Solving inverse problems for DEs using the Collage Theorem and entropy maximization

• Published in: Applied mathematics letters 2012-12, Vol.25 (12), p.2306-2311
• Main Authors: Kunze, Herb, La Torre, Davide, Vrscay, Edward R.
• Format: Article
• Language: English
• Subjects: Approximation; Collage Theorem; Entropy; Errors; Inverse problems; Inverse problems for DEs; Mathematical analysis; Mathematical models; Theorems; Weight reduction
• ISSN: 0893-9659; 1873-5452
• DOI: 10.1016/j.aml.2012.06.021

In this work, we focus on the inverse problem associated with a DE: Given a target function x, find a DE such that its solution x̄ is sufficiently close to x in the sup norm distance. We extend the previous method for solving inverse problems for DEs using the Collage Theorem along a new direction. We search for a set of coefficients that not only minimizes the collage error but also maximizes the entropy. This approach is motivated by some promising results for IFS and probabilities. In our new formulation, the minimization of the collage error can be understood as a multi-criteria problem: two different and conflicting criteria are considered, i.e., collage error and the entropy. In order to deal with this kind of scenario we propose to scalarize the model, which reduces the multi-criteria program to a single-criterion program by combining all objective functions with different trade-off weights. Numerical examples confirm the sub-optimality of the Collage Theorem and we show that, by adding the entropy term, we obtain a better approximation of the solution.