Iterated Function Systems, Multifunctions and Multimeasures: Inverse Problems and Applications

<p>The landmark papers by Hutchinson and Barnsley and Demko showed how systems of contractive maps with associated probabilities (called <em>iterated function systems</em> by the latter), acting in a parallel manner, either deterministically or probabilistically, could be used to construct fractal sets and measures.</p> <p>There is an ongoing research programme (see <a href="http://links.uwaterloo.ca">http://links.uwaterloo.ca</a>) on the construction of appropriate IFS-type operators, or generalized fractal transforms (GFT), over various spaces, i.e., function spaces and distributions, vector-valued measures, integral transforms and wavelet transforms.</p> <p>The action of a GFT on an element $u$ of the complete metric space $(X,d)$ under consideration can be summarized as follows:</p> <ol type='i'> <li>it produces a set of $N$ spatially-contracted copies of $u$,</li> <li>it then modifies the values of these copies by means of a suitable range-mapping,</li> <li>it recombines these copies using an appropriate operator to produce the element $v$ in $X$, $v = T u$.</li> </ol> <p>In each of the above-mentioned cases, the fractal transform $T$ is guaranteed to be contractive when the parameters defining it satisfy appropriate conditions specific to the metric space of concern. In this situation, Banach's fixed point theorem guarantees the existence of a unique fixed point $u = T u$.</p> <p>The inverse problem of fractal-based approximation is as follows: given an element $y$, can we find a fractal transform $T$ with fixed point $u$ so that $d(y,u)$ is sufficiently small?</p> <p>However, the search for such transforms is enormously complicated. Thanks to a simple consequence of Banach's fixed point theorem known as the Collage Theorem, most practical methods of solving the inverse problem seek to find an operator $T$ for which the collage distance $d(u,T u)$ is as small as possible. The aim of this talk is to present some recent developments and extensions of fractal transforms and show interesting applications in image processing and economics.</p> <h6>Recent references</h6> <ul> <li>La Torre, D., Vrscay, E.R., Ebrahimi A., Barnsley M., A method of fractal coding for measure-valued images, SIAM Journal on Imaging Sciences (SIIMS), revised submission.</li> <li>Kunze H., La Torre D., Vrscay E.R., Inverse problems for random differential equations using the collage method for random contraction mappings (2008) - available on line at the <a href='http://www.sciencedirect.com/science/journal/03770427'> Journal of Computational and Applied Mathematics</a>.</li> <li>Capasso V., Kunze H., La Torre D., Vrscay E.R., Parametric estimation for deterministic and stochastic differential equations with applications (2008) - Cambridge University Press - Advances in nonlinear analysis theory methods and applications (S. Sivasundaram ed.).</li> <li>La Torre, D., Mendivil, F., Iterated function systems on multifunctions and inverse problems (2008) - 340, 2, 1469-1479 - Journal of Mathematical Analysis and Applications.</li> <li>H. Kunze, D. La Torre, E. R. Vrscay, Contractive multifunctions, fixed point inclusions and iterated multifunction systems (2007) - 330, 159-173 - Journal of Mathematical Analysis and Applications.</li> <li>H. Kunze, D. La Torre, E. R. Vrscay, Random fixed point equations and inverse problems by collage theorem (2007) - 334, 1116-1129 - Journal of Mathematical Analysis and Applications.</li> </ul>