122 research outputs found
Corrector Analysis of a Heterogeneous Multi-scale Scheme for Elliptic Equations with Random Potential
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. 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 by the authors for more
general equations in the one-dimensional setting and provide indications on the
loss in accuracy that results from using coarser, and hence less
computationally intensive, algorithms
Passive Array Correlation-Based Imaging in a Random Waveguide
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.Comment: 27 pages; second version with minor revisions; an extended
explanation on stability of the imaging method
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