10,185 research outputs found
Arnold maps with noise: Differentiability and non-monotonicity of the rotation number
Arnold's standard circle maps are widely used to study the quasi-periodic
route to chaos and other phenomena associated with nonlinear dynamics in the
presence of two rationally unrelated periodicities. In particular, the El
Nino-Southern Oscillation (ENSO) phenomenon is a crucial component of climate
variability on interannual time scales and it is dominated by the seasonal
cycle, on the one hand, and an intrinsic oscillatory instability with a period
of a few years, on the other. The role of meteorological phenomena on much
shorter time scales, such as westerly wind bursts, has also been recognized and
modeled as additive noise. We consider herein Arnold maps with additive,
uniformly distributed noise. When the map's nonlinear term, scaled by the
parameter , is sufficiently small, i.e. , the map is
known to be a diffeomorphism and the rotation number is a
differentiable function of the driving frequency . We concentrate on
the rotation number's behavior as the nonlinearity becomes large, and show
rigorously that is a differentiable function of ,
even for , at every point at which the noise-perturbed map is
mixing. We also provide a formula for the derivative of the rotation number.
The reasoning relies on linear-response theory and a computer-aided proof. In
the diffeomorphism case of , the rotation number
behaves monotonically with respect to . We show, using again a
computer-aided proof, that this is not the case when and the
map is not a diffeomorphism.Comment: Electronic copy of final peer-reviewed manuscript accepted for
publication in the Journal of Statistical Physic
An algorithm for optimal transport between a simplex soup and a point cloud
We propose a numerical method to find the optimal transport map between a
measure supported on a lower-dimensional subset of R^d and a finitely supported
measure. More precisely, the source measure is assumed to be supported on a
simplex soup, i.e. on a union of simplices of arbitrary dimension between 2 and
d. As in [Aurenhammer, Hoffman, Aronov, Algorithmica 20 (1), 1998, 61--76] we
recast this optimal transport problem as the resolution of a non-linear system
where one wants to prescribe the quantity of mass in each cell of the so-called
Laguerre diagram. We prove the convergence with linear speed of a damped
Newton's algorithm to solve this non-linear system. The convergence relies on
two conditions: (i) a genericity condition on the point cloud with respect to
the simplex soup and (ii) a (strong) connectedness condition on the support of
the source measure defined on the simplex soup. Finally, we apply our algorithm
in R^3 to compute optimal transport plans between a measure supported on a
triangulation and a discrete measure. We also detail some applications such as
optimal quantization of a probability density over a surface, remeshing or
rigid point set registration on a mesh
Comparison of linear and non-linear monotononicity-based shape reconstruction using exact matrix characterizations
Detecting inhomogeneities in the electrical conductivity is a special case of
the inverse problem in electrical impedance tomography, that leads to fast
direct reconstruction methods. One such method can, under reasonable
assumptions, exactly characterize the inhomogeneities based on monotonicity
properties of either the Neumann-to-Dirichlet map (non-linear) or its Fr\'echet
derivative (linear). We give a comparison of the non-linear and linear approach
in the presence of measurement noise, and show numerically that the two methods
give essentially the same reconstruction in the unit disk domain. For a fair
comparison, exact matrix characterizations are used when probing the
monotonicity relations to avoid errors from numerical solution to PDEs and
numerical integration. Using a special factorization of the
Neumann-to-Dirichlet map also makes the non-linear method as fast as the linear
method in the unit disk geometry.Comment: 18 pages, 5 figures, 1 tabl
Alternative fidelity measure for quantum states
We propose an alternative fidelity measure (namely, a measure of the degree
of similarity) between quantum states and benchmark it against a number of
properties of the standard Uhlmann-Jozsa fidelity. This measure is a simple
function of the linear entropy and the Hilbert-Schmidt inner product between
the given states and is thus, in comparison, not as computationally demanding.
It also features several remarkable properties such as being jointly concave
and satisfying all of "Jozsa's axioms". The trade-off, however, is that it is
supermultiplicative and does not behave monotonically under quantum operations.
In addition, new metrics for the space of density matrices are identified and
the joint concavity of the Uhlmann-Jozsa fidelity for qubit states is
established.Comment: 12 pages, 3 figures. v2 includes minor changes, new references and
new numerical results (Sec. IV
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