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 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).

Quadratic convergence of Levenberg-Marquardt method for general nonlinear inverse problems with two parameters

2019 ◽  
Vol 97 (10) ◽  
pp. 1949-1966
Author(s):  
Komivi Souley Agbodjan ◽  
Osama Said Ahmed ◽  
Djemaa Messaoudi ◽  
Ting Cheng ◽  
Daijun Jiang
Keyword(s):  
Inverse Problems ◽  
Quadratic Convergence ◽  
Nonlinear Inverse Problems ◽  
Marquardt Method ◽  
Levenberg Marquardt ◽  
Two Parameters

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.

Existence of variational source conditions for nonlinear inverse problems in Banach spaces

2018 ◽  
Vol 26 (2) ◽  
pp. 277-286 ◽  
Author(s):  
Jens Flemming
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Banach Spaces ◽  
Regularization Method ◽  
Multiple Solutions ◽  
Convergence Rates ◽  
Nonlinear Inverse Problems ◽  
Range Type ◽  
Ill Posed ◽  
Tikhonov’S Regularization

AbstractVariational source conditions proved to be useful for deriving convergence rates for Tikhonov’s regularization method and also for other methods. Up to now, such conditions have been verified only for few examples or for situations which can be also handled by classical range-type source conditions. Here we show that for almost every ill-posed inverse problem variational source conditions are satisfied. Whether linear or nonlinear, whether Hilbert or Banach spaces, whether one or multiple solutions, variational source conditions are a universal tool for proving convergence rates.

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