Herb Kunze

IFSM fractal image compression with entropy and sparsity constraints: A sequential quadratic programming approach

We consider the inverse problem associated with IFSM: Given a target function f , find an IFSM, such that its fixed point f¯ is sufficiently close to f in the Lp distance. Forte and Vrscay [1] showed how to reduce this problem to a quadratic optimization model. In this paper, we extend the collage-based method developed by Kunze, La Torre and Vrscay ([2][3][4]), by proposing the minimization of the 1-norm instead of the 0-norm. In fact, optimization problems involving the 0-norm are combinatorial in nature, and hence in general NP-hard. To overcome these difficulties, we introduce the 1-norm and propose a Sequential Quadratic Programming algorithm to solve the corresponding inverse problem. As in Kunze, La Torre and Vrscay [3] in our formulation, the minimization of collage error is treated as a multi-criteria problem that includes three different and conflicting criteria i.e., collage error, entropy and sparsity. This multi-criteria program is solved by means of a scalarization technique which reduces th...

Inverse Problems: Theory and Application to Science and Engineering 2015

1 Department of Mathematics and Statistics, University of Guelph, Guelph, ON, Canada N1G 2WA 2Department of Economics, Management, and Quantitative Methods, University of Milan, 20122 Milan, Italy 3 Department of Applied Mathematics and Sciences, Khalifa University, 127788 Abu Dhabi, UAE 4Department of Mathematics and Statistics, Acadia University, Wolfville, NS, Canada B4P 2R6 5Department of Applied Mathematics, University of Granada, 18071 Granada, Spain

Collage theorem-based approaches for solving inverse problems for differential equations: A review of recent developments

In this short survey, we review the current status of fractal-based techniques and their application to the solution of inverse problems for ordinary and partial differential equations. This involves an examination of several methods which are based on the so-called Collage Theorem, a simple consequence of Banach’s Fixed Point Theorem, and its extensions.

An inverse problem for a 2-D system of steady-state reaction-diffusion equations on a perforated domain

We consider an inverse problem for a two-dimensional system of steady-state reaction-diffusions acting on a perforated domain. We establish several results that connect the parameter values for the problem on the perforated domain with the corresponding problem on the related unperforated of solid domain. This opens the possibility of estimating a solution to the inverse problem on the perforated domain by instead working with the easier-to-solve inverse problem on the solid domain. We illustrate the results with an example.