Relevant bibliographies by topics / Jacobian matrix.Levenberg-Marquardt algorithm

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Academic literature on the topic 'Jacobian matrix.Levenberg-Marquardt algorithm'

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Published: 3 June 2025
Last updated: 23 June 2025

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Journal articles on the topic "Jacobian matrix.Levenberg-Marquardt algorithm"

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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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Book chapters on the topic "Jacobian matrix.Levenberg-Marquardt algorithm"

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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." In Robotic Systems. IGI Global, 2020. http://dx.doi.org/10.4018/978-1-7998-1754-3.ch038.

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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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Conference papers on the topic "Jacobian matrix.Levenberg-Marquardt algorithm"

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Radler, Simon, and Manfred Hajek. "Periodic Free Wake Simulation Using a Numerical Optimization Method." In Vertical Flight Society 72nd Annual Forum & Technology Display. The Vertical Flight Society, 2016. http://dx.doi.org/10.4050/f-0072-2016-11373.

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A novel method has been developed for the free vortex wake simulation of rotors under the assumption of periodicity. An error measure is introduced which is based on the mismatch between prevailing convection velocities and the assumed wake element positions. The systematic reduction of this defined error results in a numerically robust method for the computation of wake geometries. Two strategies are compared for the error reduction. The first uses the repeated evaluation of the velocity field to update the wake geometry as long as the defined error decreases. In the second strategy, the application of a Levenberg-Marquardt method is added, in which the error is further reduced using an analytically defined Jacobian matrix. Results correlate well with experiments for single rotor and tandem rotor cases. While the Levenberg-Marquardt method achieves an additional error reduction, correlation with experimental data is not generally improved by including this method.
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Sancibrian, Ramon, Ana De-Juan, and Fernando Viadero. "Non-Linear Least-Square Optimization of Mechanisms Based on Levenberg-Marquardt Method." In ASME 2008 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2008. http://dx.doi.org/10.1115/detc2008-49743.

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One of the main problems to improve the convergence rate in deterministic optimization of mechanisms is to obtain the Hessian matrix. The required second-order derivatives are difficult to obtain or they are not available. Levenberg-Marquardt optimization method is a pseudo-second order method which means that uses the jacobian information to estimate the Hessian matrix. In this paper, the formulation to obtain the exact form of the jacobian matrix is presented and how can be implemented in the Levenberg-Marquardt method. This formulation gives a very effective method to optimize mechanism geometry considering a large number of prescribed positions and design variables. At the same time it is possible to have control over singularities and permits to compare the desired and generated path avoiding translation and rotation effects.
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Varziri, Saeed, and Leila Notash. "Kinematic Calibration of the Central Linkage of a Wire-Actuated Parallel Robot: A Comparison Between Two Nonlinear Approaches." In ASME 2004 International Mechanical Engineering Congress and Exposition. ASMEDC, 2004. http://dx.doi.org/10.1115/imece2004-61132.

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In this article kinematic calibration of the central linkage of a wire-actuated parallel robot, which has a parallelogram mechanism in its structure, is discussed. It is shown that the dependency between the parallelogram joint angles affect the conditioning of the identification Jacobian matrix. Through a sensitivity analysis, it is presented that some of the parallelogram link lengths are not identifiable. Two nonlinear calibration approaches, Gauss-Newton and Levenberg-Marquardt, have also been explained and their difference especially when the identification Jacobian matrix is ill-conditioned, are pointed out.
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Chang, Shih Yu, Hsiao-Chun Wu, Kun Yan, and Yiyan Wu. "Novel Extended Kalman Filter Using Matrix-Based Levenberg-Marquardt Algorithm and Its Application for Variable Bit-Rate Video Frame-Size Prediction." In 2022 IEEE International Symposium on Broadband Multimedia Systems and Broadcasting (BMSB). IEEE, 2022. http://dx.doi.org/10.1109/bmsb55706.2022.9828691.

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Groot, J. A. W. M., C. G. Giannopapa, and R. M. M. Mattheij. "A Numerical Shape Optimisation Method for Blowing Glass Bottles." In ASME 2011 Pressure Vessels and Piping Conference. ASMEDC, 2011. http://dx.doi.org/10.1115/pvp2011-57879.

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Industrial glass blowing is an essential stage of manufacturing hollow glass containers, e.g. bottles, jars. A glass preform is brought into a mould and inflated with compressed air until it reaches the mould shape. A simulation model for blowing glass containers based on finite element methods, which adopts a level set method to track the glass-air interfaces, has previously been developed [1–3]. A considerable challenge in glass blowing is the inverse problem: to determine an optimal preform from the desired container shape. In previous work of the authors [4, 5] a numerical method was introduced for optimising the shape of the preform. The optimisation method described the shape of the preform by parametric curves, e.g. Bezier-curves or splines, and employed a modified Levenberg-Marquardt algorithm to find the optimal positions of the control points of the curves. A combined finite difference and Broyden method was used to compute the Jacobian of the residual with respect to changes in the positions of the control points. The objective of this paper is to perform an error analysis of the optimisation method previously introduced and to improve its accuracy and performance. The improved optimisation method is applied to modelled containers of industrial relevance, which shows its usefulness for practical applications.
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Duraisamy, Karthikeyan, Alba Perez-Gracia, and Marco P. Schoen. "Vision-Based Kinematic Synthesis of Hand Motion." In ASME 2006 International Mechanical Engineering Congress and Exposition. ASMEDC, 2006. http://dx.doi.org/10.1115/imece2006-14870.

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The purpose of this work is to develop a human hand model that will work in conjunction with the myoelectric signals from the arm muscles, for those people who have lost their upper extremity. Though there are many prostheses available on the market with variable cost and functional accuracy, it is hard to find a prosthesis that mimics the complete functionality of the human hand, due to the complex hand motion, the complex dynamics of the myoelectric signals, and the difficulty involved in the acquisition of these signals, which complicates the implementation. In order to overcome some of these problems, the proposed hand model mimics most of the hand movements and it is used together with a kinematic synthesis process to identify the motion of the hand, obtained from visual data. In this paper, the human hand is modeled as a collection of five serial chains. For each movement performed by the joints in the finger/wrist, revolute joints are considered in different configurations, which yield movements similar to those of the human hand. The forward kinematics in matrix form is formulated using Denavit-Hartenberg parameters and expressed using Clifford Algebra exponentials. Kinematic synthesis is used to adjust the dimensions of the proposed model to the hand of the subject, and to identify the angles at each joint for a given hand motion. In the kinematic synthesis process, the forward kinematics equations of the hand are solved for both the angles of the joints and the dimensions of the hand. The synthesis equations obtained from the kinematic synthesis process are solved using a Levenberg-Marquardt nonlinear least-squares algorithm. The experimental setup for the real-time motion capturing consists of three camerasand is to be used in future work to relate the joint motion to the myoelectric signals acquired from the subject's arm.
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Omion, O. O., and A. Dosunmu. "Using Artificial Neural Network to Predict Gas Flare Volume, Gas Composition, and Temperature from Gaseous Emission." In SPE Nigeria Annual International Conference and Exhibition. SPE, 2023. http://dx.doi.org/10.2118/217203-ms.

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Abstract Gas emission from gas flaring is known to have deleterious effects on the environment and they constitute a major source of global warming. Flare gas volume, temperature, gas composition and other meteorological factors are major parameters in gas flaring processes. Using Artificial Neural Network, a model was developed to estimate gas flare volume, gas composition and flare temperature from gaseous pollutants using. Data used for the study were from oil prolific Niger-Delta region of Nigeria. Air quality index parameter, gas flared volume, temperature and composition between 2013-2017 were used in developing the ANN model using the Neural Training toolbox (nntool) of the Matrix Laboratory (R2019a MATLAB) mathematical software. An 8-6-3 network architecture was adopted. It consists of eight input parameters (suspended particulate matter, carbon monoxide, sulphur oxide, nitrogen oxide, volatile organic compounds, hydrogen sulphide, methane, and ammonia), six hidden layers and three output parameters (gas flared volume, gas composition and temperature) using 1286 dataset for each input and output parameter. Multiple-input multiple-output (MIMO) neural network using supervised learning algorithm (Levenberg-Marquardt) to train the network was adopted in model development. 75% (880 data points) of the data was set aside for the training of the model at its developmental stage, 10% for test data set and 15% for the validation data set. From the models’ prediction, it was observed that the developed model predicted excellently and performed well when tested with new set of data which was not a part of the developmental dataset with a coefficient of determination of 0.99999918, a root mean square error of 0.009029, an absolute average percentage relative error of 0.0362% for Gas Flare Volume, Composition and Temperature respectively. The outcome of this study presents a reliable and speedy tool for forecasting of gas flare volume, composition, and temperature in the absence of conventional methodologies.
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Soloviev, Arcady, Anton Bychkov, and Maria Shevtsova. "Determination of Full Set Elastic Constants for Composite Materials on Basis of Frequency Response Analysis, FEA, and GA." In ASME 2008 9th Biennial Conference on Engineering Systems Design and Analysis. ASMEDC, 2008. http://dx.doi.org/10.1115/esda2008-59556.

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The number of engineering problems includes the identification of anisotropic composite elastic constants determination. We developed an experimentally - analytical technique for identification of all elastic constants of orthotropic materials. The offered technique is substantially based on measurement of eigenfrequencies and semi quantitative analysis of natural vibration modes, instead of wave propagation speed and fields of vibrational displacement used by other acoustic methods. The developed method of the elastic composite and piezoelectric materials properties identification is implemented in linked MATLAB – Comsol Multiphysics combining the finite element analysis (FEA) of oscillations dynamics and minimization of some functional, which type is determined by particularity of a solved problem. These techniques complement the early designed by authors’ FEM-based methods for orthotropic composite static tests. The offered dynamic tests include an evaluation of specimen’s frequency response, determination of natural frequencies and vibration modes of specimens both in natural experiments and numerical finite element simulations. The identification process consists of several stages. In series of static tests are determined all allowable modules. Further a complete matrix of elastic constant is constructed, but some modules specified by approximated values (in particular, interlaminar shear modules). A series of dynamic tests executed in which the periodical excitation of samples and the frequency response is recorded by means of piezoelectric actuators and sensors. Then on basis of early defined (in static tests and with use of mix rule) modules of composite and experimentally founded eigenfrequencies by means of FEM the vibration natural modes are identified. By combination of FEM, genetic algorithm (GA) and Levenberg-Marquardt minimization method the specification of composite mechanical properties is evaluated. Application of developed technique to orthotropic composite used in aviation structures (polymeric composite spar of the helicopter main rotor blade) is explicitly illustrated. The obtained results have shown a good efficiency of proposed identification methods. We demonstrate that proposed approach provides best reliability and shows small dependence on metering equipment precision.
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