93 research outputs found
Electromagnetic wormholes and virtual magnetic monopoles
We describe new configurations of electromagnetic (EM) material parameters,
the electric permittivity and magnetic permeability , that
allow one to construct from metamaterials objects that function as invisible
tunnels. These allow EM wave propagation between two points, but the tunnels
and the regions they enclose are not detectable to EM observations. Such
devices function as wormholes with respect to Maxwell's equations and
effectively change the topology of space vis-a-vis EM wave propagation. We
suggest several applications, including devices behaving as virtual magnetic
monopoles.Comment: 4 pages, 3 figure
Full-wave invisibility of active devices at all frequencies
There has recently been considerable interest in the possibility, both
theoretical and practical, of invisibility (or "cloaking") from observation by
electromagnetic (EM) waves. Here, we prove invisibility, with respect to
solutions of the Helmholtz and Maxwell's equations, for several constructions
of cloaking devices. Previous results have either been on the level of ray
tracing [Le,PSS] or at zero frequency [GLU2,GLU3], but recent numerical [CPSSP]
and experimental [SMJCPSS] work has provided evidence for invisibility at
frequency . We give two basic constructions for cloaking a region
contained in a domain from measurements of Cauchy data of waves at \p
\Omega; we pay particular attention to cloaking not just a passive object, but
an active device within , interpreted as a collection of sources and sinks
or an internal current.Comment: Final revision; to appear in Commun. in Math. Physic
Inverse problem for wave equation with sources and observations on disjoint sets
We consider an inverse problem for a hyperbolic partial differential equation
on a compact Riemannian manifold. Assuming that and are
two disjoint open subsets of the boundary of the manifold we define the
restricted Dirichlet-to-Neumann operator . This
operator corresponds the boundary measurements when we have smooth sources
supported on and the fields produced by these sources are observed
on . We show that when and are disjoint but
their closures intersect at least at one point, then the restricted
Dirichlet-to-Neumann operator determines the
Riemannian manifold and the metric on it up to an isometry. In the Euclidian
space, the result yields that an anisotropic wave speed inside a compact body
is determined, up to a natural coordinate transformations, by measurements on
the boundary of the body even when wave sources are kept away from receivers.
Moreover, we show that if we have three arbitrary non-empty open subsets
, and of the boundary, then the restricted
Dirichlet-to-Neumann operators for determine the Riemannian manifold to an isometry. Similar result is proven
also for the finite-time boundary measurements when the hyperbolic equation
satisfies an exact controllability condition
Electromagnetic wormholes via handlebody constructions
Cloaking devices are prescriptions of electrostatic, optical or
electromagnetic parameter fields (conductivity , index of refraction
, or electric permittivity and magnetic permeability
) which are piecewise smooth on and singular on a
hypersurface , and such that objects in the region enclosed by
are not detectable to external observation by waves. Here, we give related
constructions of invisible tunnels, which allow electromagnetic waves to pass
between possibly distant points, but with only the ends of the tunnels visible
to electromagnetic imaging. Effectively, these change the topology of space
with respect to solutions of Maxwell's equations, corresponding to attaching a
handlebody to . The resulting devices thus function as
electromagnetic wormholes.Comment: 25 pages, 6 figures (some color
Bifurcations sets of the Sretensky axial symmetric gyrostat
In this paper, we perform an adapted Deprit coordinate transformation and we analyse the flow evolution on the phase space for the axial symmetric gyrostat in the Sretensky case .We give a complete description of thegeneric bifurcations of the common level sets of the first integrals. A numerical investigation of these bifurcations is considered.In this paper, we perform an adapted Deprit coordinate transformation and we analyse the flow evolution on the phase space for the axial symmetric gyrostat in the Sretensky case .We give a complete description of thegeneric bifurcations of the common level sets of the first integrals. A numerical investigation of these bifurcations is considered
Inverse problems with partial data for a magnetic Schr\"odinger operator in an infinite slab and on a bounded domain
In this paper we study inverse boundary value problems with partial data for
the magnetic Schr\"odinger operator. In the case of an infinite slab in ,
, we establish that the magnetic field and the electric potential can
be determined uniquely, when the Dirichlet and Neumann data are given either on
the different boundary hyperplanes of the slab or on the same hyperplane. This
is a generalization of the results of [41], obtained for the Schr\"odinger
operator without magnetic potentials. In the case of a bounded domain in ,
, extending the results of [2], we show the unique determination of the
magnetic field and electric potential from the Dirichlet and Neumann data,
given on two arbitrary open subsets of the boundary, provided that the magnetic
and electric potentials are known in a neighborhood of the boundary.
Generalizing the results of [31], we also obtain uniqueness results for the
magnetic Schr\"odinger operator, when the Dirichlet and Neumann data are known
on the same part of the boundary, assuming that the inaccessible part of the
boundary is a part of a hyperplane
Approximate quantum cloaking and almost trapped states
We describe families of potentials which act as approximate cloaks for matter
waves, i.e., for solutions of the time-independent Schr\"odinger equation at
energy , with applications to the design of ion traps. These are derived
from perfect cloaks for the conductivity and Helmholtz equations, by a
procedure we refer to as isotropic transformation optics. If is a potential
which is surrounded by a sequence of approximate
cloaks, then for generic , asymptotically in (i) is both
undetectable and unaltered by matter waves originating externally to the cloak;
and (ii) the combined potential does not perturb waves outside the
cloak. On the other hand, for near a discrete set of energies, cloaking
{\it per se} fails and the approximate cloaks support wave functions
concentrated, or {\it almost trapped}, inside the cloaked region and negligible
outside. Applications include ion traps, almost invisible to matter waves or
customizable to support almost trapped states of arbitrary multiplicity.
Possible uses include simulation of abstract quantum systems, magnetically
tunable quantum beam switches, and illusions of singular magnetic fields.Comment: Revised, with new figures. Single column forma
A TV-Gaussian prior for infinite-dimensional Bayesian inverse problems and its numerical implementations
Many scientific and engineering problems require to perform Bayesian
inferences in function spaces, in which the unknowns are of infinite dimension.
In such problems, choosing an appropriate prior distribution is an important
task. In particular we consider problems where the function to infer is subject
to sharp jumps which render the commonly used Gaussian measures unsuitable. On
the other hand, the so-called total variation (TV) prior can only be defined in
a finite dimensional setting, and does not lead to a well-defined posterior
measure in function spaces. In this work we present a TV-Gaussian (TG) prior to
address such problems, where the TV term is used to detect sharp jumps of the
function, and the Gaussian distribution is used as a reference measure so that
it results in a well-defined posterior measure in the function space. We also
present an efficient Markov Chain Monte Carlo (MCMC) algorithm to draw samples
from the posterior distribution of the TG prior. With numerical examples we
demonstrate the performance of the TG prior and the efficiency of the proposed
MCMC algorithm
Deep Neural Networks for Inverse Problems with Pseudodifferential Operators: An Application to Limited-Angle Tomography
We propose a novel convolutional neural network (CNN), called \Psi DONet, designed for learning pseudodifferential operators (\Psi DOs) in the context of linear inverse problems. Our starting point is the iterative soft thresholding algorithm (ISTA), a well-known algorithm to solve sparsity-promoting
minimization problems. We show that, under rather general assumptions on the forward operator, the unfolded iterations of ISTA can be interpreted as the successive layers of a CNN, which in turn provides fairly general network architectures that, for a specific choice of the parameters involved, allow us to reproduce ISTA, or a perturbation of ISTA for which we can bound the coefficients of the filters. Our case study is the limited-angle X-ray transform and its application to limited-angle computed tomography (LA-CT). In particular, we prove that, in the case of LA-CT, the operations of upscaling, downscaling, and convolution, which characterize our \Psi DONet and most deep learning schemes, can be exactly determined by combining the convolutional nature of the limited-angle Xray transform and basic properties defining an orthogonal wavelet system. We test two different implementations of \Psi DONet on simulated data from limited-angle geometry, generated from the ellipse data set. Both implementations provide equally good and noteworthy preliminary results, showing the potential of the approach we propose and paving the way to applying the same idea to other convolutional operators which are \Psi DOs or Fourier integral operators
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