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Inverse problems for ODEs using contraction maps and suboptimality of the ‘collage method’. (English) Zbl 1067.34010

Author’s abstract: Broad classes of inverse problems in differential and integral equations can be cast in the following framework: the optimal approximation of a target \(x\) of a suitable metric space \(X\) by the fixed-point \(\overline{x}\) of a contraction map \(T\) on \(X\). The ‘collage method’ attempts to solve such inverse problems by finding an operator \(T_{c}\) that maps the target \(x\) as close as possible to itself. In the case of ODEs, the appropriate contraction maps are integral Picard operators. In practice, the target solutions possibly arise from an interpolation of experimental data points. In this paper, we investigate the suboptimality of the collage method. A simple inequality that provides upper bounds on the improvement over collage coding is presented and some examples are studied. We conclude that, at worst, the collage method provides an excellent starting point for further optimization, in contrast to more traditional searching methods that must first select a good starting point.

MSC:

34A55 Inverse problems involving ordinary differential equations

Keywords:

collage method
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