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Stability theory based on a variational principle and finite-time direct-adjoint optimization commonly relies on the kinetic perturbation energy density E-1(t ) = (1/V-Omega) integral(Omega) e(x, t) d Omega (where e(x, t) = vertical bar u vertical bar(2)/2) as a measure of disturbance size. This type of optimization typically yields optimal perturbations that are global in the fluid domain Omega of volume V-Omega. This paper explores the use of p-norms in determining optimal perturbations for 'energy' growth over prescribed time intervals of length T. For p = 1 the traditional energy-based stability analysis is recovered, while for large p >> 1, localization of the optimal perturbations is observed which identifies confined regions, or 'hotspots', in the domain where significant energy growth can be expected. In addition, the p-norm optimization yields insight into the role and significance of various regions of the flow regarding the overall energy dynamics. As a canonical example, we choose to solve the infinity-norm optimal perturbation problem for the simple case of two-dimensional channel flow. For such a configuration, several solutions branches emerge, each of them identifying a different energy production zone in the flow: either the centre or the walls of the domain. We study several scenarios (involving centre or wall perturbations) leading to localized energy production for different optimization time intervals. Our investigation reveals that even for this simple two-dimensional channel flow, the mechanism for the production of a highly energetic and localized perturbation is not unique in time. We show that wall perturbations are optimal (with respect to the infinity-norm) for relatively short and long times, while the centre perturbations are preferred for very short and intermediate times. The developed p-norm framework is intended to facilitate worst-case analysis of shear flows and to identify localized regions supporting dominant energy growth.
Imaging polarimetry is an important tool for the study of cosmic magnetic fields. In our Galaxy, polarization levels of a few up to $\sim$10\% are measured in the submillimeter dust emission from molecular clouds and in the synchrotron emission from supernova remnants. Only few techniques exist to image the distribution of polarization angles, as a means of tracing the plane-of-sky projection of the magnetic field orientation. At submillimeter wavelengths, polarization is either measured as the differential total power of polarization-sensitive bolometer elements, or by modulating the polarization of the signal. Bolometer arrays such as LABOCA at the APEX telescope are used to observe the continuum emission from fields as large as $\sim0\fdg2$ in diameter. %Here we present the results from the commissioning of PolKa, a polarimeter for Here we present PolKa, a polarimeter for LABOCA with a reflection-type waveplate of at least 90\% efficiency. The modulation efficiency depends mainly on the sampling and on the angular velocity of the waveplate. For the data analysis the concept of generalized synchronous demodulation is introduced. The instrumental polarization towards a point source is at the level of $\sim0.1$\%, increasing to a few percent at the $-10$db contour of the main beam. A method to correct for its effect in observations of extended sources is presented. Our map of the polarized synchrotron emission from the Crab nebula is in agreement with structures observed at radio and optical wavelengths. The linear polarization measured in OMC1 agrees with results from previous studies, while the high sensitivity of LABOCA enables us to also map the polarized emission of the Orion Bar, a prototypical photon-dominated region.
While methods based on functional approaches for uncertainty quantification in physical models have reached maturity, multiscale stochastic models have recently been the focus of new numerical developments. Here we specifically take an interest in multiscale problems with numerous localized uncertainties at a micro level that can be associated with some variability in the operator or source terms, or even with some geometrical uncertainty. In order to handle the high dimensionality and the complexity that issue from such problems, a multiscale method based on patches has emerged as a relevant candidate for exploiting the localized side of uncertainties and has been extended to the stochastic framework in [1]. It proposes an efficient iterative global-local algorithm where the global problems at the macro level are made simple by introducing a fictitious patch that enables to define the (possibly coarse) global problem on a domain that contains no small scale geometrical details and that involves a deterministic operator. At the micro level, specific reformulations of local problems using fictitious domain methods [2] are introduced when the patch contains internal boundaries in order to formulate the local problem on a tensor product space. The global and local problems are solved using tensor based approximation methods [3] that allow the representation of high dimensional stochastic parametric solutions and at the same time make the stochastic methods non intrusive. In the present work, the approach is extended to problems with local non-linearities within the patches for which convergence properties are shown. We will also consider patches with variable positions which involve non conforming interfaces and for which rise questions of stability of approximation and optimal convergence with respect to the mesh.
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