Using the Levenberg-Marquardt Method for Solutions of Inverse Transport Problems in One- and Two-Dimensional Geometries

2011 ◽  
Vol 176 (1) ◽  
pp. 106-126 ◽  
Author(s):  
Keith C. Bledsoe ◽  
Jeffrey A. Favorite ◽  
Tunc Aldemir
Keyword(s):  
Two Dimensional ◽  
Marquardt Method ◽  
Levenberg Marquardt ◽  
Transport Problems ◽  
Inverse Transport

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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