Applied Mathematics and Numerical Analysis Seminar 

06/10/2008, 10:00 — 11:00 — Room P3.10, Mathematics Building
Davide La Torre, University of Milan, Italy

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

The landmark papers by Hutchinson and Barnsley and Demko showed how systems of contractive maps with associated probabilities (called iterated function systems by the latter), acting in a parallel manner, either deterministically or probabilistically, could be used to construct fractal sets and measures.

There is an ongoing research programme (see http://links.uwaterloo.ca) 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.

The action of a GFT on an element $u$ of the complete metric space $(X,d)$ under consideration can be summarized as follows:

1. it produces a set of $N$ spatially-contracted copies of $u$,
2. it then modifies the values of these copies by means of a suitable range-mapping,
3. it recombines these copies using an appropriate operator to produce the element $v$ in $X$, $v=Tu$.

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 $u=Tu$.

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?

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 $d(u,Tu)$ 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.

Recent references

• 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.
• 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 Journal of Computational and Applied Mathematics.
• 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.).
• La Torre, D., Mendivil, F., Iterated function systems on multifunctions and inverse problems (2008) - 340, 2, 1469-1479 - Journal of Mathematical Analysis and Applications.
• 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.
• 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.