Gaia in-orbit realignment. Overview and data analysis

The ESA Gaia spacecraft has two Shack-Hartmann wavefront sensors (WFS) on its focal plane. They are required to refocus the telescope in-orbit due to launch settings and gravity release. They require bright stars to provide good signal to noise patterns. The centroiding precision achievable poses a limit on the minimum stellar brightness required and, ultimately, on the observing time required to reconstruct the wavefront. Maximum likelihood algorithms have been developed at the Gaia SOC. They provide optimum performance according to the Cr\'amer-Rao lower bound. Detailed wavefront reconstruction procedures, dealing with partial telescope pupil sampling and partial microlens illumination have also been developed. In this work, a brief overview of the WFS and an in depth description of the centroiding and wavefront reconstruction algorithms is provided.Comment: 14 pages, 6 figures, 2 tables, proceedings of SPIE Astronomical Telescopes + Instrumentation 2012 Conference 8442 (1-6 July 2012

q-Classical polynomials and the q-Askey and Nikiforov-Uvarov tableaus

In this paper we continue the study of the q-classical (discrete) polynomials (in the Hahn's sense) started in Medem et al. (this issue, Comput. Appl. Math. 135 (2001) 157-196). Here we will compare our scheme with the well known q-Askey scheme and the Nikiforov-Uvarov tableau. Also, new families of q-polynomials are introduced.Junta de AndalucíaUnión EuropeaDirección General de Enseñanza Superio

A stochastic-dynamic model for global atmospheric mass field statistics

A model that yields the spatial correlation structure of atmospheric mass field forecast errors was developed. The model is governed by the potential vorticity equation forced by random noise. Expansion in spherical harmonics and correlation function was computed analytically using the expansion coefficients. The finite difference equivalent was solved using a fast Poisson solver and the correlation function was computed using stratified sampling of the individual realization of F(omega) and hence of phi(omega). A higher order equation for gamma was derived and solved directly in finite differences by two successive applications of the fast Poisson solver. The methods were compared for accuracy and efficiency and the third method was chosen as clearly superior. The results agree well with the latitude dependence of observed atmospheric correlation data. The value of the parameter c sub o which gives the best fit to the data is close to the value expected from dynamical considerations

The moments of Minkowski question mark function: the dyadic period function

The Minkowski question mark function ?(x) arises as a real distribution of rationals in the Farey tree. We examine the generating function of moments of ?(x). It appears that the generating function is a direct dyadic analogue of period functions for Maass wave forms and it is defined in the cut plane C(0,infinity). The exponential generating function satisfies the integral equation with kernel being the Bessel function. The solution of this integral equation leads to the definition of dyadic eigenfunctions, arising from a certain Hilbert-Schmidt operator. Finally, we describe p-adic distribution of rationals in the Stern-Brocot tree. Surprisingly, the Eisenstein series G_1(z) does manifest in both real and p-adic cases.Comment: 26 pages, 1 figure (submitted). The current paper is an essential revision of the previous version (September 2006-May 2007). Some results from an article arXiv:0801.0054 were merged into a new versio

Transport simulations in nanosystems and low-dimensional systems

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Niching in derandomized evolution strategies and its applications in quantum control

Evolutionary Algorithms (EAs), computational problem-solvers, encode complex problems into an artificial biological environment, define its genetic operators and simulate its propagation in time. Motivated by Darwinian Evolution, it is suggested that such simulations would yield an optimal solution for the given problem. The goal of this doctoral work is to extend specific variants of EAs, namely Derandomized Evolution Strategies, to subpopulations of trial solutions which evolve in parallel to various solutions of the problem. This idea stems from the evolutionary concept of organic speciation. Such techniques are called niching methods, and they are successfully developed, at several levels, throughout the first part of the thesis. Controlling the motion of atoms has been a dream since the early days of Quantum Mechanics; The foundation of the Quantum Control field in the 1980s has brought this dream to fruition. This field has experienced an amazing increase of interest during the past 10 years, in parallel to the technological developments of ultrafast laser pulse shaping capabilities. The second part of this work is devoted to the optimization of state-of-the-art Quantum Control applications, both theoretically (simulations) and experimentally (laboratory). The application of the newly developed niching techniques successfully attains multiple laser pulse conceptual designs.FOM, The Dutch Foundation for Research on Fundamental ResearchUBL - phd migration 201

The rational SPDE approach for Gaussian random fields with general smoothness

A popular approach for modeling and inference in spatial statistics is to represent Gaussian random fields as solutions to stochastic partial differential equations (SPDEs) of the form $L^{\beta}u = \mathcal{W}$, where $\mathcal{W}$ is Gaussian white noise, $L$ is a second-order differential operator, and $\beta>0$ is a parameter that determines the smoothness of $u$. However, this approach has been limited to the case $2\beta\in\mathbb{N}$, which excludes several important models and makes it necessary to keep $\beta$ fixed during inference. We propose a new method, the rational SPDE approach, which in spatial dimension $d\in\mathbb{N}$ is applicable for any $\beta>d/4$, and thus remedies the mentioned limitation. The presented scheme combines a finite element discretization with a rational approximation of the function $x^{-\beta}$ to approximate $u$. For the resulting approximation, an explicit rate of convergence to $u$ in mean-square sense is derived. Furthermore, we show that our method has the same computational benefits as in the restricted case $2\beta\in\mathbb{N}$. Several numerical experiments and a statistical application are used to illustrate the accuracy of the method, and to show that it facilitates likelihood-based inference for all model parameters including $\beta$.Comment: 28 pages, 4 figure

MDFEM: Multivariate decomposition finite element method for elliptic PDEs with lognormal diffusion coefficients using higher-order QMC and FEM

We introduce the multivariate decomposition finite element method for elliptic PDEs with lognormal diffusion coefficient $a=\exp(Z)$ where $Z$ is a Gaussian random field defined by an infinite series expansion $Z(\boldsymbol{y}) = \sum_{j \ge 1} y_j \, \phi_j$ with $y_j \sim \mathcal{N}(0,1)$ and a given sequence of functions $\{\phi_j\}_{j \ge 1}$. We use the MDFEM to approximate the expected value of a linear functional of the solution of the PDE which is an infinite-dimensional integral over the parameter space. The proposed algorithm uses the multivariate decomposition method to compute the infinite-dimensional integral by a decomposition into finite-dimensional integrals, which we resolve using quasi-Monte Carlo methods, and for which we use the finite element method to solve different instances of the PDE. We develop higher-order quasi-Monte Carlo rules for integration over the finite-dimensional Euclidean space with respect to the Gaussian distribution by use of a truncation strategy. By linear transformations of interlaced polynomial lattice rules from the unit cube to a multivariate box of the Euclidean space we achieve higher-order convergence rates for functions belonging to a class of anchored Gaussian Sobolev spaces while taking into account the truncation error. Under appropriate conditions, the MDFEM achieves higher-order convergence rates in term of error versus cost, i.e., to achieve an accuracy of $O(\epsilon)$ the computational cost is $O(\epsilon^{-1/\lambda-d'/\lambda}) = O(\epsilon^{-(p^* + d'/\tau)/(1-p^*)})$ where $\epsilon^{-1/\lambda}$ and $\epsilon^{-d'/\lambda}$ are respectively the cost of the quasi-Monte Carlo cubature and the finite element approximations, with $d' = d \, (1+\delta')$ for some $\delta' \ge 0$ and $d$ the physical dimension, and $0 < p^* \le (2 + d'/\tau)^{-1}$ is a parameter representing the sparsity of $\{\phi_j\}_{j \ge 1}$.Comment: 48 page

Perturbation Theory for Fractional Brownian Motion in Presence of Absorbing Boundaries

Fractional Brownian motion is a Gaussian process x(t) with zero mean and two-time correlations ~ t^{2H} + s^{2H} - |t-s|^{2H}, where H, with 0<H<1 is called the Hurst exponent. For H = 1/2, x(t) is a Brownian motion, while for H unequal 1/2, x(t) is a non-Markovian process. Here we study x(t) in presence of an absorbing boundary at the origin and focus on the probability density P(x,t) for the process to arrive at x at time t, starting near the origin at time 0, given that it has never crossed the origin. It has a scaling form P(x,t) ~ R(x/t^H)/t^H. Our objective is to compute the scaling function R(y), which up to now was only known for the Markov case H=1/2. We develop a systematic perturbation theory around this limit, setting H = 1/2 + epsilon, to calculate the scaling function R(y) to first order in epsilon. We find that R(y) behaves as R(y) ~ y^phi as y -> 0 (near the absorbing boundary), while R(y) ~ y^gamma exp(-y^2/2) as y -> oo, with phi = 1 - 4 epsilon + O(epsilon^2) and gamma = 1 - 2 epsilon + O(epsilon^2). Our epsilon-expansion result confirms the scaling relation phi = (1-H)/H proposed in Ref. [28]. We verify our findings via numerical simulations for H = 2/3. The tools developed here are versatile, powerful, and adaptable to different situations.Comment: 16 pages, 8 figures; revised version 2 adds discussion on spatial small-distance cutof