It seems that for a nonlinear map $f: \mathbb{R}^n \mapsto \mathbb{R}^n$ Newton's method solves (suppose f is sufficiantly smooth and solution exists) the nonlinear problem: for given $b$, find $a$ such that
$f(a)=b$.
with a initial guess $a_0$ sufficiently close to $a$.
Theoretically, the method converges quadratically to the exact solution $a$. But in actual numerical computation (Matlab etc.) if the number of unknowns $n$ is very large the method seems to be very slow and sometimes even do not converge quadratically.
Is there any recent development to overcome this shortcomings?
Matlab also has the Levenberg-Marquardt algorithm implemented which improves upon the Gauss-Newton method: http://se.mathworks.com/help/optim/ug/lsqnonlin.html
Hello Gouranga Mallik,
Actually, the most robust algorithm for identification (estimation) are meta-heuristic such as genetic algorithms and PSO. Those allow to avoid local minima because they are based on population distributed randomly (stochastic method), selection of the best individual, crossover and mutation. The limitations of these algorithm are the time consuming and the approximation of the solution. In fact, you can't obtain the right solution. You can find an explanation in my publication.
It is also true that Nelder-Mead (simplex) and Simulated-annealing algorithms are good for identification problem.
I hope these information will help you.
Bye
Conference Paper 9 Parameters Estimation of an Extended Induction Machine Mod...
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