IJPAM: Volume 60, No. 4 (2010)

SOLVING INVERSE PROBLEMS FOR THE HAMMERSTEIN
INTEGRAL EQUATION AND ITS RANDOM ANALOG
USING THE ``COLLAGE METHOD'' FOR FIXED POINTS

H.E. Kunze$^1$, D. La Torre$^{2}$, K.M. Levere$^3$, E.R. Vrscay$^4$
$^{1,3}$Department of Mathematics and Statistics
College of Physical and Engineering Science
University of Guelph
50, Stone Road East, Guelph, Ontario, N1G 2W1, CANADA
$^1$e-mail: hkunze@uoguelph.ca
$^3$e-mail: klevere@uoguelph.ca
$^2$Department of Economics, Business and Statistics
University of Milan
Milan, ITALY
e-mail: davide.latorre@unimi.it
$^4$Department of Applied Mathematics
University of Waterloo
Waterloo, Ontario, N2L 3G1, CANADA
e-mail: ervrscay@math.uwaterloo.ca


Abstract.Many inverse problems in applied mathematics can be formulated as the approximation of a target element $u$ in a complete metric space $(X,d_X)$ by the fixed point $\bar x$ of an appropriate contraction mapping $T : X \mapsto X$. The method of collage coding seeks to solve this problem by finding a contraction mapping $T$ that minimizes the so-called collage distance $d(x,Tx)$. In this paper, we develop a collage coding framework for inverse problems involving deterministic or random Hammerstein integral operators. Such operators are used to model image blurring. We illustrate the method with examples.

Received: March 6, 2010

AMS Subject Classification: 45Q05, 45R05, 60H25

Key Words and Phrases: Hammerstein integral equations, inverse problems, fixed point equations, random fixed point equations, collage theorem

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2010
Volume: 60
Issue: 4