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.

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.

Two‐dimensional velocity inversion and synthetic seismogram computation

Geophysics ◽  
1987 ◽  
Vol 52 (1) ◽  
pp. 37-50 ◽  
Author(s):  
Tianfei Zhu ◽  
Larry D. Brown
Keyword(s):  
Least Squares ◽  
Velocity Structure ◽  
Trust Region ◽  
Damping Factor ◽  
Two Dimensional ◽  
Velocity Inversion ◽  
Least Squares Problem ◽  
Velocity Models ◽  
Marquardt Method ◽  
Levenberg Marquardt

A traveltime inversion scheme has been developed to estimate velocity and interface geometries of two‐dimensional media from deep reflection data. The velocity structure is represented by finite elements, and the inversion is formulated as an iterative, constrained, linear least‐squares problem which can be solved by either the singular value truncation method or the Levenberg‐Marquardt method. The damping factor of the Levenberg‐Marquardt method is chosen by the model‐trust region approach. The traveltimes and derivative matrix required to solve the least‐squares problem are computed by ray tracing. To aid seismic interpretation, we also include in the inversion scheme a fast algorithm based on asymptotic ray theory for calculating synthetic seismograms from the derived velocity model. Numerical tests indicate that the inversion scheme is effective, and that the accuracy of inversion results depends upon both noise in the data and the aperture of recording used in data acquisition. Two real examples demonstrate that the new inversion scheme produces velocity models fitting the data better than those estimated by other approaches.

scholarly journals An Inverse Problem Involving a Viscous Eikonal Equation with Applications in Electrophysiology

2021 ◽  
Author(s):  
Karl Kunisch ◽  
Philip Trautmann
Keyword(s):  
Inverse Problem ◽  
Least Squares ◽  
Eikonal Equation ◽  
State Equation ◽  
High Accuracy ◽  
Well Posedness ◽  
Numerical Examples ◽  
Marquardt Method ◽  
Math Biol ◽  
Levenberg Marquardt

AbstractIn this work we discuss the reconstruction of cardiac activation instants based on a viscous Eikonal equation from boundary observations. The problem is formulated as a least squares problem and solved by a projected version of the Levenberg–Marquardt method. Moreover, we analyze the well-posedness of the state equation and derive the gradient of the least squares functional with respect to the activation instants. In the numerical examples we also conduct an experiment in which the location of the activation sites and the activation instants are reconstructed jointly based on an adapted version of the shape gradient method from (J. Math. Biol. 79, 2033–2068, 2019). We are able to reconstruct the activation instants as well as the locations of the activations with high accuracy relative to the noise level.

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