In this article we propose a novel strategy for choosing the Lagrange multipliers in the Levenberg-Marquardt method for solving ill-posed problems modeled by nonlinear operators acting between Hilbert spaces. Convergence analysis results are established for the proposed method, including: monotonicity of iteration error, geometrical decay of the residual, convergence for exact data, stability and semi-convergence for noisy data. Numerical experiments are presented for an elliptic parameter identification two-dimensional EIT problem. The performance of our strategy is compared with standard implementations of the Levenberg-Marquardt method (using a priori choice of the multipliers).
In this article a modified Levenberg-Marquardt method coupled with a Kaczmarz strategy for obtaining stable solutions of nonlinear systems of ill-posed operator equations is investigated. We show that the proposed method is a convergent regularization method. Numerical tests are presented for a non-linear inverse doping problem based on a bipolar model.
A q-Levenberg-Marquardt method is an iterative procedure that blends a q-steepest descent and q-Gauss-Newton methods. When the current solution is far from the correct one the algorithm acts as the q-steepest descent method. Otherwise the algorithm acts as the q-Gauss-Newton method. A damping parameter is used to interpolate between these two methods. The q-parameter is used to escape from local minima and to speed up the search process near the optimal solution.
In this paper, we consider a modified Levenberg--Marquardt method for solving an ill-posed inverse problem where the forward mapping is not Gâteaux differentiable. By relaxing the standard assumptions for the classical smooth setting, we derive asymptotic stability estimates that are then used to prove the convergence of the proposed method. This method can be applied to an inverse source problem for a non-smooth semilinear elliptic PDE where a Bouligand subdifferential can be used in place of the non-existing Fréchet derivative, and we show that the corresponding Bouligand-Levenberg-Marquardt…
We consider large-scale nonlinear least squares problems with sparse residuals, each of them depending on a small number of variables. A decoupling procedure which results in a splitting of the original problems into a sequence of independent problems of smaller sizes is proposed and analysed. The smaller size problems are modified in a way that offsets the error made by disregarding dependencies that allow us to split the original problem. The resulting method is a modification of the Levenberg-Marquardt method with smaller computational costs. Global convergence is proved as well as local li…
We develop a Levenberg-Marquardt method for minimizing the sum of a smooth nonlinear least-squar es term $f(x) = \tfrac{1}{2} \|F(x)\|_2^2$ and a nonsmooth term $h$. Both $f$ and $h$ may be nonconvex. Steps are computed by minimizing the sum of a regularized linear least-squares model and a model of $h$ using a first-order method such as the proximal gradient method. We establish global convergence to a first-order stationary point of both a trust-region and a regularization variant of the Levenberg-Marquardt method under the assumptions that $F$ and its Jacobian are Lipschitz continuous and $…
The inverse problem in Acousto-Electric tomography concerns the reconstruction of the electric conductivity in a domain from knowledge of the power density function in the interior of the body. This interior power density results from currents prescribed at boundary electrodes (and can be obtained through electro-static boundary measurements together with auxiliary acoustic measurement. In Electrical Impedance Tomography, the complete electrode model is known to be the most accurate model for the forward modelling. In this paper, the reconstruction problem of Acousto-Electric tomography is pos…
On the approximation of the solution of partial differential equations by artificial neural networks trained by a multilevel Levenberg-Marquardt method
INPT-IRIT, University of Toulouse and ENSEEIHT（INPT-IRIT，图卢兹大学和ENSEEIHT）；ISAE-SUPAERO, University of Toulouse（ISAE-SUPAERO，图卢兹大学）
Higher-Order Geometric Updates for Levenberg-Marquardt Method via Riemann Normal Coordinates
University of Science and Technology of China（中国科学技术大学）；Chinese Academy of Sciences（中国科学院）；University of Chinese Academy of Sciences（中国科学院大学）；Hefei National Laboratory（合肥国家实验室）
Do physics-informed neural networks (PINNs) need to be deep? Shallow PINNs using the Levenberg-Marquardt algorithm
organization= Department of Mathematics, Institut Teknologi Sepuluh Nopember , city= Surabaya , postcode= 60111 , country= Indonesia；organization= Department of Mathematics, Khalifa University of Science \& Technology , city= Abu Dhabi , postcode= PO Box 127788 , country= United Arab Emirates；organization= Department of Mathematics, Faculty of Mathematics
We propose two variants of Newton method for solving unconstrained minimization problem. Our method leverages optimization techniques such as penalty and augmented Lagrangian method to generate novel variants of the Newton method namely the Penalty Newton method and the Augmented Newton method. In doing so, we recover several well-known existing Newton method variants such as Damped Newton, Levenberg, and Levenberg-Marquardt methods as special cases. Moreover, the proposed Augmented Newton method can be interpreted as Newton method with adaptive heavy ball momentum. We provide global convergen…
This paper considers the generalized continuation Newton method and thetrust-region updating strategy for the underdetermined system of nonlinear equations. Moreover, in order to improve its computational efficiency, the new method will not update the Jacobian matrix when the current Jacobian matrix performs well. The numerical results show that the new method is more robust and faster than the traditional optimization method such as the Levenberg-Marquardt method (a variant of trust-region methods, the built-in subroutine fsolve.m of the MATLAB R2020a environment). The computational time of t…
