discrepancy principle.Numerical examples with application to image restoration are carried out to examine that the performance of the method. 展开更多
关键词 ill-posed problems fractional matrix Tikhonov regularization orthogonal projection operator image restoration
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An iterative virtual projection method to improve the reconstruction performance for ill-posed emission tomographic problems 被引量:1
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作者 柳华蔚 郑树 周怀春 《Chinese Physics B》 SCIE EI CAS CSCD 2015年第10期200-212,共13页
In order to improve the reconstruction performance for ill-posed emission tomographic problems with limited projections, a generalized interpolation method is proposed in this paper, in which the virtual lines of proj... In order to improve the reconstruction performance for ill-posed emission tomographic problems with limited projections, a generalized interpolation method is proposed in this paper, in which the virtual lines of projection are fabricated from, but not linearly dependent on, the measured projections. The method is called the virtual projection(VP) method.Also, an iterative correction method for the integral lengths is proposed to reduce the error brought about by the virtual lines of projection. The combination of the two methods is called the iterative virtual projection(IVP) method. Based on a scheme of equilateral triangle plane meshes and a six asymmetrically arranged detection system, numerical simulations and experimental verification are conducted. Simulation results obtained by using a non-negative linear least squares method,without any other constraints or regularization, demonstrate that the VP method can gradually reduce the reconstruction error and converges to the desired one by fabricating additional effective projections. When the mean square deviation of normal error superimposed on the simulated measured projections is smaller than 0.03, i.e., the signal-to-noise ratio(SNR)for the measured projections is higher than 30.4, the IVP method can further reduce the reconstruction error reached by the VP method apparently. In addition, as the regularization matrix in the Tikhonov regularization method is updated by an iterative correction process similar to the IVP method presented in this paper, or the Tikhonov regularization method is used in the IVP method, good improvement is achieved. 展开更多
关键词 ill-posed problem emission tomography limited projections Tikhonov regularization
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Numerical Experiments and Structural Transformation of Non-Linear Evolution in Ill-posed Systems 被引量:1
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作者 OuYang Shoucheng,Wang Liuran(Chengdu University of Information and Technology,610041,China) 《工程科学(英文版)》 2007年第3期57-63,共7页
Numerical experiments on non-linear equations of the 1st-and 3rd-order derivatives have been carried out through structural analyses in the phase space according to the numerical instability of ill-posed systems,with ... Numerical experiments on non-linear equations of the 1st-and 3rd-order derivatives have been carried out through structural analyses in the phase space according to the numerical instability of ill-posed systems,with changes of initial values and parameters,etc..The results show that the quantitative instability in an ill-posed system may reveal reversed transformation in system evolution by structural representation,and confirm A·Dauglas' theorem that "a non-linear equation does not satisfy the existence of the initial value in a linear well-posed system". 展开更多
关键词 non-linearity ill-posed SYSTEMS INSTABILITY of quantity reversed TRANSFORMATION CURVATURE
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ILL-POSEDNESS FOR THE NONLINEAR DAVEY-STEWARTSON EQUATION
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作者 沈彩霞 郭柏灵 《Acta Mathematica Scientia》 SCIE CSCD 2008年第1期117-127,共11页
The nonlinear D-S equations on R^d, with general power nonlinearity and with both the focusing and defocusing signs, are proved to be ill-posed in the Sobolev space H^s whenever the exponent s is lower than that predi... The nonlinear D-S equations on R^d, with general power nonlinearity and with both the focusing and defocusing signs, are proved to be ill-posed in the Sobolev space H^s whenever the exponent s is lower than that predicted by scaling or Galilean invariance, or when the regularity is too low to support distributional solutions. Authors analyze a class of solutions for which the zero-dispersion limit provides good approximations. 展开更多
关键词 The Davey-Stewartson equation the Cauchy problem ill-posedNESS
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Double Optimal Regularization Algorithms for Solving Ill-Posed Linear Problems under Large Noise
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作者 Chein-Shan Liu Satya N.Atluri 《Computer Modeling in Engineering & Sciences》 SCIE EI 2015年第1期1-39,共39页
A double optimal solution of an n-dimensional system of linear equations Ax=b has been derived in an affine m-dimensional Krylov subspace with m <<n.We further develop a double optimal iterative algorithm(DOIA),... A double optimal solution of an n-dimensional system of linear equations Ax=b has been derived in an affine m-dimensional Krylov subspace with m <<n.We further develop a double optimal iterative algorithm(DOIA),with the descent direction z being solved from the residual equation Az=r0 by using its double optimal solution,to solve ill-posed linear problem under large noise.The DOIA is proven to be absolutely convergent step-by-step with the square residual error ||r||^2=||b-Ax||^2 being reduced by a positive quantity ||Azk||^2 at each iteration step,which is found to be better than those algorithms based on the minimization of the square residual error in an m-dimensional Krylov subspace.In order to tackle the ill-posed linear problem under a large noise,we also propose a novel double optimal regularization algorithm(DORA)to solve it,which is an improvement of the Tikhonov regularization method.Some numerical tests reveal the high performance of DOIA and DORA against large noise.These methods are of use in the ill-posed problems of structural health-monitoring. 展开更多
关键词 ill-posed LINEAR equations system DOUBLE OPTIMAL solution Affine Krylov subspace DOUBLE OPTIMAL iterative ALGORITHM DOUBLE OPTIMAL REGULARIZATION ALGORITHM
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WEAK REGULARIZATION FOR A CLASS OF ILL-POSED CAUCHY PROBLEMS
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作者 黄永忠 郑权 《Acta Mathematica Scientia》 SCIE CSCD 2006年第3期483-490,共8页
This article is concerned with the ill-posed Cauchy problem associated with a densely defined linear operator A in a Banach space. A family of weak regularizing operators is introduced. If the spectrum of A is contain... This article is concerned with the ill-posed Cauchy problem associated with a densely defined linear operator A in a Banach space. A family of weak regularizing operators is introduced. If the spectrum of A is contained in a sector of right-half complex plane and its resolvent is polynomially bounded, the weak regularization for such ill-posed Cauchy problem can be shown by using the quasi-reversibilky method and regularized semigroups. Finally, an example is given. 展开更多
关键词 ill-posed Cauchy problem QUASI-REVERSIBILITY family of weakly regularizing operators fractional power regularized semigroup
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LAVRENTIEV'S REGULARIZATION METHOD FOR NONLINEAR ILL-POSED EQUATIONS IN BANACH SPACES
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作者 Santhosh GEORGE C.D.SREEDEEP 《Acta Mathematica Scientia》 SCIE CSCD 2018年第1期303-314,共12页
In this paper, we deal with nonlinear ill-posed problems involving m-accretive mappings in Banach spaces. We consider a derivative and inverse free method for the imple- mentation of Lavrentiev regularization method. ... In this paper, we deal with nonlinear ill-posed problems involving m-accretive mappings in Banach spaces. We consider a derivative and inverse free method for the imple- mentation of Lavrentiev regularization method. Using general HSlder type source condition we obtain an optimal order error estimate. Also we consider the adaptive parameter choice strategy proposed by Pereverzev and Schock (2005) for choosing the regularization parameter. 展开更多
关键词 nonlinear ill-posed problem Banach space Lavrentiev regularization m-accretive mappings adaptive parameter choice strategy
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Nonlinear implicit iterative method for solving nonlinear ill-posed problems
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作者 柳建军 贺国强 康传刚 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2009年第9期1183-1192,共10页
In the paper, we extend the implicit iterative method for linear ill-posed operator equations to solve nonlinear ill-posed problems. We show that under some conditions the error sequence of solutions of the nonlinear ... In the paper, we extend the implicit iterative method for linear ill-posed operator equations to solve nonlinear ill-posed problems. We show that under some conditions the error sequence of solutions of the nonlinear implicit iterative method is monotonically decreasing and, with this monotonicity, prove convergence of the new method for both the exact and perturbed equations. 展开更多
关键词 nonlinear ill-posed problem nonlinear implicit iterative method MONOTONICITY CONVERGENCE
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A Newton-type method for nonlinear ill-posed problems with A-smooth regularization
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作者 殷萍 贺国强 孟泽红 《Journal of Shanghai University(English Edition)》 CAS 2007年第5期457-463,共7页
In this paper we present a regularized Newton-type method for ill-posed problems, by using the A-smooth regularization to solve the linearized ill-posed equations. For noisy data a proper a posteriori stopping rule is... In this paper we present a regularized Newton-type method for ill-posed problems, by using the A-smooth regularization to solve the linearized ill-posed equations. For noisy data a proper a posteriori stopping rule is used that yields convergence of the Newton iteration to a solution, as the noise level goes to zero, under certain smoothness conditions on the nonlinear operator. Some appropriate assumptions on the closedness and smoothness of the starting value and the solution are shown to lead to optimal convergence rates. 展开更多
关键词 nonlinear ill-posed problems A-smooth regularization a posteriori stopping rule convergence and convergencerates.
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REGULARIZATION OF AN ILL-POSED HYPERBOLIC PROBLEM AND THE UNIQUENESS OF THE SOLUTION BY A WAVELET GALERKIN METHOD
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作者 Jos Roberto Linhares de Mattos Ernesto Prado Lopes 《Analysis in Theory and Applications》 2012年第2期125-134,共10页
We consider the problem K(x)Uxx = utt , 0 〈 x 〈 1, t 〉 0, with the boundary condition u(O,t) = g(t) E LZ(R) and ux(O,t) = 0, where K(x) is continuous and 0 〈α≤ K (x) 〈 +∞. This is an ill-posed p... We consider the problem K(x)Uxx = utt , 0 〈 x 〈 1, t 〉 0, with the boundary condition u(O,t) = g(t) E LZ(R) and ux(O,t) = 0, where K(x) is continuous and 0 〈α≤ K (x) 〈 +∞. This is an ill-posed problem in the sense that, if the solution exists, it does not depend continuously on g. Considering the existence of a solution u(x, .) E H2(R) and using a wavelet Galerkin method with Meyer multiresolution analysis, we regularize the ill-posedness of the problem. Furthermore we prove the uniqueness of the solution for this problem. 展开更多
关键词 ill-posed problem meyer wavelet hyperbolic equation REGULARIZATION
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A Regularized Randomized Kaczmarz Algorithm for Large Discrete Ill-Posed Problems
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作者 LIU Fengming WANG Zhengsheng +1 位作者 YANG Siyu XU Guili 《Transactions of Nanjing University of Aeronautics and Astronautics》 EI CSCD 2020年第5期787-795,共9页
Tikhonov regularization is a powerful tool for solving linear discrete ill-posed problems.However,effective methods for dealing with large-scale ill-posed problems are still lacking.The Kaczmarz method is an effective... Tikhonov regularization is a powerful tool for solving linear discrete ill-posed problems.However,effective methods for dealing with large-scale ill-posed problems are still lacking.The Kaczmarz method is an effective iterative projection algorithm for solving large linear equations due to its simplicity.We propose a regularized randomized extended Kaczmarz(RREK)algorithm for solving large discrete ill-posed problems via combining the Tikhonov regularization and the randomized Kaczmarz method.The convergence of the algorithm is proved.Numerical experiments illustrate that the proposed algorithm has higher accuracy and better image restoration quality compared with the existing randomized extended Kaczmarz(REK)method. 展开更多
关键词 ill-posed problem Tikhonov regularization randomized extended Kaczmarz(REK)algorithm image restoration
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The Ill-Posedness of Derivative Interpolation and Regularized Derivative Interpolation for Non-Bandlimited Functions
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作者 Weidong Chen 《Applied Mathematics》 2022年第1期87-100,共14页
In this paper, the ill-posedness of derivative interpolation is discussed, and a regularized derivative interpolation for non-bandlimited signals is presented. The convergence of the regularized derivative interpolati... In this paper, the ill-posedness of derivative interpolation is discussed, and a regularized derivative interpolation for non-bandlimited signals is presented. The convergence of the regularized derivative interpolation is studied. The numerical results are given and compared with derivative interpolation using the Tikhonov regularization method. The regularized derivative interpolation in this paper is more accurate in computation. 展开更多
关键词 Nonband-Limited Function Derivative Interpolation ill-posedNESS REGULARIZATION
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A New Regularized Solution to Ill-Posed Problem in Coordinate Transformation
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作者 Xuming Ge Jicang Wu 《International Journal of Geosciences》 2012年第1期14-20,共7页
Coordinates transformation is generally required in GPS applications. If the transformation parameters are solved with the known coordinates in a small area using the Bursa model, the precision of transformed coordina... Coordinates transformation is generally required in GPS applications. If the transformation parameters are solved with the known coordinates in a small area using the Bursa model, the precision of transformed coordinates is generally very poor. Since the translation parameters and rotation parameters are highly correlated in this case, a very large condition number of the coefficient matrix A exists in the linear system of equations. Regularization is required to reduce the effects caused by the intrinsic ill-conditioning of the problem and noises in the data, and to stabilize the solution. Based on advanced regularized methods, we propose a new regularized solution to the ill-posed coordinate transformation problem. Simulation numerical experiments of coordinate transformation are given to shed light on the relationship among different regularization approaches. The results indicate that the proposed new method can obtain a more reasonable resolution with higher precision and/or robustness. 展开更多
关键词 BURSA COORDINATE TRANSFORMATION ill-posed METHODOLOGY
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