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 Submillimeter Polarimetry with PolKa, a reflection-type modulator for the APEX telescopeImaging 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. Multiscale method with patches for the propagation of localized uncertainties in stochastic modelsWhile 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. On the verification of PGD reduced-order modelsIn current computational mechanics practice, multidimensional as well as multiscale or parametric models encountered in a wide variety of scientific and engineering fields often require either the resolution of significantly large complexity problems or the direct calculation of very numerous solutions of such complex models. In this framework, the use of model order reduction allows to dramatically reduce the computational requirements engendered by the increasing model complexity. Over the last few years, model order reduction techniques have sparked a growing interest in the whole scientific community [1, 2]. These appealing methods are based on separated variables representations of the solution of multi-parameter models lying in tensor product spaces. They enable to circumvent the terrific curse of dimensionality as the associated solution complexity scales linearly with the dimension of the tensor product space, whereas classical brute force approaches, such as mesh-based (or grid-based) approximation methods, face an exponentially growing solution complexity. Proper Generalized Decomposition (PGD) is currently one of the most popular Reduced-Order Modeling (ROM) techniques which can be interpreted as an extension of Proper Orthogonal Decomposition (POD). It allows the a priori construction of separated variables representations of the model solution without requiring any knowledge or information about this one (contrary to POD). Low-dimensional PGD reduced basis functions (or modes) are first constructed online on the fly by sequentially solving a series of few tractable simple problems. The resulting PGD-based approximate solution can then be computed offline, as it is defined explicitly in terms of all model parameters. Despite the good performances of PGD techniques, a major issue concerns the control of PGD reduced-order models and the development of robust and efficient verification tools able to assess the quality of PGD-based numerical approximations. A few works have been devoted to the development of a posteriori error estimation methods allowing to control and assess the numerical quality of PGD reduced-order models. Pioneering works provided goal-oriented error indicators designed for adaptivity purposes without furnishing reliable and strict error bounds [3]. Subsequently, guaranteed and robust error estimators have been recently introduced in [4, 5, 6] in order to control the precision of PGD reduced-order approximations for multi-parameter linear elliptic and parabolic problems depending on a moderate number of parameters. The underlying verification procedure relies on the concept of Constitutive Relation Error (CRE) [7] along with the construction of associated admissible fields. It enables to capture various error sources (space and time discretization errors, truncation error in the finite sums decomposition, etc.) and to assess their relative contributions by means of appropriate error indicators in order to drive adaptive strategies for the optimal construction of PGD reduced-order approximations. In this work, we propose an extension of this PGD-verification method to the case of large sets of model parameters. The progressive Galerkin-based PGD technique is used to build an approximate separated representation of the model solution via a greedy algorithm. Numerical experiments carried out on linear elasticity problems with numerous fluctuating model parameters illustrate the behavior and the capabilities of the verification method in the PGD framework.
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