Chaotic Optimization and the Construction of Fractals: Solution of an Inverse Problem
Giorgio Mantica
Department of Physics, Atlanta University, Atlanta, GA 30314, USA
and
School of Physics, Georgia Institute of Technology,
Atlanta, GA 30332-0430, USA
Alan Sloan
School of Mathematics, Georgia Institute of Technology,
Atlanta, GA 30332, USA
and
Iterated Systems Incorporated, 5550 P'tree Parkway, Suite 545,
Atlanta, GA 30092, USA
Abstract
An inverse problem in fractal set construction is introduced in this paper, according to the theory of iterated function systems (IFS). This theory allows the construction of a class of fractals depending on a finite number of parameters. Finding a set of parameters which reconstructs a given fractal is the goal of the inverse problem. As the solution of the inverse problem generally involves a compression of the information encoded in the fractal, complexity theory is here applied. In particular, we define the IFS-entropy to characterize the class of fractals for which the problem can be profitably solved.
The inverse problem can be reduced to the minimization of a suitable function in parameter space. We describe a new algorithm to obtain a reliable minimum solution, which originates from the theory of dynamical systems. We suggest that this algorithm should greatly improve simulated thermal annealing à la Kirkpatrick-Szu when a metric structure can be given to an optimization problem.
