535 research outputs found
Shape identification in inverse medium scattering problems with a single far-field pattern
Consider time-harmonic acoustic scattering from a bounded penetrable obstacle
embedded in a homogeneous background medium. The index
of refraction characterizing the material inside is supposed to be H\"older
continuous near the corners. If is a convex polygon, we
prove that its shape and location can be uniquely determined by the far-field
pattern incited by a single incident wave at a fixed frequency. In dimensions
, the uniqueness applies to penetrable scatterers of rectangular type
with additional assumptions on the smoothness of the contrast. Our arguments
are motivated by recent studies on the absence of non-scattering wavenumbers in
domains with corners. As a byproduct, we show that the smoothness conditions in
previous corner scattering results are only required near the corners
ADAM: a general method for using various data types in asteroid reconstruction
We introduce ADAM, the All-Data Asteroid Modelling algorithm. ADAM is simple
and universal since it handles all disk-resolved data types (adaptive optics or
other images, interferometry, and range-Doppler radar data) in a uniform manner
via the 2D Fourier transform, enabling fast convergence in model optimization.
The resolved data can be combined with disk-integrated data (photometry). In
the reconstruction process, the difference between each data type is only a few
code lines defining the particular generalized projection from 3D onto a 2D
image plane. Occultation timings can be included as sparse silhouettes, and
thermal infrared data are efficiently handled with an approximate algorithm
that is sufficient in practice due to the dominance of the high-contrast
(boundary) pixels over the low-contrast (interior) ones. This is of particular
importance to the raw ALMA data that can be directly handled by ADAM without
having to construct the standard image. We study the reliability of the
inversion by using the independent shape supports of function series and
control-point surfaces. When other data are lacking, one can carry out fast
nonconvex lightcurve-only inversion, but any shape models resulting from it
should only be taken as illustrative global-scale ones.Comment: 11 pages, submitted to A&
Corners and edges always scatter
Consider time-harmonic acoustic scattering problems governed by the Helmholtz equation in two and three dimensions. We prove that bounded penetrable obstacles with corners or edges scatter every incident wave nontrivially, provided the function of refractive index is real-analytic. Moreover, if such a penetrable obstacle is a convex polyhedron or polygon, then its shape can be uniquely determined by the far-field pattern over all observation directions incited by a single incident wave. Our arguments are elementary and rely on the expansion of solutions to the Helmholtz equation
Recovering piecewise constant refractive indices by a single far-field pattern
We are concerned with the inverse scattering problem of recovering an inhomogeneous medium by the associated acoustic wave measurement. We prove that under certain assumptions, a single far-field pattern determines the values of a perturbation to the refractive index on the corners of its support. These assumptions are satisfied, for example, in the low acoustic frequency regime. As a consequence if the perturbation is piecewise constant with either a polyhedral nest geometry or a known polyhedral cell geometry, such as a pixel or voxel array, we establish the injectivity of the perturbation to far-field map given a fixed incident wave. This is the first unique determinancy result of its type in the literature, and all of the existing results essentially make use of infinitely many measurements.Peer reviewe
Perfect, strongly eutactic lattices are periodic extreme
We introduce a parameter space for periodic point sets, given as unions of
translates of point lattices. In it we investigate the behavior of the
sphere packing density function and derive sufficient conditions for local
optimality. Using these criteria we prove that perfect, strongly eutactic
lattices cannot be locally improved to yield a periodic sphere packing with
greater density. This applies in particular to the densest known lattice sphere
packings in dimension and .Comment: 20 pages, 1 table; some corrections, incorporated referee suggestion
Fast Numerical Methods for Non-local Operators
[no abstract available
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