A Levenberg–Marquardt method for large nonlinear least-squares problems with dynamic accuracy in functions and gradients

Bellavia, Stefania; Gratton, Serge; Riccietti, Elisa

doi:10.1007/s00211-018-0977-z

Cited by 49 publications

(38 citation statements)

References 25 publications

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“…When both the model operator and the gradient are not available exactly, inexact methods need to be used. In [18,23] a Levenberg-Marquardt method is proposed and investigated for dealing with inexact gradients and Jacobians.…”

Section: Hybrid Methods and Inexact Approachesmentioning

confidence: 99%

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Numerical linear algebra in data assimilation

Freitag

2020

GAMM-Mitteilungen

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Data assimilation is a method that combines observations (ie, real world data) of a state of a system with model output for that system in order to improve the estimate of the state of the system and thereby the model output. The model is usually represented by a discretized partial differential equation. The data assimilation problem can be formulated as a large scale Bayesian inverse problem. Based on this interpretation we will derive the most important variational and sequential data assimilation approaches, in particular three‐dimensional and four‐dimensional variational data assimilation (3D‐Var and 4D‐Var) and the Kalman filter. We will then consider more advanced methods which are extensions of the Kalman filter and variational data assimilation and pay particular attention to their advantages and disadvantages. The data assimilation problem usually results in a very large optimization problem and/or a very large linear system to solve (due to inclusion of time and space dimensions). Therefore, the second part of this article aims to review advances and challenges, in particular from the numerical linear algebra perspective, within the various data assimilation approaches.

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Section: Hybrid Methods and Inexact Approachesmentioning

confidence: 99%

“…For nonlinear problems, ∇ 2 J i (x i ) −1 is an approximation to the posterior covariance. Solving (18) iteratively is more advantageous than computing the Kalman gain directly for very large problems with sparsity structure. More details on this idea can be found in [13].…”

Section: Compute the Forecast Error Covariancementioning

confidence: 99%

Numerical linear algebra in data assimilation

Freitag

2020

GAMM-Mitteilungen

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“…In our setting, the Levenberg-Marquardt algorithm is used to update the parameter vector ψ iteratively by solving the nonlinear optimization problem described in (15). The algorithm adaptively updates the parameter estimates by combining the gradient descent update and the Gauss-Newton update [27] by tuning a damping parameter λ. The Marquardt's update equation is given by:…”

Section: Parameter Estimationmentioning

confidence: 99%

“…The damping factor λ is adjusted by checking the values obtained with the new parameter set against the previous values. One possible way to do this is by using a ρ factor [27,29,30] defined in (19).…”

Section: Parameter Estimationmentioning

confidence: 99%

A Virtual Sensor for Electric Vehicles’ State of Charge Estimation

et al. 2020

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The estimation of the state of charge is a critical function in the operation of electric vehicles. The battery management system must provide accurate information about the battery state, even in the presence of failures in the vehicle sensors. This article presents a new methodology for the state of charge estimation (SOC) in electric vehicles without the use of a battery current sensor, relying on a virtual sensor, based on other available vehicle measurements, such as speed, battery voltage and acceleration pedal position. The estimator was derived from experimental data, employing support vector regression (SVR), principal component analysis (PCA) and a dual polarization (DP) battery model (BM). It is shown that the obtained model is able to predict the state of charge of the battery with acceptable precision in the case of a failure of the current sensor.

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“…e effectiveness of the algorithm was verified by numerical examples. Bellavia et al [31] improved the approximation function by controlling the accuracy level when the accuracy was too low to be optimized, and then proposed the LM method based on the dynamic precision relationship between the evaluation function and gradient for solving large-scale nonlinear least squares problems. ey proved the global and local convergence and complexity of this method.…”

Section: Introductionmentioning

confidence: 99%

Efficient Parameters Estimation Method for the Separable Nonlinear Least Squares Problem

et al. 2020

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In this work, we combine the special structure of the separable nonlinear least squares problem with a variable projection algorithm based on singular value decomposition to separate linear and nonlinear parameters. en, we propose finding the nonlinear parameters using the Levenberg-Marquart (LM) algorithm and either solve the linear parameters using the least squares method directly or by using an iteration method that corrects the characteristic values based on the L-curve, according to whether or not the nonlinear function coefficient matrix is ill posed. To prove the feasibility of the proposed method, we compared its performance on three examples with that of the LM method without parameter separation. e results show that (1) the parameter separation method reduces the number of iterations and improves computational efficiency by reducing the parameter dimensions and (2) when the coefficient matrix of the linear parameters is well-posed, using the least squares method to solve the fitting problem provides the highest fitting accuracy. When the coefficient matrix is ill posed, the method of correcting characteristic values based on the L-curve provides the most accurate solution to the fitting problem.

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A Levenberg–Marquardt method for large nonlinear least-squares problems with dynamic accuracy in functions and gradients

Cited by 49 publications

References 25 publications

Numerical linear algebra in data assimilation

Numerical linear algebra in data assimilation

A Virtual Sensor for Electric Vehicles’ State of Charge Estimation

Efficient Parameters Estimation Method for the Separable Nonlinear Least Squares Problem

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