The focus of this work is on rigorous mathematical analysis of the topological derivative based detection algorithms for the localization of an elastic inclusion of vanishing characteristic size. A filtered quadratic misfit is considered, and the performance of the topological derivative imaging functional resulting therefrom is analyzed. Our analysis reveals that the imaging functional may not attain its maximum at the location of the inclusion. Moreover, the resolution of the image is below the diffraction limit. Both phenomena are due to the coupling of pressure and shear waves propagating with different wave speeds and polarization directions. A novel imaging functional based on the weighted Helmholtz decomposition of the topological derivative is, therefore, introduced. It is thereby substantiated that the maximum of the imaging functional is attained at the location of the inclusion and the resolution is enhanced and proves to be the diffraction limit. Finally, we investigate the stability of the proposed imaging functionals with respect to measurement and medium noises.

This paper is devoted to finding the highest possible focus order of planar polynomial differential equations. The results consist of two parts: (i) we explicitly construct a class of concrete systems of degree n, where n+1 is a prime p or a power of a prime p^k, and show that these systems can have a focus order n^2-n; (ii) we theoretically prove the existence of polynomial systems of degree n having a focus order n^2-1 for any even number n. Corresponding results for odd n and more concrete examples having higher focus orders are given too.

We consider reflector imaging in a weakly random waveguide. We address the situation in which the source is farther from the reflector to be imaged than the energy equipartition distance, but the receiver array is closer to the reflector to be imaged than the energy equipartition distance. As a consequence, the reflector is illuminated by a partially coherent field and the signals recorded by the receiver array are noisy. This paper shows that migration of the recorded signals cannot give a good image, but an appropriate migration of the cross correlations of the recorded signals can give a very good image. The resolution and stability analysis of this original functional shows that the reflector can be localized with an accuracy of the order of the wavelength even when the receiver array has small aperture, and that broadband sources are necessary to ensure statistical stability, whatever the aperture of the array.

No abstract uploaded!

This paper analyzes the random fluctuations obtained by a heterogeneous multi-scale first-order finite element method applied to solve elliptic equations with a random potential. Several multiscale numerical algorithms have been shown to correctly capture the homogenized limit of solutions of elliptic equations with coefficients modeled as stationary and ergodic random fields. Because theoretical results are available in the continuum setting for such equations, we consider here the case of a secondorder elliptic equations with random potential in two dimensions of space. We show that the random fluctuations of such solutions are correctly estimated by the heterogeneous multi-scale algorithm when appropriate fine-scale problems are solved on subsets that cover the whole computational domain. However, when the fine-scale problems are solved over patches that do not cover the entire domain, the random fluctuations may or may not be estimated accurately. In the case of random potentials with short-range interactions, the variance of the random fluctuations is amplified as the inverse of the fraction of the medium covered by the patches. In the case of random potentials with long-range interactions, however, such an amplification does not occur and random fluctuations are correctly captured independent of the (macroscopic) size of the patches. These results are consistent with those obtained in [9] for more general equations in the one-dimensional setting and provide indications on the loss in accuracy that results from using coarser, and hence computationally less intensive, algorithms.