scholarly journals On the global convergence of a parameter-adjusting Levenberg-Marquardt method

2015 ◽  
Vol 5 (1) ◽  
pp. 25-36 ◽  
Author(s):  
Liyan Qi ◽  
◽  
Xiantao Xiao ◽  
Liwei Zhang
Keyword(s):  
Global Convergence ◽  
Marquardt Method ◽  
Levenberg Marquardt

scholarly journals An inexact Levenberg-Marquardt method for large sparse nonlinear least squres

1985 ◽  
Vol 26 (4) ◽  
pp. 387-403 ◽  
Author(s):  
S. J. Wright ◽  
J. N. Holt
Keyword(s):  
Global Convergence ◽  
Least Squares ◽  
Jacobian Matrix ◽  
Numerical Test ◽  
Convergence Result ◽  
Test Results ◽  
Linear Least Squares ◽  
Optimal Point ◽  
Marquardt Method ◽  
Levenberg Marquardt

AbstractA method for solving problems of the form is presented. The approach of Levenberg and Marquardt is used, except that the linear least squares subproblem arising at each iteration is not solved exactly, but only to within a certain tolerance. The method is most suited to problems in which the Jacobian matrix is sparse. Use is made of the iterative algorithm LSQR of Paige and Saunders for sparse linear least squares.A global convergence result can be proven, and under certain conditions it can be shown that the method converges quadratically when the sum of squares at the optimal point is zero.Numerical test results for problems of varying residual size are given.

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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