scholarly journals VIGOROUS 3D ANGULAR RESECTION MODEL USING LEVENBERG – MARQUARDT METHOD

2020 ◽  
Vol 1 (1) ◽  
pp. 08-13
Author(s):  
Yaseen Mustafa
Keyword(s):  
Global Minimum ◽  
Least Squares Method ◽  
Reference Points ◽  
Least Square ◽  
Starting Values ◽  
Marquardt Method ◽  
3D Space ◽  
Levenberg Marquardt ◽  
Minimum Solution ◽  
Spherical Triangles

The resection in 3D space is a common problem in surveying engineering and photogrammetry based on observed distances, angles, and coordinates. This resection problem is nonlinear and comprises redundant observations which is normally solved using the least-squares method in an iterative approach. In this paper, we introduce a vigorous angular based resection method that converges to the global minimum even with very challenging starting values of the unknowns. The method is based on deriving oblique angles from the measured horizontal and vertical angles by solving spherical triangles. The derived oblique angles tightly connected the rays enclosed between the resection point and the reference points. Both techniques of the nonlinear least square adjustment either using the Gauss-Newton or Levenberg – Marquardt are applied in two 3D resection experiments. In both numerical methods, the results converged steadily to the global minimum using the proposed angular resection even with improper starting values. However, applying the Levenberg – Marquardt method proved to reach the global minimum solution in all the challenging situations and outperformed the Gauss-Newton method.

Calculating the Phase Equilibria of Al Rich Al-Cu-Mg Alloys

2011 ◽  
Vol 287-290 ◽  
pp. 2411-2414
Author(s):  
Zhi He ◽  
Lan Yun Li ◽  
Yong Qin Liu
Keyword(s):  
Phase Equilibria ◽  
Linear Equations ◽  
Least Square ◽  
Mg Alloys ◽  
Experimental Results ◽  
Ternary Alloys ◽  
Calculated Phase ◽  
Marquardt Method ◽  
Levenberg Marquardt ◽  
Non Linear Equations

This paper investigates a new method, the Levenberg-Marquardt method, to calculate the phase equilibria of the Al-Cu-Mg ternary alloys. The Levenberg-Marquardt method is the best algorithm to obtain the least-square solution of non-linear equations. Its application to ternary Al-Cu-Mg system is executed in detail in this paper. The calculated phase equilibria agrees well with the experimental results. Furthermore, the Levenberg-Marquardt method is not sensitive to the initial values.

Non-Linear Least-Square Optimization of Mechanisms Based on Levenberg-Marquardt Method

2008 ◽  
Author(s):  
Ramon Sancibrian ◽  
Ana De-Juan ◽  
Fernando Viadero
Keyword(s):  
Jacobian Matrix ◽  
Hessian Matrix ◽  
Optimization Method ◽  
Second Order ◽  
Least Square ◽  
Marquardt Method ◽  
Design Variables ◽  
Levenberg Marquardt ◽  
Pseudo Second Order ◽  
Linear Least Square

One of the main problems to improve the convergence rate in deterministic optimization of mechanisms is to obtain the Hessian matrix. The required second-order derivatives are difficult to obtain or they are not available. Levenberg-Marquardt optimization method is a pseudo-second order method which means that uses the jacobian information to estimate the Hessian matrix. In this paper, the formulation to obtain the exact form of the jacobian matrix is presented and how can be implemented in the Levenberg-Marquardt method. This formulation gives a very effective method to optimize mechanism geometry considering a large number of prescribed positions and design variables. At the same time it is possible to have control over singularities and permits to compare the desired and generated path avoiding translation and rotation effects.

Inverse problem of Mueller polarimetry for metrological applications

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Tatiana Novikova ◽  
Pavel Bulkin
Keyword(s):  
Inverse Problem ◽  
Nonlinear Problem ◽  
Optimization Problem ◽  
Global Minimum ◽  
Mueller Matrix ◽  
Least Square ◽  
Marquardt Algorithm ◽  
Geometrical Features ◽  
Levenberg Marquardt

Abstract Inverse problem of Mueller polarimetry is defined as a determination of geometrical features of the metrological structures (i.e. 1D diffraction gratings) from its experimental Mueller polarimetric signature. This nonlinear problem was considered as an optimization problem in a multi-parametric space using the least square criterion and the Levenberg–Marquardt algorithm. We demonstrated that solving optimization problem with the experimental Mueller matrix spectra taken in conical diffraction configuration helps finding a global minimum and results in smaller variance values of reconstructed dimensions of the grating profile.

scholarly journals Refining the Orbit of Visual Double Stars

1988 ◽  
Vol 98 ◽  
pp. 133-133
Author(s):  
Edgar Soulie
Keyword(s):  
Iterative Method ◽  
Least Square ◽  
Orbital Parameters ◽  
Marquardt Method ◽  
Double Stars ◽  
Visual Double Stars ◽  
Levenberg Marquardt

AbstractAn iterative method of refining the orbital parameters of visual double stars was described. The sum of the least-square differences is minimized by the Levenberg-Marquardt method. The application to two examples was described, including one highly inclined orbit, ADS 8862 = Hussey 664 (i = 94.3 degrees).

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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