Mathematical Methods in Waves-Based Inverse Problems at Different Scales

Dear Colleagues, 

This Special Issue focuses on the mathematical analysis of inverse problems related to wave propagation-based modern imaging modalities with motivations coming from real-world applications. Such modalities can apply at different scales: larges scales (as in geophysics), moderate scales (as in tomography in the broader sense) and small scales (as in microscopy). We welcome contributions to the modeling, analysis and computational aspects of these inverse problems. We expect contributions from authors coming from different backgrounds to discuss different aspects of such imaging modalities.

The following tentative of classifying these inverse problems, according to the related scales, can be instructive:

At the large scale, we expect to estimate the first-order modes of the object to the image as ‘simple’ equivalent shapes or/and averages of the material parameters. Here, the mathematical techniques are related to the asymptotic modeling and analysis of the related functionals in terms of the large-scale parameters.

At the moderate scales, which are also related to the resonance regimes, different approaches are expected. Here, we deal with the intermediate modes of the objects to image. Without being exhaustive, we can cite techniques based on regularizations, optimizations, localized perturbations, Carleman estimates, spectral theory, and the use of exponential-type solutions or singular (or Green-type) solutions.

At a small scale, at least two approaches can be discussed. In the first one, we use higher-order functionals to detect inhomogeneities with finer details. In the second one, we use small perturbations (as contrasting agents) to highlight the small contrasts of the internal images. Therefore, one can extract higher-order modes of the object to the image (modeling finer details). The mathematical methods are related to the asymptotic modeling and analysis of the related functionals in terms of the small-scale parameters.

Of course, these scales are not sharply distinguishable. The techniques developed for some modalities (at a certain scale) might be used to initiate or improve the results related to the others. In addition, combinations of these techniques, whenever they are applicable, might be fruitful.

Prof. Dr. Mourad Sini
Guest Editor

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