Chaotic behavior in an algorithm to escape from poor local minima in lens design.
In lens design, damped least-squares methods are typically used to find the nearest local minimum to a starting configuration in the merit function landscape. In this paper, we explore the use of such a method for a purpose that goes beyond local optimization. The merit function barrier, which separates an unsatisfactory solution from a neighboring one that is better, can be overcome by using low damping and by allowing the merit function to temporarily increase. However, such an algorithm displays chaos, chaotic transients and other types of complex behavior. A successful escape of the iteration trajectory from a poor local minimum to a better one is associated with a crisis phenomenon that transforms a chaotic attractor into a chaotic saddle. The present analysis also enables a better understanding of peculiarities encountered with damped least-squares algorithms in conventional local optimization tasks.
Department of Imaging Science and Technology, Faculty of Applied Sciences, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands. mvanturnhout@gmail.com
Maarten
M
van Turnhout
Florian
F
Bociort
D000465:Algorithms; D003198:Computer Simulation; D017076:Computer-Aided Design; D004867:Equipment Design; D019544:Equipment Failure Analysis; D007089:Image Enhancement / Q000295:instrumentation*; D007909:Lenses; D008027:Light; D008962:Models, Theoretical; D017711:Nonlinear Dynamics; D015203:Reproducibility of Results; D012542:Scattering, Radiation; D012680:Sensitivity and Specificity