<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
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
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
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
Spectral conjugate gradient method is an algorithm obtained by combination of spectral gradient method and conjugate gradient method,which is characterized with global convergence and simplicity of spectral gradient m...Spectral conjugate gradient method is an algorithm obtained by combination of spectral gradient method and conjugate gradient method,which is characterized with global convergence and simplicity of spectral gradient method,and small storage of conjugate gradient method.Besides,the spectral conjugate gradient method was proved that the search direction at each iteration is a descent direction of objective function even without relying on any line search method.Spectral conjugate gradient method is applied to full waveform inversion for numerical tests on Marmousi model.The authors give a comparison on numerical results obtained by steepest descent method,conjugate gradient method and spectral conjugate gradient method,which shows that the spectral conjugate gradient method is superior to the other two methods.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
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 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.
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.
文摘Spectral conjugate gradient method is an algorithm obtained by combination of spectral gradient method and conjugate gradient method,which is characterized with global convergence and simplicity of spectral gradient method,and small storage of conjugate gradient method.Besides,the spectral conjugate gradient method was proved that the search direction at each iteration is a descent direction of objective function even without relying on any line search method.Spectral conjugate gradient method is applied to full waveform inversion for numerical tests on Marmousi model.The authors give a comparison on numerical results obtained by steepest descent method,conjugate gradient method and spectral conjugate gradient method,which shows that the spectral conjugate gradient method is superior to the other two methods.
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.
基金Basic Research Project of Institute of Earthquake Science,China Earthquake Administrator(No.2013IES010103,2012IES010301)International Science and Technology Cooperation Program of China(No.2012DFR20440)
