High dynamic conditions impose critical challenges on Global Navigation Satellite System(GNSS)receivers,leading to large tracking errors or even loss of tracking.Current methods that intend to improve receivers’adapt...High dynamic conditions impose critical challenges on Global Navigation Satellite System(GNSS)receivers,leading to large tracking errors or even loss of tracking.Current methods that intend to improve receivers’adaptability for high dynamics require either complicated structures or prior statistical information of noises.This paper proposes a high dynamics algorithm based on steepest ascent method that can circumvent the deficiencies of existing methods.First,the relationship between the error of carrier tracking and the maximum of Fast Fourier Transform(FFT)outputs is established,and a performance function based on the steepest ascent method is designed.It can keep stable in high dynamics.Second,a new carrier-tracking loop is constructed by deploying the performance function.When the variation of GPS receiver acceleration ranges from 10 g to 100 g,compared with the PLL that either loses lock or keeps tracking accuracy less than 33.89 Hz,the experimental results show that the proposed method can not only keep tracking,but also achieve tracking accuracy more than 2.77 Hz.展开更多
Some properties of a class of quasi-differentiable functions（the difference of two finite convex functions） are considered in this paper. And the convergence of the steepest descent algorithm for unconstrained and c...Some properties of a class of quasi-differentiable functions（the difference of two finite convex functions） are considered in this paper. And the convergence of the steepest descent algorithm for unconstrained and constrained quasi-differentiable programming is proved.展开更多
<div style="text-align:justify;"> In order to speed up the global optimization-based mesh smoothing, an enhanced steepest descent method is presented in the paper. Numerical experiment results show tha...<div style="text-align:justify;"> In order to speed up the global optimization-based mesh smoothing, an enhanced steepest descent method is presented in the paper. Numerical experiment results show that the method performs better than the steepest descent method in the global smoothing. We also presented a physically-based interpretation to explain why the method works better than the steepest descent method. </div>展开更多
The classical variational inequality problem with a Lipschitzian and strongly monotone operator on a nonempty closed convex subset in a real Hilbert space is studied. A new three-step relaxed hybrid steepest-descent m...The classical variational inequality problem with a Lipschitzian and strongly monotone operator on a nonempty closed convex subset in a real Hilbert space is studied. A new three-step relaxed hybrid steepest-descent method for this class of variational inequalities is introduced. Strong convergence of this method is established under suitable assumptions imposed on the algorithm parameters.展开更多
The steepest descent method is the simplest gradient method for optimization. It is well known that exact line searches along each steepest descent direction may converge very slowly. An important result was given by ...The steepest descent method is the simplest gradient method for optimization. It is well known that exact line searches along each steepest descent direction may converge very slowly. An important result was given by Barzilar and Borwein, which is proved to be superlinearly convergent for convex quadratic in two dimensional space, and performs quite well for high dimensional problems. The BB method is not monotone, thus it is not easy to be generalized for general nonlinear functions unless certain non-monotone techniques being applied. Therefore, it is very desirable to find stepsize formulae which enable fast convergence and possess the monotone property. Such a stepsize αk for the steepest descent method is suggested in this paper. An algorithm with this new stepsize in even iterations and exact line search in odd iterations is proposed. Numerical results are presented, which confirm that the new method can find the exact solution within 3 iteration for two dimensional problems. The new method is very efficient for small scale problems. A modified version of the new method is also presented, where the new technique for selecting the stepsize is used after every two exact line searches. The modified algorithm is comparable to the Barzilar-Borwein method for large scale problems and better for small scale problems.展开更多
We introduce the fractional-order global optimal backpropagation machine,which is trained by an improved fractionalorder steepest descent method(FSDM).This is a fractional-order backpropagation neural network(FBPNN),a...We introduce the fractional-order global optimal backpropagation machine,which is trained by an improved fractionalorder steepest descent method(FSDM).This is a fractional-order backpropagation neural network(FBPNN),a state-of-the-art fractional-order branch of the family of backpropagation neural networks(BPNNs),different from the majority of the previous classic first-order BPNNs which are trained by the traditional first-order steepest descent method.The reverse incremental search of the proposed FBPNN is in the negative directions of the approximate fractional-order partial derivatives of the square error.First,the theoretical concept of an FBPNN trained by an improved FSDM is described mathematically.Then,the mathematical proof of fractional-order global optimal convergence,an assumption of the structure,and fractional-order multi-scale global optimization of the FBPNN are analyzed in detail.Finally,we perform three(types of)experiments to compare the performances of an FBPNN and a classic first-order BPNN,i.e.,example function approximation,fractional-order multi-scale global optimization,and comparison of global search and error fitting abilities with real data.The higher optimal search ability of an FBPNN to determine the global optimal solution is the major advantage that makes the FBPNN superior to a classic first-order BPNN.展开更多
Multipath interference seriously degrades the performance of Global Navigation Satellite System(GNSS)positioning in an urban canyon.Most current multipath mitigation algorithms suffer from heavy computational load or ...Multipath interference seriously degrades the performance of Global Navigation Satellite System(GNSS)positioning in an urban canyon.Most current multipath mitigation algorithms suffer from heavy computational load or need external assistance.We propose a multipath mitigation algorithm based on the steepest descent approach,which has the merits of less computational load and no need for external aid.A new ranging code tracking loop is designed based on the steepest descent method,which can save an early branch or a late branch compared with the narrow-spacing correlation method.The power of the Non-Line-of-Sight(NLOS)signal is weaker than that of the Line-of-Sight(LOS)signal when the LOS signal is not obstructed and with a relatively high Carrier Noise Ratio(CNR).The peak position in the X-axis of the ranging code autocorrelation function does not move with the NLOS interference.Meanwhile,the cost function is designed according to this phenomenon.The results demonstrate that the proposed algorithm outperforms the narrow-spacing correlation and the Multipath Estimated Delay Locked Loop(MEDLL)in terms of the code multipath mitigation and computation time.The Standard Deviation(STD)of the tracking error with the proposed algorithm is less than 0.016 chips.Moreover,the computation time of the proposed algorithm in a software defined receiver is shortened by 24.21%compared with the narrow-spacing correlation.展开更多
In this paper, we consider the generalized variational inequality GVI（F, g, C）, where F and g are mappings from a Hilbert space into itself and C is the fixed point set of a nonexpansive mapping. We propose two iter...In this paper, we consider the generalized variational inequality GVI（F, g, C）, where F and g are mappings from a Hilbert space into itself and C is the fixed point set of a nonexpansive mapping. We propose two iterative algorithms to find approximate solutions of the GVI（F,g, C）. Strong convergence results are established and applications to constrained generalized pseudo-inverse are included.展开更多
By extending the classical analysis techniques due to Samokish, Faddeev and Faddee- va, and Longsine and McCormick among others, we prove the convergence of the precon- ditioned steepest descent with implicit deflati...By extending the classical analysis techniques due to Samokish, Faddeev and Faddee- va, and Longsine and McCormick among others, we prove the convergence of the precon- ditioned steepest descent with implicit deflation （PSD-id） method for solving Hermitian- definite generalized eigenvalue problems. Furthermore, we derive a nonasymptotie estimate of the rate of convergence of the PSD-id method. We show that with a proper choice of the shift, the indefinite shift-and-invert preconditioner is a locally accelerated preconditioner, and is asymptotically optimal which leads to superlinear convergence Numerical examples are presented to verify the theoretical results on the convergence behavior of the PSD- id method for solving ill-conditioned Hermitian-definite generalized eigenvalue problems arising from electronic structure calculations. While rigorous and full-scale convergence proofs of preconditioned block steepest descent methods in practical use still largely eludes us, we believe the theoretical results presented in this paper shed light on an improved understanding of the convergence behavior of these block methods.展开更多
For the maximal space-like hypersurface defined on 2-dimensional space forms,based on the regularity and the strict convexity of the level sets,the steepest descents are well defined.In this paper,we come to estimate ...For the maximal space-like hypersurface defined on 2-dimensional space forms,based on the regularity and the strict convexity of the level sets,the steepest descents are well defined.In this paper,we come to estimate the curvature of its steepest descents by deriving a differential equality.展开更多
基金funded by National Natural Science Foundation of China(Nos.61533008,61603181,61673208,61873125)。
文摘High dynamic conditions impose critical challenges on Global Navigation Satellite System(GNSS)receivers,leading to large tracking errors or even loss of tracking.Current methods that intend to improve receivers’adaptability for high dynamics require either complicated structures or prior statistical information of noises.This paper proposes a high dynamics algorithm based on steepest ascent method that can circumvent the deficiencies of existing methods.First,the relationship between the error of carrier tracking and the maximum of Fast Fourier Transform(FFT)outputs is established,and a performance function based on the steepest ascent method is designed.It can keep stable in high dynamics.Second,a new carrier-tracking loop is constructed by deploying the performance function.When the variation of GPS receiver acceleration ranges from 10 g to 100 g,compared with the PLL that either loses lock or keeps tracking accuracy less than 33.89 Hz,the experimental results show that the proposed method can not only keep tracking,but also achieve tracking accuracy more than 2.77 Hz.
基金Supported by the State Foundations of Ph.D.Units(20020141013)Supported by the NSF of China(10001007)
文摘Some properties of a class of quasi-differentiable functions（the difference of two finite convex functions） are considered in this paper. And the convergence of the steepest descent algorithm for unconstrained and constrained quasi-differentiable programming is proved.
文摘<div style="text-align:justify;"> In order to speed up the global optimization-based mesh smoothing, an enhanced steepest descent method is presented in the paper. Numerical experiment results show that the method performs better than the steepest descent method in the global smoothing. We also presented a physically-based interpretation to explain why the method works better than the steepest descent method. </div>
基金Project supported by the Key Science Foundation of Education Department of Sichuan Province of China (No.2003A081)Sichuan Province Leading Academic Discipline Project (No.SZD0406)
文摘The classical variational inequality problem with a Lipschitzian and strongly monotone operator on a nonempty closed convex subset in a real Hilbert space is studied. A new three-step relaxed hybrid steepest-descent method for this class of variational inequalities is introduced. Strong convergence of this method is established under suitable assumptions imposed on the algorithm parameters.
基金Acknowledgment： This work was partly supported by the National Natural Science Foundation of China（60672150） and Science and Technology Planning Project of Shenzhen, China （szkj0706）.
文摘The steepest descent method is the simplest gradient method for optimization. It is well known that exact line searches along each steepest descent direction may converge very slowly. An important result was given by Barzilar and Borwein, which is proved to be superlinearly convergent for convex quadratic in two dimensional space, and performs quite well for high dimensional problems. The BB method is not monotone, thus it is not easy to be generalized for general nonlinear functions unless certain non-monotone techniques being applied. Therefore, it is very desirable to find stepsize formulae which enable fast convergence and possess the monotone property. Such a stepsize αk for the steepest descent method is suggested in this paper. An algorithm with this new stepsize in even iterations and exact line search in odd iterations is proposed. Numerical results are presented, which confirm that the new method can find the exact solution within 3 iteration for two dimensional problems. The new method is very efficient for small scale problems. A modified version of the new method is also presented, where the new technique for selecting the stepsize is used after every two exact line searches. The modified algorithm is comparable to the Barzilar-Borwein method for large scale problems and better for small scale problems.
基金Project supported by the National Key Research and Development Program of China(No.2018YFC0830300)the National Natural Science Foundation of China(No.61571312)。
文摘We introduce the fractional-order global optimal backpropagation machine,which is trained by an improved fractionalorder steepest descent method(FSDM).This is a fractional-order backpropagation neural network(FBPNN),a state-of-the-art fractional-order branch of the family of backpropagation neural networks(BPNNs),different from the majority of the previous classic first-order BPNNs which are trained by the traditional first-order steepest descent method.The reverse incremental search of the proposed FBPNN is in the negative directions of the approximate fractional-order partial derivatives of the square error.First,the theoretical concept of an FBPNN trained by an improved FSDM is described mathematically.Then,the mathematical proof of fractional-order global optimal convergence,an assumption of the structure,and fractional-order multi-scale global optimization of the FBPNN are analyzed in detail.Finally,we perform three(types of)experiments to compare the performances of an FBPNN and a classic first-order BPNN,i.e.,example function approximation,fractional-order multi-scale global optimization,and comparison of global search and error fitting abilities with real data.The higher optimal search ability of an FBPNN to determine the global optimal solution is the major advantage that makes the FBPNN superior to a classic first-order BPNN.
基金National Natural Science Foundation of China(NO.61533008,61603181,61673208,61873125).
文摘Multipath interference seriously degrades the performance of Global Navigation Satellite System(GNSS)positioning in an urban canyon.Most current multipath mitigation algorithms suffer from heavy computational load or need external assistance.We propose a multipath mitigation algorithm based on the steepest descent approach,which has the merits of less computational load and no need for external aid.A new ranging code tracking loop is designed based on the steepest descent method,which can save an early branch or a late branch compared with the narrow-spacing correlation method.The power of the Non-Line-of-Sight(NLOS)signal is weaker than that of the Line-of-Sight(LOS)signal when the LOS signal is not obstructed and with a relatively high Carrier Noise Ratio(CNR).The peak position in the X-axis of the ranging code autocorrelation function does not move with the NLOS interference.Meanwhile,the cost function is designed according to this phenomenon.The results demonstrate that the proposed algorithm outperforms the narrow-spacing correlation and the Multipath Estimated Delay Locked Loop(MEDLL)in terms of the code multipath mitigation and computation time.The Standard Deviation(STD)of the tracking error with the proposed algorithm is less than 0.016 chips.Moreover,the computation time of the proposed algorithm in a software defined receiver is shortened by 24.21%compared with the narrow-spacing correlation.
文摘In this paper, we consider the generalized variational inequality GVI（F, g, C）, where F and g are mappings from a Hilbert space into itself and C is the fixed point set of a nonexpansive mapping. We propose two iterative algorithms to find approximate solutions of the GVI（F,g, C）. Strong convergence results are established and applications to constrained generalized pseudo-inverse are included.
文摘By extending the classical analysis techniques due to Samokish, Faddeev and Faddee- va, and Longsine and McCormick among others, we prove the convergence of the precon- ditioned steepest descent with implicit deflation （PSD-id） method for solving Hermitian- definite generalized eigenvalue problems. Furthermore, we derive a nonasymptotie estimate of the rate of convergence of the PSD-id method. We show that with a proper choice of the shift, the indefinite shift-and-invert preconditioner is a locally accelerated preconditioner, and is asymptotically optimal which leads to superlinear convergence Numerical examples are presented to verify the theoretical results on the convergence behavior of the PSD- id method for solving ill-conditioned Hermitian-definite generalized eigenvalue problems arising from electronic structure calculations. While rigorous and full-scale convergence proofs of preconditioned block steepest descent methods in practical use still largely eludes us, we believe the theoretical results presented in this paper shed light on an improved understanding of the convergence behavior of these block methods.
基金the National Natural Science Foundation of China(Grant No.11471188)the STPF of Shandong Province(No.J17KA161).
文摘For the maximal space-like hypersurface defined on 2-dimensional space forms,based on the regularity and the strict convexity of the level sets,the steepest descents are well defined.In this paper,we come to estimate the curvature of its steepest descents by deriving a differential equality.
