scholarly journals Quadratic convergence of Levenberg-Marquardt method for elliptic and parabolic inverse robin problems

2018 ◽  
Vol 52 (3) ◽  
pp. 1085-1107 ◽  
Author(s):  
Daijun Jiang ◽  
Hui Feng ◽  
Jun Zou
Keyword(s):  
Quadratic Convergence ◽  
Parabolic Systems ◽  
Adaptive Strategy ◽  
Nonlinear Elliptic ◽  
Highly Nonlinear ◽  
Regularization Parameters ◽  
Levenberg Marquardt ◽  
Ill Posed ◽  
First Time ◽  
Elliptic And Parabolic Systems

We study the Levenberg-Marquardt (L-M) method for solving the highly nonlinear and ill-posed inverse problem of identifying the Robin coefficients in elliptic and parabolic systems. The L-M method transforms the Tikhonov regularized nonlinear non-convex minimizations into convex minimizations. And the quadratic convergence of the L-M method is rigorously established for the nonlinear elliptic and parabolic inverse problems for the first time, under a simple novel adaptive strategy for selecting regularization parameters during the L-M iteration. Then the surrogate functional approach is adopted to solve the strongly ill-conditioned convex minimizations, resulting in an explicit solution of the minimisation at each L-M iteration for both the elliptic and parabolic cases. Numerical experiments are provided to demonstrate the accuracy, efficiency and quadratic convergence of the methods.

Domain Decomposition Methods for Recovering Robin Coefficients in Elliptic and Parabolic Systems

2018 ◽  
Vol 18 (2) ◽  
pp. 257-274
Author(s):  
Daijun Jiang ◽  
Hui Feng
Keyword(s):  
Domain Decomposition ◽  
Parabolic Systems ◽  
Decomposition Methods ◽  
Domain Decomposition Methods ◽  
Nonlinear Inverse Problem ◽  
Local Minimizers ◽  
Marquardt Method ◽  
Output Least Squares ◽  
Ill Posed ◽  
Elliptic And Parabolic Systems

AbstractWe shall derive and propose several efficient domain decomposition methods for solving the nonlinear inverse problem of identifying the Robin coefficients in elliptic and parabolic systems. The highly ill-posed inverse problems are transformed into output least-squares nonlinear and non-convex minimizations with classical Tikhonov regularization. The Levenberg–Marquardt method is applied to transform the non-convex minimizations into convex minimizations, which will be solved by several efficient domain decomposition methods. The methods are completely local and the local minimizers have explicit expressions within the subdomains. Several numerical experiments are presented to show the accuracy and efficiency of the methods; in particular, the convergence seems nearly optimal in the sense that the iteration number of the methods is nearly independent of mesh sizes.

scholarly journals Robust Bayesian Joint Inversion of Gravimetric and Muographic Data for the Density Imaging of the Puy de Dôme Volcano (France)

2021 ◽  
Vol 8 ◽  
Author(s):  
Anne Barnoud ◽  
Valérie Cayol ◽  
Peter G. Lelièvre ◽  
Angélie Portal ◽  
Philippe Labazuy ◽  
...  
Keyword(s):  
Joint Inversion ◽  
A Priori ◽  
Constant Density ◽  
Acquisition Duration ◽  
Regularization Parameters ◽  
Absolute Density ◽  
Gravimetric Data ◽  
Ill Posed ◽  
Leave One Out

Imaging the internal structure of volcanoes helps highlighting magma pathways and monitoring potential structural weaknesses. We jointly invert gravimetric and muographic data to determine the most precise image of the 3D density structure of the Puy de Dôme volcano (Chaîne des Puys, France) ever obtained. With rock thickness of up to 1,600 m along the muon lines of sight, it is, to our knowledge, the largest volcano ever imaged by combining muography and gravimetry. The inversion of gravimetric data is an ill-posed problem with a non-unique solution and a sensitivity rapidly decreasing with depth. Muography has the potential to constrain the absolute density of the studied structures but the use of the method is limited by the possible number of acquisition view points, by the long acquisition duration and by the noise contained in the data. To take advantage of both types of data in a joint inversion scheme, we develop a robust method adapted to the specificities of both the gravimetric and muographic data. Our method is based on a Bayesian formalism. It includes a smoothing relying on two regularization parameters (an a priori density standard deviation and an isotropic correlation length) which are automatically determined using a leave one out criterion. This smoothing overcomes artifacts linked to the data acquisition geometry of each dataset. A possible constant density offset between both datasets is also determined by least-squares. The potential of the method is shown using the Puy de Dôme volcano as case study as high quality gravimetric and muographic data are both available. Our results show that the dome is dry and permeable. Thanks to the muographic data, we better delineate a trachytic dense core surrounded by a less dense talus.

scholarly journals Range-relaxed criteria for choosing the Lagrange multipliers in nonstationary iterated Tikhonov method

2018 ◽  
Vol 40 (1) ◽  
pp. 606-627 ◽  
Author(s):  
R Boiger ◽  
A Leitão ◽  
B F Svaiter
Keyword(s):  
Lagrange Multipliers ◽  
Approximate Solutions ◽  
Identification Problem ◽  
Linear Operators ◽  
Type Method ◽  
Geometrical Properties ◽  
Regularization Parameters ◽  
New Strategy ◽  
Parameter Identification Problem ◽  
Ill Posed

Abstract In this article we propose a novel nonstationary iterated Tikhonov (NIT)-type method for obtaining stable approximate solutions to ill-posed operator equations modeled by linear operators acting between Hilbert spaces. Geometrical properties of the problem are used to derive a new strategy for choosing the sequence of regularization parameters (Lagrange multipliers) for the NIT iteration. Convergence analysis for this new method is provided. Numerical experiments are presented for two distinct applications: (I) a two-dimensional elliptic parameter identification problem (inverse potential problem); and (II) an image-deblurring problem. The results obtained validate the efficiency of our method compared with standard implementations of the NIT method (where a geometrical choice is typically used for the sequence of Lagrange multipliers).

scholarly journals Two-Parameter Regularization Method for a Nonlinear Backward Heat Problem with a Conformable Derivative

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Vu Ho ◽  
Donal O’Regan ◽  
Hoa Ngo Van
Keyword(s):  
Integral Equation ◽  
Regularization Method ◽  
Time Variable ◽  
Conformable Derivative ◽  
Heat Problem ◽  
Regularization Parameters ◽  
Ill Posed ◽  
Parameter Regularization ◽  
Two Parameter ◽  
Backward Heat Problem

In this paper, we consider the nonlinear inverse-time heat problem with a conformable derivative concerning the time variable. This problem is severely ill posed. A new method on the modified integral equation based on two regularization parameters is proposed to regularize this problem. Numerical results are presented to illustrate the efficiency of the proposed method.

scholarly journals On Levenberg-Marquardt-Kaczmarz iterative methods for solving systems of nonlinear ill-posed equations

2010 ◽  
Vol 4 (3) ◽  
pp. 335-350 ◽  
Author(s):  
Johann Baumeister ◽  
◽  
Barbara Kaltenbacher ◽  
Antonio Leitão ◽  
◽  
...  
Keyword(s):  
Iterative Methods ◽  
Levenberg Marquardt ◽  
Ill Posed

Research of Screening and Grading Method of Ageing Sensitive Objectives in Nuclear Power Plant

2014 ◽  
Vol 543-547 ◽  
pp. 858-861
Author(s):  
Xiao Tian Liu ◽  
Yong Wang ◽  
Shao Rui Niu ◽  
Yan Zhao Zhang ◽  
Zhen Hao Shi ◽  
...  
Keyword(s):  
Fuzzy Logic ◽  
Power Plant ◽  
Nuclear Power Plant ◽  
Nuclear Power ◽  
Power Plants ◽  
Lessons Learned ◽  
Nonlinear Coupling ◽  
Highly Nonlinear ◽  
Ageing Management ◽  
First Time

This first step of ageing management in nuclear power plant is to determine the objectives and their priorities. The characteristics of the objectives are complex and highly nonlinear coupling. A fuzzy logic based screening and grading method have been developed in this research for the first time which combined the genetic ageing lessons learned and field expert experience to resolve the problem. The method have been approved of highly applicability and applied to ageing management in multiple nuclear power plants.

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