Reverse Iteration in Chemical Process Simulation

Abstract

A new technique for generating approximations of Julia sets, called reverse iteration, is proposed. The concept of reverse iteration is simple and based on the idea of solving an iterative map of the general form Z k = G(Z k - 1, p) for Z k - 1 instead of Z k . It is shown that if the process of reverse iteration is initiated at any singular point, the collection of inverse images must be members of the Julia set since singular points are in the Julia set and the Julia set is closed. This sequence of reverse iterates, say {Z k - 1}, is necessarily distributed throughout the basin boundaries. It is also shown that reverse iteration can have multiple inverse images and a tree structure for the Julia set but that the associated potential combinatorial computational demand is easily resolved by exploiting the fractal nature of any Julia set. From this, practical ways generating initial values that converge to solutions to the given model equations are proposed. Several examples and geometric illustrations are used to elucidate key concepts.