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Ordinary differential equations solution in kernel space

NEURAL COMPUTING & APPLICATIONS 2012年第s1期21卷 S79-S85页

作者： Yazdi, Hadi Sadoghi Modaghegh, Hamed Pakdaman, Morteza Islamic Azad Univ Mashhad Branch Sama Tech & Vocat Training Coll Mashhad Iran Ferdowsi Univ Mashhad Dept Engn Mashhad Iran

This paper presents a new method based on the use of an optimization approach along with kernel least mean square (KLMS) algorithm for solving ordinary differential equations (ODEs). The new approach in comparison with the other existing methods (such as numerical methods and the methods that are based on neural networks) has more advantages such as simple implementation, fast convergence, and also little error. In this paper, we use the ability of KLMS in prediction by applying an optimization method to predict the solution of ODE. The basic idea is that first a trial solution of the ODE is written by using the KLMS structure, and then by defining an error function and minimizing it via an optimization algorithm (in this paper, we used the quasi-Newton bfgs method), the parameters of KLMS are adjusted such that the trial solution satisfies the DE. After the optimization step, the achieved optimal parameters of the KLMS model are replaced in the trial solution. The accuracy of the method is illustrated by solving several problems.

关键词： Least mean square Kernel least mean square Ordinary differential equation bfgs algorithm

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Improved analysis-error covariance matrix for high-dimensional variational inversions: application to source estimation using a 3D atmospheric transport model

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QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY 2015年第690期141卷 1906-1921页

作者： Bousserez, N. Henze, D. K. Perkins, A. Bowman, K. W. Lee, M. Liu, J. Deng, F. Jones, D. B. A. Univ Colorado Dept Mech Engn Boulder CO 80309 USA CALTECH Jet Prop Lab Pasadena CA USA Univ Toronto Dept Phys Toronto ON M5S 1A1 Canada Univ Calif Los Angeles JIFRESSE Los Angeles CA USA

Variational methods are widely used to solve geophysical inverse problems. Although gradient-based minimization algorithms are available for high-dimensional problems (dimension >10(6)), they do not provide an estimate of the errors in the optimal solution. In this study, we assess the performance of several numerical methods to approximate the analysis-error covariance matrix, assuming reasonably linear models. The evaluation is performed for a CO2 flux estimation problem using synthetic remote-sensing observations of CO2 columns. A low-dimensional experiment is considered in order to compare the analysis error approximations to a full-rank finite-difference inverse Hessian estimate, followed by a realistic high-dimensional application. Two stochastic approaches, a Monte-Carlo simulation and a method based on random gradients of the cost function, produced analysis error variances with a relative error <10%. The long-distance error correlations due to sampling noise are significantly less pronounced for the gradient-based randomization, which is also particularly attractive when implemented in parallel. Deterministic evaluations of the inverse Hessian using the Broyden-Fletcher-Goldfarb-Shanno (bfgs) algorithm are also tested. While existing bfgs preconditioning techniques yield poor approximations of the error variances (relative error >120%), a new preconditioner that efficiently accumulates information on the diagonal of the inverse Hessian dramatically improves the results (relative error <50%). Furthermore, performing several cycles of the bfgs algorithm using the same gradient and vector pairs enhances its performance (relative error <30%) and is necessary to obtain convergence. Leveraging those findings, we proposed a bfgs hybrid approach which combines the new preconditioner with several bfgs cycles using information from a few (3-5) Monte-Carlo simulations. Its performance is comparable to the stochastic approximations for the low-dimensional case, while g

关键词： data assimilation variational methods uncertainty quantification analysis error bfgs algorithm

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Cooperative Localization of 3D Mobile Networks via Relative Distance and Velocity Measurement

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IEEE COMMUNICATIONS LETTERS 2021年第9期25卷 2899-2903页

作者： Fan, Yingsheng Ding, Kai Qi, Xiaogang Liu, Lifang Xidian Univ Sch Math & Stat Xian 710071 Peoples R China Zhejiang Police Coll Basic Courses Dept Hangzhou 310053 Peoples R China First Sci Res Inst Wuxi Sci & Technol Near Surface Detect Lab Wuxi 214035 Jiangsu Peoples R China Xidian Univ Sch Comp Sci & Technol Xian 710071 Peoples R China

This letter mainly considers the cooperative localization problem of 3D mobile cluster network. With the development of Doppler frequency shift measurement technique, high-precision relative velocity measurement has become possible. In this letter, we first incorporate the relative velocity information into the 3D cooperative localization framework. Then the quasi-Newton algorithm bfgs is used to solve the framework, and the vbfgs algorithm suitable for solving the relative position of the mobile cluster network is constructed. Simulation experiments show that using relative velocity measurement to assist positioning can significantly improve positioning accuracy. Compared with the state-of-the-art algorithm, vbfgs has higher positioning accuracy and faster convergence rate for low-connectivity networks, and has better numerical stability and robustness for high-speed moving models.

关键词： Velocity measurement Optimization Three-dimensional displays Clustering algorithms Location awareness Estimation Convergence Cooperative localization 3D mobile networks relative velocity measurement bfgs algorithm root mean square error

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Optimization of high speed EMU suspension parameters for vibration reduction

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JOURNAL OF MECHANICAL SCIENCE AND TECHNOLOGY 2013年第2期27卷 305-311页

作者： Baek, Seung-Guk Shin, Bumsik Lee, Sang Won Choi, Yeon-Sun Kim, Jinhyun Koo, Ja Choon Sungkyunkwan Univ Sch Mech Engn Suwon 440746 South Korea Seoul Natl Univ Sci & Technol Seoul 139743 South Korea Sungkyunkwan Univ Sch Engn Suwon 440746 South Korea

Various designs of high speed EMU (electric multiple units) operating at maximum speed of 400 km/h or above are under active development in many countries. Driving at such extreme high speed, the EMUs that are equipped with multiple power units and non articulated bogies normally experience severe level of vibration. Noting one of the primary design issues of the vertical EMU vibration, the present work tries an optimization of vehicle suspension system using ten degree of freedom analytical EMU model. The objective function for the work is consisted of the rail power spectral density (PSD), transfer function of car body, and frequency dependent weighting function which may represent human riding comfort. The transfer function is formulated using an autocorrelation of the model in frequency domain. For the enhanced reliability of the optimization work, both the US FRA (Federal Railway Administration) rail PSD and the actual rail PSD acquired from Kyeong-bu high speed rail in Korea on which the particular EMU will be driven are used. The well proven bfgs (Broyden-Fletcher-Goldfarb-Shanno) method is adopted for the optimization. The optimized design is verified using a commercial dynamic system simulation code at various operating conditions.

关键词： Optimization High speed train Suspension Frequency domain bfgs algorithm Electric multiple units

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Improved learning algorithms for mixture of experts in multiclass classification

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NEURAL NETWORKS 1999年第9期12卷 1229-1252页

作者： Chen, K Xu, L Chi, H Peking Univ Natl Lab Machine Percept Beijing 100871 Peoples R China Peking Univ Ctr Informat Sci Beijing 100871 Peoples R China Chinese Univ Hong Kong Dept Comp Sci & Engn Hong Kong Peoples R China

Mixture of experts (ME) is a modular neural network architecture for supervised learning. A double-loop Expectation-Maximization (EM) algorithm has been introduced to the ME architecture for adjusting the parameters and the iteratively reweighted least squares (IRLS) algorithm is used to perform maximization in the inner loop [Jordan, M.L, Jacobs, R.A. (1994). Hierarchical mixture of experts and the EM algorithm, Neural Computation, 6(2), 181-214]. However, it is reported in literature that the IRLS algorithm is of instability and the ME architecture trained by the EM algorithm, where IRLS algorithm is used in the inner loop, often produces the poor performance in multiclass classification. In this paper, the reason of this instability is explored. We find out that due to an implicitly imposed incorrect assumption on parameter independence in multiclass classification, an incomplete Hessian matrix is used in that IRLS algorithm. Based on this finding, we apply the Newton-Raphson method to the inner loop of the EM algorithm in the case of multiclass classification, where the exact Hessian matrix is adopted. To tackle the expensive computation of the Hessian matrix and its inverse, we propose an approximation to the Newton-Raphson algorithm based on a so-called generalized Bernoulli density. The Newton-Raphson algorithm and its approximation have been applied to synthetic data, benchmark, and real-world multiclass classification tasks, For comparison, the IRLS algorithm and a quasi-Newton algorithm called bfgs have also been applied to the same tasks. Simulation results have shown that the use of the proposed learning algorithms avoids the instability problem and makes the ME architecture produce good performance in multiclass classification. In particular, our approximation algorithm leads to fast learning. In addition, the limitation of our approximation algorithm is also empirically investigated in this paper. (C) 1999 Published by Elsevier Science Ltd. All rights

关键词： mixture of experts multiclass classification multinomial density generalized Bernoulli density Expectation-Maximization (EM) algorithm Newton-Raphson method iterative reweighted least squares (IRLS) algorithm bfgs algorithm

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ON THE CONVERGENCE OF PARALLEL bfgs METHOD

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Acta Mathematica Scientia 1995年第3期15卷 283-294页

作者：陈忠费浦生 WUHAN UNIV DEPT MATHWUHAN 430072PEOPLES R CHINA JIANGHAN PETR INST JINSHA 434102PEOPLES R CHINA

According to the sequential bfgs method, in this paper we present an asynchronous parallel bfgs method in the case when the gradient information about the function is inexact. We assume that we have p + q processors, which are divided-into two groups, the first group has p processors, the second group has q processors, the two groups are asynchronous. parallel, If we assume the objective function is twice continuously differentiable and uniformly convex, we prove the iteration converge globally to the solution, and under some additional conditions we show the method is superlinearly convergent. Finally, we show the numerical results of this algorithm.

关键词： bfgs algorithm superlinear convergence parallel method

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A modified bfgs method and its superlinear convergence in nonconvex minimization with general line search rule

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JOURNAL OF APPLIED MATHEMATICS AND COMPUTING 2008年第1-2期28卷 435-446页

作者： Guo, Qiang Liu, Jian-Guo Wang, Dan-Hong Dalian Natl Univ Sch Sci Dalian 116600 Peoples R China Dalian Univ Technol Inst Syst Engn Dalian 116024 Peoples R China Harbin Normal Univ Harbin 150080 Peoples R China

In this paper, a modification of the bfgs algorithm for unconstrained non-convex optimization is proposed. The idea of the algorithm is to modify the approximate Hessian matrix for obtaining the descent direction and guaranteeing the efficacious of the new quasi-Newton iteration equation B(k+1)s(k) = y(k)*, where y(k)* is the sum of y(k) and t(k) parallel to g(x(k))parallel to s(k). The global convergence property of the algorithm associated with the general line search rule is prove.

关键词： Unconstrained programme Non-convex function bfgs algorithm Global convergenc Quasi-Newton method

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Linear scaling geometry optimisation and transition state search in hybrid delocalised internal coordinates

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PHYSICAL CHEMISTRY CHEMICAL PHYSICS 2000年第10期2卷 2177-2186页

作者： Billeter, SR Turner, AJ Thiel, W Univ Zurich Inst Organ Chem CH-8057 Zurich Switzerland Max Planck Inst Kohlenforsch D-45470 Mulheim Germany

An algorithm for linear scaling geometry optimisation and transition state search using hybrid delocalised internal coordinates (HDLC) has been developed and implemented in the context of a semiempirical quantum-chemistry program (MNDO) and a modular quantum-mechanical/molecular-mechanical (QM/MM) package (ChemShell). Linear scaling is achieved by a divide-and-conquer approach: the system is partitioned into user-defined fragments, and all coordinate manipulations are performed exclusively within these fragments. The optimiser employs a limited-memory quasi-Newton algorithm (L-bfgs) for energy minimisation, and a microiterative scheme for transition state search using a Hessian eigenmode-following algorithm (P-RFO) for the reaction core and the L-bfgs algorithm for the environment. There are automatic procedures for generating redundant sets of internal coordinates and non-redundant sets of HDLC from Cartesian coordinates. The input to the optimiser consists of the initial Cartesian geometry, the fragmentation of the system, the choice of the working coordinate system, and any constraints to be imposed in Cartesian and/or internal coordinates. The optimiser requires an external function that provides the energy and gradient at a given Cartesian geometry. Systems with thousands of atoms have been optimised, and transition states of a model enzymatic reaction have been determined.

关键词： .QM/MM Linear scaling Cartesian internal coordinates energy minimisation non-redundant Geometry Optimisation Zu rich HDLc ChemShell optimiser microiterative reaction core bfgs algorithm

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NUMERICAL COMPARISON OF SEVERAL VARIABLE-METRIC algorithmS

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1978年第4期25卷 507-518页

作者： SHANNO, DF PHUA, KH NANYANG UNIV DEPT COMP SCI SINGAPORE SINGAPORE

The paper compares the numerical performances of the LDL′ decomposition of the bfgs variable-metric algorithm, the Dennis-Mei dogleg algorithm on the bfgs update, and Davidon's projections with the bfgs update with the straight bfgs update on a number of standard test problems. Numerical results indicate that the standard bfgs algorithm is superior to all of the more complex strategies.

关键词： Unconstrained optimization LDL′ decomposition Davidon projections dogleg algorithm bfgs algorithm

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GLOBAL CONVERGENCE PROPERTIES OF TWO MODIFIED bfgs-TYPE METHODS

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JOURNAL OF APPLIED MATHEMATICS AND COMPUTING 2007年第1-2期23卷 311-319页

作者： Guo, Qiang Liu, Jian-Guo Dalian Nationalities Univ Coll Sci Dalian 116600 Peoples R China Dalian Univ Technol Inst Syst Engn Dalian 116024 Peoples R China

This article studies a modified bfgs algorithm for solving smooth unconstrained strongly convex minimization problem. The modified bfgs method is based on the new quasi-Newton equation Bk+1 s(k) = y(k) where y(k)* = y(k) + A(k)s(k) and A(k) is a matrix. Wei, Li and Qi [WLQ] have proven that the average performance of two of those algorithms is better than that of the classical one. In this paper, we prove the global convergence of these algorithms associated to a general line search rule.

关键词： Unconstrained programme bfgs algorithm global convergence quasi-Newton method

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