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.

scholarly journals Approximate Hessian matrices and second-order optimality conditions for nonlinear programming problems with C1-data

1999 ◽  
Vol 40 (3) ◽  
pp. 403-420 ◽  
Author(s):  
V. Jeyakumar ◽  
X. Wang
Keyword(s):  
Nonlinear Programming ◽  
Optimality Conditions ◽  
Jacobian Matrix ◽  
Hessian Matrix ◽  
Second Order ◽  
Second Order Optimality Conditions ◽  
Differentiable Functions ◽  
Continuous Maps ◽  
Mathematical Programming Problems ◽  
Approximate Hessian

AbstractIn this paper, we present generalizations of the Jacobian matrix and the Hessian matrix to continuous maps and continuously differentiable functions respectively. We then establish second-order optimality conditions for mathematical programming problems with continuously differentiable functions. The results also sharpen the corresponding results for problems involving C1.1-functions.

WEIGHT GROUPINGS IN SECOND ORDER TRAINING METHODS FOR RECURRENT NETWORKS

2001 ◽  
Vol 11 (04) ◽  
pp. 379-387
Author(s):  
Lai-Wan CHAN ◽  
Chi-Cheong SZETO
Keyword(s):  
Hessian Matrix ◽  
Substantial Improvement ◽  
Training Methods ◽  
Training Time ◽  
Marquardt Method ◽  
Levenberg Marquardt ◽  
Diagonal Approximation ◽  
Time Variations ◽  
Two Parameters ◽  
Block Diagonal

In this paper, we use block-diagonal matrix to approximate the Hessian matrix in the Levenberg Marquardt method during the training of recurrent neural networks. We analyze the weight updating strategies and the groupings of the weights associated with the approximation. Two weight updating strategies, namely asynchronous and synchronous updating methods are investigated. Asynchronous method updated weights of one block at a time while synchronous method updates all weights at the same time. Variations of these two methods, which involve the determination of two parameters μ and λ, are examined. Four weight grouping methods, correlation blocks, k-unit blocks, layer blocks and arbitrary blocks are investigated and compared. Their computational complexity, approximation ability, and training time is analyzed. Comparing with the original Levenberg Marquardt method, the block-diagonal approximation methods give substantial improvement in training time without degrading the generalization ability.

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 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.

scholarly journals FLUORIDE REMOVAL USING ACTIVATED ALUMINA: A CASE STUDY OF BHOOMA CHOTA WATER

2013 ◽  
Vol 21 (21) ◽  
pp. 61-77
Author(s):  
R. AGRAWAL ◽  
M. K. MISHRA ◽  
K. MARGANDAN ◽  
K. SINGH ◽  
R. ACHARYA ◽  
...  
Keyword(s):  
Adsorption Capacity ◽  
Absorption Capacity ◽  
Intraparticle Diffusion ◽  
Second Order ◽  
Least Square ◽  
Fluoride Removal ◽  
Activated Alumina ◽  
Surface Absorption ◽  
Fluoride Adsorption ◽  
Pseudo Second Order

The adsorption of fluoride, from a fluoride, contaminated groundwater sample from the village, Bhooma Chota, District Sikar, in the State of Rajasthan, India, has been studied using alumina grade DF-101. The fluoride adsorption capacity (q1) has been fitted into the pseudo-first-order adsorption, pseudo-second-order adsorption, Elovich, and intraparticle diffusion models. It has been found that the kinetic data fits best in the pseudo-second-order rate equation giving a very high correlation coefficient (R2 = 0.991). the modeled fluoride absorption capacity (q1) has been calculated from the various equations using the constants derived from the least square regression plots. The calculated q1 values, model the experimental data very well, for the pseudo-second-order and Elovich equations, as is evident from the sum of square error calculations. Fluoride removal is through a combination of surface absorption and intraparticle diffusion. A study of the fluoride removal process with increasing dosage of activated alumina reveals that though the percentage of fluoride removal increases with activated alumina, the adsorption capacity at equilibrium decreases. The minimum dosage of activated alumina which causes the maximum percentage removal of fluoride from water, while at the same time brings forth its highest equilibrium absorption capacity has been determined.

COUPLED ALGORITHMS FOR PIEZOELECTRIC ACTUATOR DESIGN OPTIMIZATION FOR SHAPE CONTROL OF SMART STRUCTURES

2009 ◽  
Vol 06 (04) ◽  
pp. 501-519 ◽  
Author(s):  
QUAN NGUYEN ◽  
LIYONG TONG
Keyword(s):  
Genetic Algorithm ◽  
Design Optimization ◽  
Piezoelectric Actuator ◽  
Shape Control ◽  
Convergence Condition ◽  
Numerical Results ◽  
Least Square ◽  
Design Variables ◽  
Linear Least Square ◽  
Actuator Design

This paper presents two coupled algorithms for piezoelectric actuator design optimization for shape control of structures with both applied voltages and the shapes of actuators being treated as design variables. The optimum values for the applied voltages to actuators can be determined using the Linear Least Square (LLS) method, whereas the shapes of actuators can be optimized using the Genetic Algorithm (GA). These algorithms are combined together to develop a GA+LLS coupled algorithm, for design optimization of both actuators' shapes and voltages in either an alternating or a concurrent manner. In the alternating approach, LLS is utilized to determine the optimum voltages with given actuator geometry, and then GA is used to determine the optimal actuator shapes with given voltages; the alternating calculations continue until the selected convergence condition is met. In the concurrent approach, the LLS is embedded in GA to determine optimum voltages and then to modify the associated strings for each individual population. Numerical results are presented to validate the proposed algorithms. It is found that the concurrent GALLS algorithm appears to be most efficient and effective.

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).

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