A globally convergent Levenberg–Marquardt method for the least -norm solution of nonlinear inequalities

2008 ◽  
Vol 206 (1) ◽  
pp. 133-140 ◽  
Author(s):  
Changfeng Ma
Keyword(s):  
Marquardt Method ◽  
Globally Convergent ◽  
Levenberg Marquardt

scholarly journals Levenberg-Marquardt Method for the Eigenvalue Complementarity Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Yuan-yuan Chen ◽  
Yan Gao
Keyword(s):  
Complementarity Problem ◽  
Numerical Experiments ◽  
Nonlinear Complementarity Problem ◽  
Mathematical Programming Problem ◽  
Complementarity Constraints ◽  
Marquardt Method ◽  
Globally Convergent ◽  
Eigenvalue Complementarity Problem ◽  
Semismooth Newton ◽  
Levenberg Marquardt

The eigenvalue complementarity problem (EiCP) is a kind of very useful model, which is widely used in the study of many problems in mechanics, engineering, and economics. The EiCP was shown to be equivalent to a special nonlinear complementarity problem or a mathematical programming problem with complementarity constraints. The existing methods for solving the EiCP are all nonsmooth methods, including nonsmooth or semismooth Newton type methods. In this paper, we reformulate the EiCP as a system of continuously differentiable equations and give the Levenberg-Marquardt method to solve them. Under mild assumptions, the method is proved globally convergent. Finally, some numerical results and the extensions of the method are also given. The numerical experiments highlight the efficiency of the method.

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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