Using collage coding to solve inverse problems
Many problems in the mathematical sciences involve the approximate determination of physically meaningful parameters from observed data. Many such problems can be formulated as inverse problems in which one attempts to recover the vector field of a system of ordinary differential equations. The parameter estimation literature contains many methods for handling such problems, with varying degrees of rigorous justification. This thesis presents a rigorous framework for solving such problems referred to as 'collage coding' due to its roots in fractal imaging. We present a comprehensive discussion of both the theory and historical development of this fractal-based method. Numerous examples are presented, including a particularly challenging parameter estimation problem involving a noisy partial data set originating from a chaotic system of ordinary differential equations.