46 research outputs found
The role of phase dynamics in a stochastic model of a passively advected scalar
Collective synchronous motion of the phases is introduced in a model for the
stochastic passive advection-diffusion of a scalar with external forcing. The
model for the phase coupling dynamics follows the well known Kuramoto model
paradigm of limit-cycle oscillators. The natural frequencies in the Kuramoto
model are assumed to obey a given scale dependence through a dispersion
relation of the drift-wave form , where is a
constant representing the typical strength of the gradient. The present aim is
to study the importance of collective phase dynamics on the characteristic time
evolution of the fluctuation energy and the formation of coherent structures.
Our results show that the assumption of a fully stochastic phase state of
turbulence is more relevant for high values of , where we find that the
energy spectrum follows a scaling. Whereas for lower there
is a significant difference between a-synchronised and synchronised phase
states, and one could expect the formation of coherent modulations in the
latter case.Comment: Accepted for publication in Physics of Plasma
Role of phase synchronisation in turbulence
The role of the phase dynamics in turbulence is investigated. As a demonstration of the importance of the phase dynamics, a simplified system is used, namely the one-dimensional Burgers equation, which is evolved numerically. The system is forced via a known external force, with two components that are added into the evolution equations of the amplitudes and the phase of the Fourier modes, separately. In this way, we are able to control the impact of the force on the dynamics of the phases. In the absence of the direct forcing in the phase equation, it is observed that the phases are not stochastic as assumed in the Random Phase Approximation (RPA) models, and in contrast, the non-linear couplings result in intermittent locking of the phases to ± π/2. The impact of the force, applied purely on the phases, is to increase the occurrence of the phase locking events in which the phases of the modes in a wide k range are now locked to ± π/2, leading to a change in the dynamics of both phases and amplitudes, with a significant localization of the real space flow structures
Greedy optimal control for elliptic problems and its application to turnpike problems
This is a post-peer-review, pre-copyedit version of an article published in Numerische Mathematik. The final authenticated version is available online at: https://doi.org/10.1007/s00211-018-1005-zWe adapt and apply greedy methods to approximate in an efficient way the optimal controls for parameterized elliptic control problems. Our results yield an optimal approximation procedure that, in particular, performs better than simply sampling the parameter-space to compute controls for each parameter value. The same method can be adapted for parabolic control problems, but this leads to greedy selections of the realizations of the parameters that depend on the initial datum under consideration. The turnpike property (which ensures that parabolic optimal control problems behave nearly in a static manner when the control horizon is long enough) allows using the elliptic greedy choice of the parameters in the parabolic setting too. We present various numerical experiments and an extensive discussion of the efficiency of our methodology for parabolic control and indicate a number of open problems arising when analyzing the convergence of the proposed algorithmsThis project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 694126-DyCon). Part of this research was done while the second author visited DeustoTech and Univesity of Deusto with the support of the DyCon project. The second author was also partially supported by Croatian Science Foundation under ConDyS Project, IP-2016-06-2468. The work of the third author was partially supported by the Grants MTM2014-52347, MTM2017-92996 of MINECO (Spain) and ICON of the French AN
Arrows of Time and Chaotic Properties of the Cosmic Background Radiation
We advance a new viewpoint on the connection between the the thermodynamical
and cosmological arrows of time, which can be traced via the properties of
Cosmic Microwave Background (CMB) radiation. We show that in the
Friedmann-Robertson-Walker Universe with negative curvature there is a
necessary ingredient for the existence of the thermodynamical arrow of time. It
is based on the dynamical instability of motion along null geodesics in a
hyperbolic space. Together with special (de-correlated) initial conditions,
this mechanism is sufficient for the thermodynamical arrow, whereas the special
initial conditions alone are able to generate only a pre-arrow of time. Since
the negatively curved space will expand forever, this provides a direct
connection between the thermodynamical and cosmological arrows of time. The
structural stability of the geodesic flows on hyperbolic spaces and hence the
robustness of the proposed mechanism is especially stressed. We then point out
that the main relations of equilibrium statistical thermodynamics (including
the second law) do not necessarily depend on any arrow of time. Finally we
formulate a {\it curvature anthropic principle}, which stipulates the negative
curvature as a necessary condition for the time asymmetric Universe with an
observer. CMB has to carry the signature of this principle as well
Proximal mappings and moreau envelopes of single-variable convex piecewise cubic functions and multivariable gauge functions
This work presents a collection of useful properties of the Moreau envelope for finite-dimensional, proper, lower semicontinuous, convex functions. In particular, gauge functions and piecewise cubic functions are investigated and their Moreau envelopes categorized. Characterizations of convex Moreau envelopes are established; topics include strict convexity, strong convexity, and Lipschitz continuity