94 research outputs found
The Convex Geometry of Linear Inverse Problems
In applications throughout science and engineering one is often faced with
the challenge of solving an ill-posed inverse problem, where the number of
available measurements is smaller than the dimension of the model to be
estimated. However in many practical situations of interest, models are
constrained structurally so that they only have a few degrees of freedom
relative to their ambient dimension. This paper provides a general framework to
convert notions of simplicity into convex penalty functions, resulting in
convex optimization solutions to linear, underdetermined inverse problems. The
class of simple models considered are those formed as the sum of a few atoms
from some (possibly infinite) elementary atomic set; examples include
well-studied cases such as sparse vectors and low-rank matrices, as well as
several others including sums of a few permutations matrices, low-rank tensors,
orthogonal matrices, and atomic measures. The convex programming formulation is
based on minimizing the norm induced by the convex hull of the atomic set; this
norm is referred to as the atomic norm. The facial structure of the atomic norm
ball carries a number of favorable properties that are useful for recovering
simple models, and an analysis of the underlying convex geometry provides sharp
estimates of the number of generic measurements required for exact and robust
recovery of models from partial information. These estimates are based on
computing the Gaussian widths of tangent cones to the atomic norm ball. When
the atomic set has algebraic structure the resulting optimization problems can
be solved or approximated via semidefinite programming. The quality of these
approximations affects the number of measurements required for recovery. Thus
this work extends the catalog of simple models that can be recovered from
limited linear information via tractable convex programming
Première démonstration de génération d'impulsions paraboliques dans une fibre à dispersion décroissante avec un profil en peigne
International audienc
Soliton spectral tunnelling in photonic crystal fibre with sub-wavelength core defect
International audienceA numerical study of soliton spectral tunnelling in an index-guiding photonic crystal fibre with a sub-wavelength central air core defect is presented. Specifically, for a fibre design where two regions of anomalous dispersion bracket a 90 nm wide region of normal dispersion, simulations show tunnelling from 1300 to 1475 nm across the forbidden normal dispersion regime, with near unity efficiency. In addition, this effect can be exploited to obtain over a factor of two nonlinear pulse compression by interrupting propagation at the distance at which soliton tunnelling occurs
Génération de nanoparticules de niobate de lithium par ablation laser femtoseconde
International audienc
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