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1
Fairbank, Michael, and Eduardo Alonso. "Efficient Calculation of the Gauss-Newton Approximation of the Hessian Matrix in Neural Networks." Neural Computation 24, no. 3 (2012): 607–10. http://dx.doi.org/10.1162/neco_a_00248.
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The Levenberg-Marquardt (LM) learning algorithm is a popular algorithm for training neural networks; however, for large neural networks, it becomes prohibitively expensive in terms of running time and memory requirements. The most time-critical step of the algorithm is the calculation of the Gauss-Newton matrix, which is formed by multiplying two large Jacobian matrices together. We propose a method that uses backpropagation to reduce the time of this matrix-matrix multiplication. This reduces the overall asymptotic running time of the LM algorithm by a factor of the order of the number of output nodes in the neural network.
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Wright, S. J., and J. N. Holt. "An inexact Levenberg-Marquardt method for large sparse nonlinear least squres." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 26, no. 4 (1985): 387–403. http://dx.doi.org/10.1017/s0334270000004604.
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AbstractA method for solving problems of the form is presented. The approach of Levenberg and Marquardt is used, except that the linear least squares subproblem arising at each iteration is not solved exactly, but only to within a certain tolerance. The method is most suited to problems in which the Jacobian matrix is sparse. Use is made of the iterative algorithm LSQR of Paige and Saunders for sparse linear least squares.A global convergence result can be proven, and under certain conditions it can be shown that the method converges quadratically when the sum of squares at the optimal point is zero.Numerical test results for problems of varying residual size are given.
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Al-Abrahemee, Khalid Mindeel M., and Rana T. Shwayaa. "Modification of Levenberg-Marquardt Algorithm for Solve Two Dimension Partial Differential Equation." JOURNAL OF UNIVERSITY OF BABYLON for Pure and Applied Sciences 26, no. 7 (2018): 107–17. http://dx.doi.org/10.29196/jubpas.v26i7.1417.
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In this paper we presented a new way based on neural network has been developed for solutione of two dimension partial differential equations . A modified neural network use to over passing the Disadvantages of LM algorithm, in the beginning we suggest signaler value decompositions of Jacobin matrix (J) and inverse of Jacobin matrix( J-1), if a matrix rectangular or singular Secondly, we suggest new calculation of μk , that ismk=|| E (w)||2 look the nonlinear execution equations E(w) = 0 has not empty solution W* and we refer to the second norm in all cases ,whereE(w): is continuously differentiable and E(x) is Lipeschitz continuous, that is=|| E(w 2)- E(w 1)||£ L|| w 2- w 1|| ,where L is Lipeschitz constant.
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Abdelrahman, Moataz Mohamed Gomaa, Norbert Péter Szabó, and Mihály Dobróka. "Meta-algorithm assisted interval inversion for petrophysical properties prediction." Multidiszciplináris tudományok 12, no. 4 (2022): 242–60. http://dx.doi.org/10.35925/j.multi.2022.4.26.
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Well logging inversion was carried out using Levenberg-Marquardt (LM) and Singular Value Decomposition (SVD) techniques for the determination of petrophysical parameters, respectively. In this research, synthetic data contaminated with 5% Gaussian noise, and field data were used to compare the results from the two inversion methods. MATLAB software has been developed to solve the overdetermined inverse problem. The estimated petrophysical parameters from both inversion methods had been compared to one another in terms of robustness to noise, rock interface differentiation, different fluid prediction, and the accuracy of the estimated parameters. This research returns the reason to the inner iterative loop which is considered more about the Jacobian matrix sensitivity. The inversion results showed that both methods can be used in petrophysical data estimation for a reliable well-log data interpretation.
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Žic, Mark, Vanja Subotić, Sergei Pereverzyev, and Iztok Fajfar. "Solving CNLS problems using Levenberg-Marquardt algorithm: A new fitting strategy combining limits and a symbolic Jacobian matrix." Journal of Electroanalytical Chemistry 866 (June 2020): 114171. http://dx.doi.org/10.1016/j.jelechem.2020.114171.
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Chi, Zhongyuan, Yuzhang Ji, Ningning Liu, Tianchi Jiang, Xin Liu, and Weijun Zhang. "Algorithms for Solving the Equilibrium Composition Model of Arc Plasma." Entropy 27, no. 1 (2024): 24. https://doi.org/10.3390/e27010024.
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In the present study, the Homotopy Levenberg−Marquardt Algorithm (HLMA) and the Parameter Variation Levenberg–Marquardt Algorithm (PV–LMA), both developed in the context of high-temperature composition, are proposed to address the equilibrium composition model of plasma under the condition of local thermodynamic and chemical equilibrium. This model is essentially a nonlinear system of weakly singular Jacobian matrices. The model was formulated on the basis of the Saha and Guldberg–Waage equations, integrated with Dalton’s law of partial pressures, stoichiometric equilibrium, and the law of conservation of charge, resulting in a nonlinear system of equations with a weakly singular Jacobian matrix. This weak singularity primarily arises due to significant discrepancies in the coefficients between the Saha equation and the Guldberg–Waage equation, attributed to differing chemical reaction energies. By contrast, the coefficients in the equations derived from the other three principles within the equilibrium composition model are predominantly single−digit constants, further contributing to the system’s weak singularity. The key to finding the numerical solution to nonlinear equations is to set reasonable initial values for the iterative solution process. Subsequently, the principle and process of the HLMA and PV–LMA algorithms are analyzed, alongside an analysis of the unique characteristics of plasma equilibrium composition at high temperatures. Finally, a solving method for an arc plasma equilibrium composition model based on high temperature composition is obtained. The results show that both HLMA and PV–LMA can solve the plasma equilibrium composition model. The fundamental principle underlying the homotopy calculation of the (n−1) −th iteration, which provides a reliable initial value for the n−th LM iteration, is particularly well suited for the solution of nonlinear equations. A comparison of the computational efficiency of HLMA and PV–LMA reveals that the latter exhibits superior performance. Both HLMA and PV–LMA demonstrate high computational accuracy, as evidenced by the fact that the variance of the system of equations ||F|| < 1 × 10−15. This finding serves to substantiate the accuracy and feasibility of the method proposed in this paper.
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Fu, Zhengqing, and Lanlan Guo. "Tikhonov Regularized Variable Projection Algorithms for Separable Nonlinear Least Squares Problems." Complexity 2019 (November 25, 2019): 1–9. http://dx.doi.org/10.1155/2019/4861708.
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This paper considers the classical separable nonlinear least squares problem. Such problems can be expressed as a linear combination of nonlinear functions, and both linear and nonlinear parameters are to be estimated. Among the existing results, ill-conditioned problems are less often considered. Hence, this paper focuses on an algorithm for ill-conditioned problems. In the proposed linear parameter estimation process, the sensitivity of the model to disturbance is reduced using Tikhonov regularisation. The Levenberg–Marquardt algorithm is used to estimate the nonlinear parameters. The Jacobian matrix required by LM is calculated by the Golub and Pereyra, Kaufman, and Ruano methods. Combining the nonlinear and linear parameter estimation methods, three estimation models are obtained and the feasibility and stability of the model estimation are demonstrated. The model is validated by simulation data and real data. The experimental results also illustrate the feasibility and stability of the model.
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Žic, M., L. Vlašić, V. Subotić, S. Pereverzyev, I. Fajfar, and M. Kunaver. "Extraction of Distribution Function of Relaxation Times by using Levenberg-Marquardt Algorithm: A New Approach to Apply a Discretization Error Free Jacobian Matrix." Journal of The Electrochemical Society 169, no. 3 (2022): 030508. http://dx.doi.org/10.1149/1945-7111/ac55c9.
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Nowadays, Electrochemical Impedance Spectroscopy is attracting more attention due to an increasing production of power sources. One of highly popular tools to diagnose diverse power sources is Distribution Function of Relaxation Times (DRT). Because of that, there are numerous approaches to extract DRT from impedance data. The majority of them are based on the numerical approximation of integral. However, herein we have applied an analytical approximation of the EIS integral. For the first time, we have employed Levenberg-Marquardt algorithm (LMA) to extract the applicable DRT from impedance data by using the Jacobian matrix that was obtained without any discretization errors. Although LMA was previously used to fit EIS data by DRT characteristics, the DRT profile was not applicable due to discretization errors. In this work, LMA was applied as it has an automatic update of the regularization (λ) parameter. The tests conducted in this work have shown that LMA is capable of extracting DRT from ZARC and FRAC synthetic data.
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Tang, Jinping, Bo Han, Weimin Han, Bo Bi, and Li Li. "Mixed Total Variation and L1 Regularization Method for Optical Tomography Based on Radiative Transfer Equation." Computational and Mathematical Methods in Medicine 2017 (2017): 1–15. http://dx.doi.org/10.1155/2017/2953560.
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Optical tomography is an emerging and important molecular imaging modality. The aim of optical tomography is to reconstruct optical properties of human tissues. In this paper, we focus on reconstructing the absorption coefficient based on the radiative transfer equation (RTE). It is an ill-posed parameter identification problem. Regularization methods have been broadly applied to reconstruct the optical coefficients, such as the total variation (TV) regularization and the L1 regularization. In order to better reconstruct the piecewise constant and sparse coefficient distributions, TV and L1 norms are combined as the regularization. The forward problem is discretized with the discontinuous Galerkin method on the spatial space and the finite element method on the angular space. The minimization problem is solved by a Jacobian-based Levenberg-Marquardt type method which is equipped with a split Bregman algorithms for the L1 regularization. We use the adjoint method to compute the Jacobian matrix which dramatically improves the computation efficiency. By comparing with the other imaging reconstruction methods based on TV and L1 regularizations, the simulation results show the validity and efficiency of the proposed method.
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Huang, Chien-Lin, Nien-Sheng Hsu, Fu-Jian Hsu, Gene J. Y. You, and Chun-Hao Yao. "Symmetrical Rank-Three Vectorized Loading Scores Quasi-Newton for Identification of Hydrogeological Parameters and Spatiotemporal Recharges." Water 12, no. 4 (2020): 995. http://dx.doi.org/10.3390/w12040995.
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In a multi-layered groundwater model, achieving accurate spatiotemporal identification and solving the ill-posed problem is the vital topic for model calibration. This study proposes a symmetry rank three vectorized loading scores (SR3 VLS) quasi-Newton algorithm by modifying the Levenberg–Marquardt algorithm and importing a rank three structure from Broyden–Fletcher–Goldfarb–Shanno algorithm for identification of hydrogeological parameters and spatiotemporal recharge simultaneously. To accelerate directional convergence and approach a global optimum, this study uses a vectorized limited switchable step size in the transmissive groundwater inverse problem. The Hessian approximation rank three uses high and low-rank factor loading scores analyzed from simulated storage fluctuation between adjacent iterations for calculation and matrix correction. Two numerical experiments were designed to validate the proposing algorithm, showing the SR3 VLS quasi-Newton reduced the error percentages of the identified parameters by 1.63% and 9.65% compared to the Jacobian quasi-Newton. The proposing method is applied to the Chou-Shui River alluvial fan groundwater system in Taiwan. Results show that the simulated storage error decreased rapidly in six iterations, and has good head convergence as small as 0.11% with a root-mean-square-error (RMSE) of 0.134 m, indicating that the proposing algorithm reduces the computational cost to converge to the true solution.
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Dulęba, Ignacy, and Michał Opałka. "A comparison of Jacobian-based methods of inverse kinematics for serial robot manipulators." International Journal of Applied Mathematics and Computer Science 23, no. 2 (2013): 373–82. http://dx.doi.org/10.2478/amcs-2013-0028.
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The objective of this paper is to present and make a comparative study of several inverse kinematics methods for serial manipulators, based on the Jacobian matrix. Besides the well-known Jacobian transpose and Jacobian pseudo-inverse methods, three others, borrowed from numerical analysis, are presented. Among them, two approximation methods avoid the explicit manipulability matrix inversion, while the third one is a slightly modified version of the Levenberg-Marquardt method (mLM). Their comparison is based on the evaluation of a short distance approaching the goal point and on their computational complexity. As the reference method, the Jacobian pseudo-inverse is utilized. Simulation results reveal that the modified Levenberg-Marquardt method is promising, while the first order approximation method is reliable and requires mild computational costs. Some hints are formulated concerning the application of Jacobian-based methods in practice.
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Zhu, Dingyu, Yueting Yang, and Mingyuan Cao. "An accelerated adaptive two-step Levenberg–Marquardt method with the modified Metropolis criterion." AIMS Mathematics 9, no. 9 (2024): 24610–35. http://dx.doi.org/10.3934/math.20241199.
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<p>In this paper, aiming at the nonlinear equations, a new two-step Levenberg–Marquardt method was proposed. We presented a new Levenberg–Marquardt parameter to obtain the trial step. A new modified Metropolis criterion was used to adjust the upper bound of the approximate step. The convergence of the method was analyzed under the H$ \ddot{\rm o} $lderian local error bound condition and the H$ \ddot\rm o $lderian continuity of the Jacobian. Numerical experiments showed that the new algorithm is effective and competitive in the numbers of functions, Jacobian evaluations and iterations.</p>
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Zhao, Ligang, Hua Zheng, Hongyue Zhen, Li Xie, Yuan Xu, and Xianchao Huang. "Improvement of Fuzzy Newton Power Flow Convergence." Energies 16, no. 24 (2023): 8044. http://dx.doi.org/10.3390/en16248044.
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In order to address the convergence issue in fuzzy power flow calculations, this paper proposes an analytical approach based on the Levenberg–Marquardt method, aiming to improve the convergence of the fuzzy Newton power flow method. Firstly, a detailed analysis is conducted on the convergence theorem and convergence behavior of the fuzzy Newton method, revealing its poor convergence when the initial values are not properly selected. The Levenberg–Marquardt method is then selected as a means to enhance the convergence of the fuzzy Newton power flow calculations, specifically to tackle the problem of initial value deviation. Since the Jacobian matrix has a significant impact on the convergence region of the power flow, this paper reconstructs the Jacobian matrix based on the Levenberg–Marquardt method, effectively enlarging the convergence region. Through validation experiments on the IEEE 118 standard nodes and simulation comparative analysis, the results confirm the method’s effectiveness in resolving the problem of initial value deviation and notably enlarging the convergence region, thereby improving the convergence of power flow calculations.
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Zheng, Lin, Liang Chen, and Yangxin Tang. "Convergence rate of the modified Levenberg-Marquardt method under Hölderian local error bound." Open Mathematics 20, no. 1 (2022): 998–1012. http://dx.doi.org/10.1515/math-2022-0485.
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Abstract In this article, we analyze the convergence rate of the modified Levenberg-Marquardt (MLM) method under the Hölderian local error bound condition and the Hölderian continuity of the Jacobian, which are more general than the local error bound condition and the Lipschitz continuity of the Jacobian. Under special circumstances, the convergence rate of the MLM method coincides with the results presented by Fan. A globally convergent MLM algorithm by the trust region technique will also be given.
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Parker, Danny L., William G. Frazier, and Mathew A. Gray. "Damage Localization Using Levenberg-Marquardt Optimization." Key Engineering Materials 347 (September 2007): 95–100. http://dx.doi.org/10.4028/www.scientific.net/kem.347.95.
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In this paper, an optimal solution method is proposed for determining the location of change, i.e. damage, within a perturbed system utilizing a nonlinear pseudo-second order search algorithm based on function evaluations and gradient information. This method is applied to damped vibrating systems and utilizes stiffness matrix sensitivities to determine the direction of search within the estimation. The site of damage (location of change) is the solution which minimizes the error between the predicted and measured change. A by-product of the Levenberg- Marquardt algorithm is an estimation of the magnitude of the change within the system which correlates to damage extent. A second-order model of a dynamic system is used, and an approximation is developed to describe small perturbations within the system.
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Xia, Jianghai, Richard D. Miller, and Choon B. Park. "Estimation of near‐surface shear‐wave velocity by inversion of Rayleigh waves." GEOPHYSICS 64, no. 3 (1999): 691–700. http://dx.doi.org/10.1190/1.1444578.
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The shear‐wave (S-wave) velocity of near‐surface materials (soil, rocks, pavement) and its effect on seismic‐wave propagation are of fundamental interest in many groundwater, engineering, and environmental studies. Rayleigh‐wave phase velocity of a layered‐earth model is a function of frequency and four groups of earth properties: P-wave velocity, S-wave velocity, density, and thickness of layers. Analysis of the Jacobian matrix provides a measure of dispersion‐curve sensitivity to earth properties. S-wave velocities are the dominant influence on a dispersion curve in a high‐frequency range (>5 Hz) followed by layer thickness. An iterative solution technique to the weighted equation proved very effective in the high‐frequency range when using the Levenberg‐Marquardt and singular‐value decomposition techniques. Convergence of the weighted solution is guaranteed through selection of the damping factor using the Levenberg‐Marquardt method. Synthetic examples demonstrated calculation efficiency and stability of inverse procedures. We verify our method using borehole S-wave velocity measurements.
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Smolik, Waldemar, and Jacek Kryszyn. "LINEAR OVER RANGES ITERATIVE ALGORITHMS FOR IMAGE RECONSTRUCTION IN ELECTRICAL CAPACITANCE TOMOGRAPHY." Informatics Control Measurement in Economy and Environment Protection 7, no. 1 (2017): 115–20. http://dx.doi.org/10.5604/01.3001.0010.4598.
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The paper concerns the non-linear algorithms for image reconstruction in electrical capacitance tomography for which Jacobi matrix computation time is very long. The paper presents the idea of an iterative linearization in nonlinear problems, which leads to a reduction in the number of steps calculating Jacobi matrix. The linear Landweber algorithm with sensitivity matrix updating and non-linear Levenberg-Marquardt algorithm with Jacobi matrix updating in selected steps only were presented.
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Abdelrahman, Moataz Mohamed Gomaa, Norbert Péter Szabó, and Mihály Dobróka. "Petrophysical parameters estimation using Levenberg-Marquardt and Singular Value Decomposition inversion schemes." Multidiszciplináris tudományok 11, no. 5 (2021): 24–38. http://dx.doi.org/10.35925/j.multi.2021.5.3.
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Well logging inversion was carried out using Levenberg-Marquardt (LM) and Singular Value Decomposition (SVD) techniques for the determination of petrophysical parameters, respectively. In this research, synthetic data contaminated with 5% Gaussian noise, and field data were used to compare the results from the two inversion methods. MATLAB software has been developed to solve the overdetermined inverse problem. The estimated petrophysical parameters from both inversion methods had been compared to one another in terms of robustness to noise, rock interface differentiation, different fluid prediction, and the accuracy of the estimated parameters. This research returns the reason to the inner iterative loop which is considered more about the Jacobian matrix sensitivity. The inversion results showed that both methods can be used in petrophysical data estimation for a reliable well-log data interpretation.
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Dillak, Rocky Yefrenes. "IDENTIFIKASI IRIS MENGGUNAKAN IMPROVED LEVENBERG-MARQUARDT." Jurnal Ilmiah Flash 1, no. 1 (2015): 10. http://dx.doi.org/10.32511/jiflash.v1i1.12.
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Sistem biometrika adalah suatu sistem pengenalan diri menggunakan bagian tubuh atau perilaku manusia seperti sidik jari, telapak tangan, telinga, retina, iris mata, wajah, suhu tubuh, tanda tangan, dll. Iris mata merupakan salah satu biometrika yang sangat stabil, handal, akurat dan merupakan metode autentikasi biometrika tercepat oleh karena itu merupakan suatu topik penelitian yang sangat diminati oleh banyak peneliti. Penelitian ini bertujuan untuk mengembangkan suatu metode yang dapat digunakan untuk mengidentifikasi secara otomatis seseorang berdasarkan citra iris mata miliknya menggunakan jaringan syaraf tiruan levenberg-marquardt. Penelitian ini menggunakan metode deteksi tepi cany dan circular hough transform untuk segmentasi wilayah iris yang terletak diantara pupil dan sclera serta metode ekstraksi ciri gray level cooccurence matrix (GLCM) yang digunakan untuk ekstraksi ciri. Ciri-ciri tersebut adalah maximum probability, correlation, contrast, energy, homogeneity, dan entropy. Ciri-ciri tersebut kemudian dilatih menggunakan jaringan syaraf tiruan dengan aturan pembelajaran levenberg–marquardt algorithm untuk mengidentifikasi seseorang berdasarkan citra irisnya. Penelitian ini menggunakan 150 data citra iris yang masing-masing terbagi atas 100 data citra iris untuk pelatihan dan 50 data citra iris untuk pengujian. Berdasarkan hasil pengujian yang dilakukan diperoleh correct recognition rate (CRR) sebesar 99.98% yang menunjukkan bahwa metode ini dapat digunakan untuk mengidentifikasi secara otomatis seseorang berdasarkan citra iris mata miliknya.
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Kandel, Saugat, S. Maddali, Youssef S. G. Nashed, Stephan O. Hruszkewycz, Chris Jacobsen, and Marc Allain. "Efficient ptychographic phase retrieval via a matrix-free Levenberg-Marquardt algorithm." Optics Express 29, no. 15 (2021): 23019. http://dx.doi.org/10.1364/oe.422768.
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Zhang, Lei. "Optimization of Projection Matrix between Cameras Based on Levenberg-marquardt Algorithm." Journal of Information and Computational Science 12, no. 4 (2015): 1607–14. http://dx.doi.org/10.12733/jics20105445.
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Blom, Philip S. "What else can we do with auxiliary parameters in ray tracing? How about back projection for localization?" Journal of the Acoustical Society of America 155, no. 3_Supplement (2024): A72. http://dx.doi.org/10.1121/10.0026846.
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Auxiliary parameters describing variations in ray path geometry with respect to initial launch angles have been leveraged in computing the Jacobian determinant needed to solve the transport equation as well as to build a Levenberg–Marquardt algorithm for identification of source-receiver paths (termed eigenrays) in a 3D inhomogeneous moving atmosphere. Building on these applications, recent investigations have demonstrated that these parameters can be computed along back projected ray paths from infrasonic detections to improve Bayesian localization capabilities. An overview of the auxiliary parameters as introduced in previous work will be provided along with a summary of current localization development. Example applications of the method will be presented and compared with existing Bayesian infrasonic localization methods using more general propagation models.
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Xie, Changping, Xinjian Fang, and Xu Yang. "Improved Kalman Filtering Algorithm Based on Levenberg–Marquart Algorithm in Ultra-Wideband Indoor Positioning." Sensors 24, no. 22 (2024): 7213. http://dx.doi.org/10.3390/s24227213.
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To improve the current indoor positioning algorithms, which have insufficient positioning accuracy, an ultra-wideband (UWB) positioning algorithm based on the Levenberg–Marquardt algorithm with improved Kalman filtering is proposed. An alternative double-sided two-way ranging (ADS-TWR) algorithm is used to obtain the distance from the UWB tag to each base station and calculate the initial position of the tag by the least squares method. The Levenberg–Marquardt algorithm is used to correct the covariance matrix of the Kalman filter, and the improved Kalman filtering algorithm is used to filter the initial position to obtain the final position of the tag. The feasibility and effectiveness of the algorithm are verified by MATLAB simulation. Finally, the UWB positioning system is constructed, and the improved Kalman filter algorithm is experimentally verified in LOS and NLOS environments. The average X-axis and the Y-axis positioning errors in the LOS environment are 6.9 mm and 5.4 mm, respectively, with a root mean square error of 10.8 mm. The average positioning errors for the X-axis and Y-axis in the NLOS environment are 20.8 mm and 18.0 mm, respectively, while the root mean square error is 28.9 mm. The experimental results show that the improved algorithm has high accuracy and good stability. At the same time, it can effectively improve the convergence speed of the Kalman filter.
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Huang, Li Xin, Xiang Wu Guo, Bo Tao Du, Xiao Jun Zhou, and Yu Yin Liu. "Optimal Measurement Placement for Material Parameter Identification of Orthotropic Composites by the Finite Element Method." Applied Mechanics and Materials 94-96 (September 2011): 1723–28. http://dx.doi.org/10.4028/www.scientific.net/amm.94-96.1723.
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An algorithm of the optimal measurement placement is proposed for the material parameter identification of two-dimensional orthotropic composites, which is modeled by the finite element. From the analysis of the system sensitivity matrix of the parameter identification processes using the Levenberg-Marquardt method, A-optimality criterion related with the Fisher Information Matrix (FIM) is selected for the criterion of the optimal measurement placement. Thus, the algorithm for selecting the optimal measurement placement can be constructed. A numerical example is given to demonstrate the effectiveness of the proposed algorithm. The example reveals that the measurement placement has a significant influence on the identification results.
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Xiao, Zhuolei, Yerong Zhang, Kaixuan Zhang, Dongxu Zhao, and Guan Gui. "GARLM: Greedy Autocorrelation Retrieval Levenberg–Marquardt Algorithm for Improving Sparse Phase Retrieval." Applied Sciences 8, no. 10 (2018): 1797. http://dx.doi.org/10.3390/app8101797.
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The goal of phase retrieval is to recover an unknown signal from the random measurements consisting of the magnitude of its Fourier transform. Due to the loss of the phase information, phase retrieval is considered as an ill-posed problem. Conventional greedy algorithms, e.g., greedy spare phase retrieval (GESPAR), were developed to solve this problem by using prior knowledge of the unknown signal. However, due to the defect of the Gauss–Newton method in the local convergence problem, especially when the residual is large, it is very difficult to use this method in GESPAR to efficiently solve the non-convex optimization problem. In order to improve the performance of the greedy algorithm, we propose an improved phase retrieval algorithm, which is called the greedy autocorrelation retrieval Levenberg–Marquardt (GARLM) algorithm. Specifically, the proposed GARLM algorithm is a local search iterative algorithm to recover the sparse signal from its Fourier transform magnitude. The proposed algorithm is preferred to existing greedy methods of phase retrieval, since at each iteration the problem of minimizing the objective function over a given support is solved by using the improved Levenberg–Marquardt (ILM) method and matrix transform. A local search procedure such as the 2-opt method is then invoked to get the optimal estimation. Simulation results are given to show that the proposed algorithm performs better than the conventional GESPAR algorithm.
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OLIVEIRA, J. A. F., M. M. L. DUARTE, E. L. FOLETTO, and O. CHIAVONE-FILHO. "LEVENBERG-MARQUARDT METHOD APPLIED TO THE DETERMINATION OF VAPOR-LIQUID EQUILIBRIUM MODEL PARAMETERS." Latin American Applied Research - An international journal 44, no. 4 (2014): 319–24. http://dx.doi.org/10.52292/j.laar.2014.460.
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In order to correlate and optimize experimental data either from the laboratory or industry, one needs a robust method of data regression. Among the non-linear parameter estimation methods it may be pointed out of Levenberg, which applies the conversion of an arbitrary matrix into a positive definite one. Later, Marquardt applied the same procedure, calculating parameter in an iterative form. The Levenberg-Marquardt algorithm is described and two routine for correlating vaporliquid equilibrium data for pure component and mixtures, based on this efficient method, have been applied. The routines have been written with an interface very accessible for both users and programmers, using Python language. The flexibility of the developed programs for introducing the desired details is quite interesting for both process simulators and modeling properties. Furthermore, for mixtures with electrolytes, it was obtained a coherent and compatible relation for the structural parameters of the salt species, with the aid of the method and the graphical interface designed.
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Foroutan, Armina, Sagnik Basumallik, and Anurag Srivastava. "Estimating and Calibrating DER Model Parameters Using Levenberg–Marquardt Algorithm in Renewable Rich Power Grid." Energies 16, no. 8 (2023): 3512. http://dx.doi.org/10.3390/en16083512.
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The proliferation of inverter-based distributed energy resources (IBDERs) has increased the number of control variables and dynamic interactions, leading to new grid control challenges. For stability analysis and designing appropriate protection controls, it is important that IBDER models are accurate. This paper focuses on the accurate estimation and parameter calibration of DER_A, a recently proposed aggregated IBDER model. In particular, we focus on the parameters of the reactive power–voltage regulation module. We formulate the problem of parameter tuning as a non-linear least square minimization problem and solve it using the Levenberg–Marquardt (LM) method. The LM method is primarily chosen due to its flexibility in adaptively selecting between the steepest descent and Gauss–Newton methods through a damping parameter. The LM approach is used to minimize the error between the actual measurements and the estimated response of the model. Further, the computational challenges posed by the numerical calculation of the Jacobian are tackled using a quasi-Newton root-finding approach. The proposed method is validated on a real feeder model in the northeastern part of the United States. The feeder is modeled in OpenDSS and the measurements thus obtained are fed to the DER_A model for calibration. The simulation results indicate that our approach is able to successfully calibrate the relevant model parameters quickly and with high accuracy, with a total sum of square error of 3.57×10−7.
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Wang, Wei, Yunming Pu, and Wang Li. "A Parallel Levenberg-Marquardt Algorithm for Recursive Neural Network in a Robot Control System." International Journal of Cognitive Informatics and Natural Intelligence 12, no. 2 (2018): 32–47. http://dx.doi.org/10.4018/ijcini.2018040103.
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This article has the purpose of overcoming the shortcomings of the recursive neural network learning algorithm and the inherent delay problem on the manipulator master control system. This is by analyzing the shortcomings of LM learning algorithms based on DRNN network, an improved parallel LM algorithm is proposed. The parallel search of the damping coefficient β is found in order to reduce the number of iterations of the loop, and the algorithm is used to decompose the parameter operation and the matrix operation into the processor (core), thereby improve the learning convergence speed, and control the scale of the delay. The simulation results show that the pro-posed algorithm is feasible.
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Zhu, Tianfei, and Larry D. Brown. "Two‐dimensional velocity inversion and synthetic seismogram computation." GEOPHYSICS 52, no. 1 (1987): 37–50. http://dx.doi.org/10.1190/1.1442239.
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A traveltime inversion scheme has been developed to estimate velocity and interface geometries of two‐dimensional media from deep reflection data. The velocity structure is represented by finite elements, and the inversion is formulated as an iterative, constrained, linear least‐squares problem which can be solved by either the singular value truncation method or the Levenberg‐Marquardt method. The damping factor of the Levenberg‐Marquardt method is chosen by the model‐trust region approach. The traveltimes and derivative matrix required to solve the least‐squares problem are computed by ray tracing. To aid seismic interpretation, we also include in the inversion scheme a fast algorithm based on asymptotic ray theory for calculating synthetic seismograms from the derived velocity model. Numerical tests indicate that the inversion scheme is effective, and that the accuracy of inversion results depends upon both noise in the data and the aperture of recording used in data acquisition. Two real examples demonstrate that the new inversion scheme produces velocity models fitting the data better than those estimated by other approaches.
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Uygur, I., A. Cicek, E. Toklu, R. Kara, and S. Saridemir. "Fatigue Life Predictions of Metal Matrix Composites Using Artificial Neural Networks." Archives of Metallurgy and Materials 59, no. 1 (2014): 97–103. http://dx.doi.org/10.2478/amm-2014-0016.
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Abstract In this study, fatigue life predictions for the various metal matrix composites, R ratios, notch geometries, and different temperatures have been performed by using artificial neural networks (ANN) approach. Input parameters of the model comprise various materials (M), such as particle size and volume fraction of reinforcement, stress concentration factor (Kt), R ratio (R), peak stress (S), temperatures (T), whereas, output of the ANN model consist of number of failure cycles. ANN controller was trained with Levenberg-Marquardt (LM) learning algorithm. The tested actual data and predicted data were simulated by a computer program developed on MATLAB platform. It is shown that the model provides intimate fatigue life estimations compared with actual tested data.
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JEMAI, OLFA, MOURAD ZAIED, CHOKRI BEN AMAR, and MOHAMED ADEL ALIMI. "FAST LEARNING ALGORITHM OF WAVELET NETWORK BASED ON FAST WAVELET TRANSFORM." International Journal of Pattern Recognition and Artificial Intelligence 25, no. 08 (2011): 1297–319. http://dx.doi.org/10.1142/s0218001411009111.
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In this paper, a novel learning algorithm of wavelet networks based on the Fast Wavelet Transform (FWT) is proposed. It has many advantages compared to other algorithms, in which we solve the problem in previous works, when the weights of the hidden layer to the output layer are determined by applying the back propagation algorithm or by direct solution which requires to compute the matrix inversion, this may cause intensive computation when the learning data is too large. However, the new algorithm is realized by iterative application of FWT to compute the connection weights. Furthermore, we have extended the novel learning algorithm by using Levenberg–Marquardt method to optimize the learning functions. The experimental results have demonstrated that our model is remarkably more refreshing than some of the previously established models in terms of both speed and efficiency.
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Smirnova, Alexandra, Benjamin Sirb, and Gerardo Chowell. "On Stable Parameter Estimation and Forecasting in Epidemiology by the Levenberg–Marquardt Algorithm with Broyden’s Rank-one Updates for the Jacobian Operator." Bulletin of Mathematical Biology 81, no. 10 (2019): 4210–32. http://dx.doi.org/10.1007/s11538-019-00650-9.
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Wang, Xuetao, Yijun Gao, Dawei Lu, Yanbo Li, Kai Du, and Weiyu Liu. "Lithium Battery SoC Estimation Based on Improved Iterated Extended Kalman Filter." Applied Sciences 14, no. 13 (2024): 5868. http://dx.doi.org/10.3390/app14135868.
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With the application of lithium batteries more and more widely, in order to accurately estimate the state of charge (SoC) of the battery, this paper uses the iterated extended Kalman filter (IEKF) algorithm to estimate the SoC. The Levenberg–Marquardt (LM) method is used to optimize the error covariance matrix of IKEF. Based on the hybrid pulse power characteristics experiment, a second-order Thevenin model with variable parameters is established on the MATLAB platform. The experimental results show that the proposed model is effective under the constant current discharge condition, the Federal Urban Driving Schedule (FUDS) condition, and the Beijing dynamic stress test (BJDST) condition. The results show that the simulation error of the improved LM-IEKF algorithm is less than 2% under different working conditions, which is lower than that of the IKEF algorithm. The improved algorithm has a fast convergence speed to the true value, and it has a good estimation accuracy in the case of large changes in external input current. Additionally, the fluctuation of error is relatively stable, which proves the reliability of the algorithm.
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Sadeghi, Mohsen, and Mohammad Farrokhi. "Online identification of non-linear dynamic systems by Wiener model using subspace method and neural networks." Transactions of the Institute of Measurement and Control 40, no. 2 (2016): 666–74. http://dx.doi.org/10.1177/0142331216663618.
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This paper presents a method for online identification of non-linear dynamic systems using the Wiener model. For the linear dynamic part the subspace identification method with the multivariable output-error state-space algorithm is employed, whereas for the non-linear static part the multi-layer perceptron neural network with Levenberg–Marquardt algorithm is used. The stability and convergence of the proposed method is shown using the Lyapunov direct method and the region solution of the linear matrix inequality (LMI) approach. The proposed method is tested by simulations performed on the continuous stirred tank reactor (CSTR) plant, which is presented by non-linear differential equations. Moreover, the method is applied on the input–output data that are obtained from a practical system of the CSTR plant as well as the pH neutralization plant. The results show significant improvements in online identification of the non-linear dynamic systems compared with the recently reported methods in literature.
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Bera, Tushar Kanti, Samir Kumar Biswas, K. Rajan, and J. Nagaraju. "Improving Conductivity Image Quality Using Block Matrix-based Multiple Regularization (BMMR) Technique in EIT: A Simulation Study." Journal of Electrical Bioimpedance 2, no. 1 (2019): 33–47. http://dx.doi.org/10.5617/jeb.170.
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Abstract A Block Matrix based Multiple Regularization (BMMR) technique is proposed for improving conductivity image quality in Electrical Impedance Tomography (EIT). The response matrix (JTJ) has been partitioned into several sub-block matrices and the largest element of each sub-block matrix has been chosen as regularization parameter for the nodes contained by that sub-block. Simulated boundary data are generated for circular domains with circular inhomogeneities of different geometry and the conductivity images are reconstructed in a Model Based Iterative Image Reconstruction (MoBIIR) algorithm. Conductivity images are reconstructed with BMMR technique and the results are compared with the Single-step Tikhonov Regularization (STR) and modified Levenberg-Marquardt Regularization (LMR) methods. Results show that the BMMR technique improves the impedance image and its spatial resolution for single and multiple inhomogeneity phantoms of different geometries. It is observed that the BMMR technique reduces the projection error as well as the solution error and improves the conductivity reconstruction in EIT. Results also show that the BMMR method improves the image contrast and inhomogeneity conductivity profile by reducing background noise for all the phantom configurations.
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Song, Songbai, Yan Kang, Xiaoyan Song, and Vijay P. Singh. "MLE-Based Parameter Estimation for Four-Parameter Exponential Gamma Distribution and Asymptotic Variance of Its Quantiles." Water 13, no. 15 (2021): 2092. http://dx.doi.org/10.3390/w13152092.
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The choice of a probability distribution function and confidence interval of estimated design values have long been of interest in flood frequency analysis. Although the four-parameter exponential gamma (FPEG) distribution has been developed for application in hydrology, its maximum likelihood estimation (MLE)-based parameter estimation method and asymptotic variance of its quantiles have not been well documented. In this study, the MLE method was used to estimate the parameters and confidence intervals of quantiles of the FPEG distribution. This method entails parameter estimation and asymptotic variances of quantile estimators. The parameter estimation consisted of a set of four equations which, after algebraic simplification, were solved using a three dimensional Levenberg-Marquardt algorithm. Based on sample information matrix and Fisher’s expected information matrix, derivatives of the design quantile with respect to the parameters were derived. The method of estimation was applied to annual precipitation data from the Weihe watershed, China and confidence intervals for quantiles were determined. Results showed that the FPEG was a good candidate to model annual precipitation data and can provide guidance for estimating design values.
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Duda, Piotr, and Mariusz Konieczny. "An Adaptive Matrix Method for the Solution of a Nonlinear Inverse Heat Transfer Problem and Its Experimental Verification." Energies 16, no. 6 (2023): 2649. http://dx.doi.org/10.3390/en16062649.
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An adaptive matrix inverse (AMI) method is presented to identify the temperature and unknown boundary heat flux in a domain of a regular or irregular shape with temperature-dependent properties. The nonlinear problem is broken down into a number of linear submodels, and for each submodel, the temperature is obtained in measuring points. Next, based on the matching degree between the temperatures measured and calculated by each prediction submodel, the submodels are weighted and combined to create the full model for the solution of an inverse nonlinear heat transfer problem. Comparisons are also made with the existing multiple model adaptive inverse (MMAI) algorithm and method based on the Levenberg–Marquardt algorithm (LMA). The results of the presented numerical tests for undisturbed and disturbed “measuring” data indicate that the heat fluxes identified by the AMI method are close to the exact values. The application of the presented method for bodies with an irregular shape is also demonstrated. The AMI method has been experimentally verified during the thick-walled cylinder cooling process. The proposed method can be applied in online diagnostic systems for thermal state monitoring.
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Yin, Changchun, and Greg Hodges. "Simulated annealing for airborne EM inversion." GEOPHYSICS 72, no. 4 (2007): F189—F195. http://dx.doi.org/10.1190/1.2736195.
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The traditional algorithms for airborne electromagnetic (EM) inversion, e.g., the Marquardt-Levenberg method, generally run only a downhill search. Consequently, the model solutions are strongly dependent on the starting model and are easily trapped in local minima. Simulated annealing (SA) starts from the Boltzmann distribution and runs both downhill and uphill searches, rendering the searching process to easily jump out of local minima and converge to a global minimum. In the SA process, the calculation of Jacobian derivatives can be avoided because no preferred searching direction is required as in the case of the traditional algorithms. We apply SA technology for airborne EM inversion by comparing the inversion with a thermodynamic process, and we discuss specifically the SA procedure with respect to model configuration, random walk for model updates, objective function, and annealing schedule. We demonstrate the SA flexibility for starting models by allowing the model parameters to vary in a large range (far away from the true model). Further, we choose a temperature-dependent random walk for model updates and an exponential cooling schedule for the SA searching process. The initial temperature for the SA cooling scheme is chosen differently for different model parameters according to their resolvabilities. We examine the effectiveness of the algorithm for airborne EM by inverting both theoretical and survey data and by comparing the results with those from the traditional algorithms.
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Wu, Xinzhao, Peiqing Li, Qipeng Li, and Zhuoran Li. "Two-dimensional-simultaneous Localisation and Mapping Study Based on Factor Graph Elimination Optimisation." Sustainability 15, no. 2 (2023): 1172. http://dx.doi.org/10.3390/su15021172.
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A robust multi-sensor fusion simultaneous localization and mapping (SLAM) algorithm for complex road surfaces is proposed to improve recognition accuracy and reduce system memory occupation, aiming to enhance the computational efficiency of light detection and ranging in complex environments. First, a weighted signed distance function (W-SDF) map-based SLAM method is proposed. It uses a W-SDF map to capture the environment with less accuracy than the raster size but with high localization accuracy. The Levenberg–Marquardt method is used to solve the scan-matching problem in laser SLAM; it effectively alleviates the limitations of the Gaussian–Newton method that may lead to insufficient local accuracy, and reduces localisation errors. Second, ground constraint factors are added to the factor graph, and a multi-sensor fusion localisation algorithm is proposed based on factor graph elimination optimisation. A sliding window is added to the chain factor graph model to retain the historical state information within the window and avoid high-dimensional matrix operations. An elimination algorithm is introduced to transform the factor graph into a Bayesian network to marginalize the historical states and reduce the matrix dimensionality, thereby improving the algorithm localisation accuracy and reducing the memory occupation. Finally, the proposed algorithm is compared and validated with two traditional algorithms based on an unmanned cart. Experiments show that the proposed algorithm reduces memory consumption and improves localisation accuracy compared to the Hector algorithm and Cartographer algorithm, has good performance in terms of accuracy, reliability and computational efficiency in complex pavement environments, and is better utilised in practical environments.
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Cefalu, A., and D. Fritsch. "Non-Incremental Derivation of Scale and Pose from a Network of Relative Orientations." ISPRS - International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences XL-3 (August 11, 2014): 53–59. http://dx.doi.org/10.5194/isprsarchives-xl-3-53-2014.
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The majority of approaches to Structure from Motion apply an incremental triangulate-and-resect strategy in order to reconstruct camera motion and scene structure in a common reference frame. The sequential addition of images may cause a drifting behaviour during the reconstruction, in some cases causing the process to fail. Over the last decade, more attention has come to non-incremental approaches, which exploit the network characteristics arising from the 2- or 3-view relations, given for a set of images through relative orientations. Most approaches employ rotation registration, followed by translation registration. The latter being carried out with or without simultaneous scene reconstruction. We suggest an approach which starts by estimation of relative scales, followed by simultaneous registration of rotation and translation. The latter is achieved by employing a path-finding algorithm based on Ant- Colony-Optimization. For scale estimation we propose a window-search adaption of Levenberg-Marquardt, which avoids unnecessary matrix inversions. We also suggest a simple method for detection of outlier orientations.
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Silva, W. P., C. M. D. P. S. e. Silva, J. P. Gomes, N. C. Santos, A. J. M. Queiroz, and R. M. F. de Figuiredo. "Calculation of the Thermal Properties (and Their Uncertainties) of Strawberry During Its Cooling Under Natural Convection." Journal of Agricultural Science 11, no. 5 (2019): 114. http://dx.doi.org/10.5539/jas.v11n5p114.
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Many times, the thermal properties of a product are determined but their uncertainties (and, mainly, the covariance matrix) are not provided. Thus, in the simulations, it is not possible to establish a confidence band for a transient state described through the values obtained for these properties. In this article, a model was proposed to determine thermal diffusivity and convective heat transfer coefficient, providing the above-mentioned lack of information, for a product with spherical geometry during its cooling. The proposed model involved: 1) an experimental data set of the cooling kinetics in a point within the product; 2) a one-dimensional numerical solution of the heat conduction equation; 3) an optimizer based on the Levenberg-Marquardt algorithm to determine the thermal properties, their uncertainties, and the covariance between the parameters. Model was applied for determining thermal properties of strawberries, using an equivalent sphere to represent the geometry of the product, and the obtained results were compatible with the literature results.
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Qin, Yanding, Pengxiu Geng, Bowen Lv, Yiyang Meng, Zhichao Song, and Jianda Han. "Simultaneous Calibration of the Hand-Eye, Flange-Tool and Robot-Robot Relationship in Dual-Robot Collaboration Systems." Sensors 22, no. 5 (2022): 1861. http://dx.doi.org/10.3390/s22051861.
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A multi-robot collaboration system can complete more complex tasks than a single robot system. Ensuring the calibration accuracy between robots in the system is a prerequisite for the effective inter-robot cooperation. This paper presents a dual-robot system for orthopedic surgeries, where the relationships between hand-eye, flange-tool, and robot-robot need to be calibrated. This calibration problem can be summarized to the solution of the matrix equation of AXB=YCZ. A combined solution is proposed to solve the unknown parameters in the equation of AXB=YCZ, which consists of the dual quaternion closed-form method and the iterative method based on Levenberg–Marquardt (LM) algorithm. The closed-form method is used to quickly obtain the initial value for the iterative method so as to increase the convergence speed and calibration accuracy of the iterative method. Simulation and experimental analyses are carried out to verify the accuracy and effectiveness of the proposed method.
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Reuter, Thomas, and Christof Hurschler. "Comparison of biphasic material properties of equine articular cartilage from stress relaxation indentation tests with and without tension-compression nonlinearity." Current Directions in Biomedical Engineering 4, no. 1 (2018): 485–88. http://dx.doi.org/10.1515/cdbme-2018-0116.
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AbstractThe mechanical parameters of articular cartilage estimated from indentation tests depend on the constitutive model adopted to analyze the data. In this study, we present a 3D-FE-based method to determine the biomechanical properties of equine articular cartilage from stress relaxation indentation tests (ε = 6 %, t = 1000 s) whereby articular cartilage is modeled as a biphasic material without (BM) and with tension-compression nonlinearity (BMTCN). The FEmodel computation was optimized by exploiting the axial symmetry and mesh resolution. Parameter identification was executed with the Levenberg-Marquardt-algorithm. The R² of the fit results varies between 0.695 and 0.930 for the BMmodel and between 0.877 and 0.958 for the BMTCN-model. The differences of the R² occur from the more exact description of the initial stress relaxation behaviour by the fiber modulus from the BMTCN-model. The fiber modulus defines the collagen matrix of cartilage. Furthermore, for both models the determined values of Young’s modulus and permeability were in the same order of magnitude.
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Zhang, Huizhen, Gang Cheng, Xianlei Shan, and Feng Guo. "Kinematic accuracy research of 2(3HUS+S) parallel manipulator for simulation of hip joint motion." Robotica 36, no. 9 (2018): 1386–401. http://dx.doi.org/10.1017/s0263574718000073.
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SUMMARYIn this paper, the kinematic accuracy problem caused by geometric errors of a 2(3HUS+S) parallel manipulator is described. The kinematic equation of the manipulator is obtained by establishing a D–H (Denavit–Hartenberg) coordinate system. A D–H transformation matrix is used as the error-modeling tool, and the kinematic error model of the manipulator integrating manufacturing and assembly errors is established based on the perturbation theory. The iterative Levenberg–Marquardt algorithm is used to identify the geometric errors in the error model. According to the experimentally measured attitudes, the kinematic calibration process is simulated using MATLAB software. The simulation and experiment results show that the attitude errors of the moving platforms after calibration are reduced compared with before the calibration, and the kinematic accuracy of the manipulator is significantly improved. The correctness and effectiveness of the error model and the kinematic calibration method of the 2(3HUS+S) parallel manipulator for simulation of hip joint motion are verified.
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Aybek, Unsal, Lutfu Namli, Mustafa Ozbey, and Bekir Dogan. "Artificial neural network assisted multi-objective optimization of a methane-fed DIR-SOFC system with waste heat recovery." Thermal Science 27, no. 4 Part B (2023): 3413–22. http://dx.doi.org/10.2298/tsci2304413a.
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The main purpose of this study is to enhance the performance of solid oxide fuel cell systems. For this purpose, a mathematical model of a direct internal reforming (DIR) methane-fed solid oxide fuel cell system with waste heat recovery was designed in the engineering equation solver program. We optimised the performance of the solid oxide fuel cell using a genetic algorithm and TOPSIS technique considering exergy, power, and environmental analyzes. An ANN working with the Levenberg-Marquardt training function was designed in the MATLprogram to create the decision matrix to which the TOPSIS method will be applied. According to the power optimization, 786 kW net power was obtained from the system. In exergetic optimization, the exergy efficiency was found to be 57.6%. In environmental optimization, the environmental impact was determined as 330.6 kgCO2/MWh. According to the multi-objective optimization results, the exergy efficiency, the net power of the solid oxide fuel cell system, and the environmental impact were 504.1 kW, 40.08%, and 475.4 kgCO2/MWh.
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Tavakoli, Reza, and Albert C. Reynolds. "History Matching With Parameterization Based on the Singular Value Decomposition of a Dimensionless Sensitivity Matrix." SPE Journal 15, no. 02 (2009): 495–508. http://dx.doi.org/10.2118/118952-pa.
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Summary In gradient-based automatic history matching, calculation of the derivatives (sensitivities) of all production data with respect to gridblock rock properties and other model parameters is not feasible for large-scale problems. Thus, the Gauss-Newton (GN) method and Levenberg-Marquardt (LM) algorithm, which require calculation of all sensitivities to form the Hessian, are seldom viable. For such problems, the quasi-Newton and nonlinear conjugate gradient algorithms present reasonable alternatives because these two methods do not require explicit calculation of the complete sensitivity matrix or the Hessian. Another possibility, the one explored here, is to define a new parameterization to radically reduce the number of model parameters. We provide a theoretical argument that indicates that reparameterization based on the principal right singular vectors of the dimensionless sensitivity matrix provides an optimal basis for reparameterization of the vector of model parameters. We develop and illustrate algorithms based on this parameterization. Like limited-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS), these algorithms avoid explicit computation of individual sensitivity coefficients. Explicit computation of the sensitivities is avoided by using a partial singular value decomposition (SVD) based on a form of the Lanczos algorithm. At least for all synthetic problems that we have considered, the reliability, computational efficiency, and robustness of the methods presented here are as good as those obtained with quasi-Newton methods.
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Costa, Raniere Fernandes, Alysson Dantas Ferreira, José Jefferson da Silva Nascimento, et al. "Obtaining thermal and mass diffusivity of the drying process on ceramic plates with the addition of diatomite tailings." Research, Society and Development 11, no. 10 (2022): e410111032174. http://dx.doi.org/10.33448/rsd-v11i10.32174.
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The drying stage is the most relevant procedure in the industrial production of ceramic bricks, since the final quality of the product is directly related to the success of this operation. Knowledge of the material's thermal and mass diffusivity facilitates mathematical modeling, which brings us a greater domain of the process. With this purpose, the drying of solid ceramic bricks with different percentages of diatomite tailings was experimentally carried out in an oven with forced air circulation. The experiments were performed based on a 2k experimental design with three factors: the drying temperatures (333 and 383 K), the homogenization time of the mixture (30 and 60 min), and the percentage of tailings (10 and 30%). From the results matrix, we applied the Levenberg-Marquardt algorithm to minimize the difference between the theoretical and the experimental drying and heating curves, thus obtaining the thermal and mass diffusivities. Still in possession of the results matrix, we applied a regression model in order to obtain an equation to estimate the diffusivity values. The drying and heating curves that were built from the estimated diffusivities showed good agreement with the experimental data, and were validated within a 99% confidence interval.
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Alp, Harun, Mehmet Cüneyd Demirel, and Ömer Levend Aşıkoğlu. "Effect of Model Structure and Calibration Algorithm on Discharge Simulation in the Acısu Basin, Turkey." Climate 10, no. 12 (2022): 196. http://dx.doi.org/10.3390/cli10120196.
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In this study, the Acısu Basin—viz., the headwater of the Gediz Basin—in Turkey, was modelled using three types of hydrological models and three different calibration algorithms. A well-known lumped model (GR4J), a commonly used semi-distributed (SWAT+) model, and a skillful distributed (mHM) hydrological model were built and integrated with the Parameter Estimation Tool (PEST). PEST is a model-independent calibration tool including three algorithms—namely, Levenberg Marquardt (L-M), Shuffled Complex Evolution (SCE), and Covariance Matrix Adoption Evolution Strategy (CMA-ES). The calibration period was 1991–2000, and the validation results were obtained for 2002–2005. The effect of the model structure and calibration algorithm selection on the discharge simulation was evaluated via comparison of nine different model-algorithm combinations. Results have shown that mHM and CMA-ES combination performed the best discharge simulation according to NSE values (calibration: 0.67, validation: 0.60). Although statistically the model results were classified as acceptable, the models mostly missed the peak values in the hydrograph. This problem may be related to the interventions made in 2000–2001 and may be overcome by changing the calibration and validation periods, increasing the number of iterations, or using the naturalized gauge data.
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Rajesh, Ravichandran, and Pudureddiyur Venkataraman Manivannan. "Robust Calibration Technique for Precise Transformation of Low-Resolution 2D LiDAR Points to Camera Image Pixels in Intelligent Autonomous Driving Systems." Vehicles 6, no. 2 (2024): 711–27. http://dx.doi.org/10.3390/vehicles6020033.
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In the context of autonomous driving, the fusion of LiDAR and camera sensors is essential for robust obstacle detection and distance estimation. However, accurately estimating the transformation matrix between cost-effective low-resolution LiDAR and cameras presents challenges due to the generation of uncertain points by low-resolution LiDAR. In the present work, a new calibration technique is developed to accurately transform low-resolution 2D LiDAR points into camera pixels by utilizing both static and dynamic calibration patterns. Initially, the key corresponding points are identified at the intersection of 2D LiDAR points and calibration patterns. Subsequently, interpolation is applied to generate additional corresponding points for estimating the homography matrix. The homography matrix is then optimized using the Levenberg–Marquardt algorithm to minimize the rotation error, followed by a Procrustes analysis to minimize the translation error. The accuracy of the developed calibration technique is validated through various experiments (varying distances and orientations). The experimental findings demonstrate that the developed calibration technique significantly reduces the mean reprojection error by 0.45 pixels, rotation error by 65.08%, and distance error by 71.93% compared to the standard homography technique. Thus, the developed calibration technique promises the accurate transformation of low-resolution LiDAR points into camera pixels, thereby contributing to improved obstacle perception in intelligent autonomous driving systems.
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Gao, Guohua, and Albert C. Reynolds. "An Improved Implementation of the LBFGS Algorithm for Automatic History Matching." SPE Journal 11, no. 01 (2006): 5–17. http://dx.doi.org/10.2118/90058-pa.
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Summary For large scale history matching problems, where it is not feasible to compute individual sensitivity coefficients, the limited memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) is an efficient optimization algorithm, (Zhang and Reynolds, 2002; Zhang, 2002). However, computational experiments reveal that application of the original implementation of LBFGS may encounter the following problems:converge to a model which gives an unacceptable match of production data;generate a bad search direction that either leads to false convergence or a restart with the steepest descent direction which radically reduces the convergence rate;exhibit overshooting and undershooting, i.e., converge to a vector of model parameters which contains some abnormally high or low values of model parameters which are physically unreasonable. Overshooting and undershooting can occur even though all history matching problems are formulated in a Bayesian framework with a prior model providing regularization. We show that the rate of convergence and the robustness of the algorithm can be significantly improved by:a more robust line search algorithm motivated by the theoretical result that the Wolfe conditions should be satisfied;an application of a data damping procedure at early iterations orenforcing constraints on the model parameters. Computational experiments also indicate thata simple rescaling of model parameters prior to application of the optimization algorithm can improve the convergence properties of the algorithm although the scaling procedure used can not be theoretically validated. Introduction Minimization of a smooth objective function is customarily done using a gradient based optimization algorithm such as the Gauss- Newton (GN) method or Levenberg-Marquardt (LM) algorithm. The standard implementations of these algorithms (Tan and Kalogerakis, 1991; Wu et al., 1999; Li et al., 2003), however, require the computation of all sensitivity coefficients in order to formulate the Hessian matrix. We are interested in history matching problems where the number of data to be matched ranges from a few hundred to several thousand and the number of reservoir variables or model parameters to be estimated or simulated ranges from a few hundred to a hundred thousand or more. For the larger problems in this range, the computer resources required to compute all sensitivity coefficients would prohibit the use of the standard Gauss- Newton and Levenberg-Marquardt algorithms. Even for the smallest problems in this range, computation of all sensitivity coefficients may not be feasible as the resulting GN and LM algorithms may require the equivalent of several hundred simulation runs. The relative computational efficiency of GN, LM, nonlinear conjugate gradient and quasi-Newton methods have been discussed in some detail by Zhang and Reynolds (2002) and Zhang (2002).