scholarly journals Range-relaxed criteria for choosing the Lagrange multipliers in the Levenberg–Marquardt method

2020 ◽  
Author(s):  
A Leitão ◽  
F Margotti ◽  
B F Svaiter
Keyword(s):  
Hilbert Spaces ◽  
Lagrange Multipliers ◽  
A Priori ◽  
Impedance Tomography ◽  
Nonlinear Operators ◽  
Marquardt Method ◽  
Exact Data ◽  
Levenberg Marquardt ◽  
Ill Posed ◽  
Novel Strategy

Abstract 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 electrical impedance tomography problem. The performance of our strategy is compared with standard implementations of the Levenberg–Marquardt method (using a priori choice of the multipliers).

Levenberg–Marquardt method for ill-posed inverse problems with possibly non-smooth forward mappings between Banach spaces

2021 ◽  
Author(s):  
Huu Nhu Vu
Keyword(s):  
Inverse Problems ◽  
Heat Source ◽  
Banach Spaces ◽  
Elliptic Boundary ◽  
Uniformly Convex ◽  
Bounded Operators ◽  
Marquardt Method ◽  
Levenberg Marquardt ◽  
Ill Posed ◽  
Forward Mapping

Abstract In this paper, we consider a Levenberg–Marquardt method with general regularization terms that are uniformly convex on bounded sets to solve the ill-posed inverse problems in Banach spaces, where the forward mapping might not Gˆateaux differentiable and the image space is unnecessarily reflexive. The method therefore extends the one proposed by Jin and Yang in (Numer. Math. 133:655–684, 2016) for smooth inverse problem setting with globally uniformly convex regularization terms. We prove a novel convergence analysis of the proposed method under some standing assumptions, in particular, the generalized tangential cone condition and a compactness assumption. All these assumptions are fulfilled when investigating the identification of the heat source for semilinear elliptic boundary-value problems with a Robin boundary condition, a heat source acting on the boundary, and a possibly non-smooth nonlinearity. Therein, the Clarke subdifferential of the non-smooth nonlinearity is employed to construct the family of bounded operators that is a replacement for the nonexisting Gˆateaux derivative of the forward mapping. The efficiency of the proposed method is illustrated with a numerical example.

scholarly journals Discussion on the damping factor and improvement of the Levenberg-Marquardt method

2021 ◽  
Author(s):  
Wenwu Zhu
Keyword(s):  
Gravity Model ◽  
Positive Constant ◽  
Damping Factor ◽  
Geophysical Inversion ◽  
Marquardt Method ◽  
The Third ◽  
Levenberg Marquardt ◽  
Ill Posed ◽  
Oscillation Problem

Abstract The ill-posed problem is the key obstacle to obtain the accurate inversion results in the geophysical inversion field, and the Levenberg-Marquardt1, 2(hereinafter referred to as the L-M method) method has been widely used as it can effectively improve the ill-posed problems. However, the inversion results obtained by the L-M method are usually stable but incorrect, the reason is that the damping factor in the L-M method is difficult to solve, and it is usually approximated with a positive constant by experience or through some fitting methods. This paper uses the binary gravity model to demonstrate that the damping factor in the L-M method cannot be regarded as a positive constant only, it should have the following characteristics: (i) the damping factor is a vector, not just a constant; (ii) the values of the vector are composed of both positive and negative constants, not just positive constants; (iii) the corresponding value in the vector is close or equal to ∞ when the corresponding density block’s value is close or equal to zero. Even if the above characteristics have been found in the L-M method, it is difficult to reasonably estimate the damping factor as the damping factor oscillate severely due to the third characteristic, and the improved L-M method is proposed which effectively avoids the damping factor’s severe oscillation problem. The strategy of obtaining the reasonable damping factor is given finally.

A study of frozen iteratively regularized Gauss–Newton algorithm for nonlinear ill-posed problems under generalized normal solvability condition

2020 ◽  
Vol 28 (2) ◽  
pp. 275-286
Author(s):  
Anatoly Bakushinsky ◽  
Alexandra Smirnova
Keyword(s):  
Solvability Condition ◽  
A Priori ◽  
Nonlinear Least Squares ◽  
Normal Solvability ◽  
Nonlinear Operators ◽  
Penalty Term ◽  
Free Case ◽  
Ill Posed ◽  
Priori Information ◽  
Theoretical Results

AbstractA parameter identification inverse problem in the form of nonlinear least squares is considered. In the lack of stability, the frozen iteratively regularized Gauss–Newton (FIRGN) algorithm is proposed and its convergence is justified under what we call a generalized normal solvability condition. The penalty term is constructed based on a semi-norm generated by a linear operator yielding a greater flexibility in the use of qualitative and quantitative a priori information available for each particular model. Unlike previously known theoretical results on the FIRGN method, our convergence analysis does not rely on any nonlinearity conditions and it is applicable to a large class of nonlinear operators. In our study, we leverage the nature of ill-posedness in order to establish convergence in the noise-free case. For noise contaminated data, we show that, at least theoretically, the process does not require a stopping rule and is no longer semi-convergent. Numerical simulations for a parameter estimation problem in epidemiology illustrate the efficiency of the algorithm.

Accuracy estimates of Gauss–Newton-type iterative regularization methods for nonlinear equations with operators having normally solvable derivative at the solution

2016 ◽  
Vol 24 (4) ◽  
Author(s):  
Mikhail Y. Kokurin
Keyword(s):  
Hilbert Spaces ◽  
Total Error ◽  
A Priori ◽  
Stopping Rules ◽  
Regularization Methods ◽  
Iterative Regularization ◽  
Operator Equations ◽  
Ill Posed ◽  
Irregular Operator ◽  
The Right

AbstractWe investigate a class of iterative regularization methods for solving nonlinear irregular operator equations in Hilbert spaces. The operator of an equation is supposed to have a normally solvable derivative at the desired solution. The operators and right parts of equations can be given with errors. A priori and a posteriori stopping rules for the iterations are analyzed. We prove that the accuracy of delivered approximations is proportional to the total error level in the operator and the right part of an equation. The obtained results improve known accuracy estimates for the class of iterative regularization methods, as applied to general irregular operator equations. The results also extend previous similar estimates related to regularization methods for linear ill-posed equations with normally solvable operators.

scholarly journals The Levenberg–Marquardt Method for Acousto-Electric Tomography on Different Conductivity Contrast

2020 ◽  
Vol 10 (10) ◽  
pp. 3482
Author(s):  
Changyou Li ◽  
Kang An ◽  
Kuisong Zheng
Keyword(s):  
Imaging Modality ◽  
Iteration Scheme ◽  
Stability And Convergence ◽  
Reconstruction Method ◽  
Impedance Tomography ◽  
Marquardt Method ◽  
Levenberg Marquardt ◽  
The Stability ◽  
Different Levels ◽  
Conductivity Contrast

The stability and convergence performance of Levenberg–Marquardt method for acousto-electric tomography (AET) applied to different levels of conductivity contrast is studied in this paper. As a multi-physical imaging modality, acousto-electric tomography (AET) provides high spatial imaging resolution while also conserving the high contrast property of electrical impedance tomography. The Levenberg–Marquardt method is a well known iteration scheme which can be applied for the nonlinear problem of AET. However, the influence of the conductivity contrast on the stability and convergence performances of this conductivity reconstruction method is rarely discussed in the literature. In this paper, the performance of the Tikhonov regularization-based Levenberg–Marquardt method is applied to reconstruct conductivity map with different conductivity contrast between different regions of the domain of interest (DOI). Numerical investigations are carried out for phantoms with different conductivity contrast. Reconstructed results with different levels of noise are presented and discussed in detail.

scholarly journals Complex wavefront sensing based on coherent diffraction imaging using vortex modulation

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Rujia Li ◽  
Liangcai Cao
Keyword(s):  
Phase Retrieval ◽  
Dynamic Range ◽  
A Priori ◽  
Measured Intensity ◽  
Physical Constraints ◽  
Coherent Diffraction Imaging ◽  
Effective Dynamic ◽  
Ill Posed ◽  
Retrieval Problem ◽  
Diffraction Imaging

AbstractPhase retrieval seeks to reconstruct the phase from the measured intensity, which is an ill-posed problem. A phase retrieval problem can be solved with physical constraints by modulating the investigated complex wavefront. Orbital angular momentum has been recently employed as a type of reliable modulation. The topological charge l is robust during propagation when there is atmospheric turbulence. In this work, topological modulation is used to solve the phase retrieval problem. Topological modulation offers an effective dynamic range of intensity constraints for reconstruction. The maximum intensity value of the spectrum is reduced by a factor of 173 under topological modulation when l is 50. The phase is iteratively reconstructed without a priori knowledge. The stagnation problem during the iteration can be avoided using multiple topological modulations.

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