Spectral element modeling of three dimensional wave propagation in a self-gravitating Earth with an arbitrarily stratified outer core

This paper deals with the spectral element modeling of seismic wave propagation at the global scale. Two aspects relevant to low-frequency studies are particularly emphasized. First, the method is generalized beyond the Cowling approximation in order to fully account for the effects of self-gravitation. In particular, the perturbation of the gravity field outside the Earth is handled by a projection of the spectral element solution onto the basis of spherical harmonics. Second, we propose a new formulation inside the fluid which allows to account for an arbitrary density stratification. It is based upon a decomposition of the displacement into two scalar potentials, and results in a fully explicit fluid-solid coupling strategy. The implementation of the method is carefully detailed and its accuracy is demonstrated through a series of benchmark tests.Comment: Sent to Geophysical Journal International on July 29, 200

An algorithm for computing the 2D structure of fast rotating stars

Stars may be understood as self-gravitating masses of a compressible fluid whose radiative cooling is compensated by nuclear reactions or gravitational contraction. The understanding of their time evolution requires the use of detailed models that account for a complex microphysics including that of opacities, equation of state and nuclear reactions. The present stellar models are essentially one-dimensional, namely spherically symmetric. However, the interpretation of recent data like the surface abundances of elements or the distribution of internal rotation have reached the limits of validity of one-dimensional models because of their very simplified representation of large-scale fluid flows. In this article, we describe the ESTER code, which is the first code able to compute in a consistent way a two-dimensional model of a fast rotating star including its large-scale flows. Compared to classical 1D stellar evolution codes, many numerical innovations have been introduced to deal with this complex problem. First, the spectral discretization based on spherical harmonics and Chebyshev polynomials is used to represent the 2D axisymmetric fields. A nonlinear mapping maps the spheroidal star and allows a smooth spectral representation of the fields. The properties of Picard and Newton iterations for solving the nonlinear partial differential equations of the problem are discussed. It turns out that the Picard scheme is efficient on the computation of the simple polytropic stars, but Newton algorithm is unsurpassed when stellar models include complex microphysics. Finally, we discuss the numerical efficiency of our solver of Newton iterations. This linear solver combines the iterative Conjugate Gradient Squared algorithm together with an LU-factorization serving as a preconditionner of the Jacobian matrix.Comment: 40 pages, 12 figures, accepted in J. Comput. Physic

Recent advances in directional statistics

Mainstream statistical methodology is generally applicable to data observed in Euclidean space. There are, however, numerous contexts of considerable scientific interest in which the natural supports for the data under consideration are Riemannian manifolds like the unit circle, torus, sphere and their extensions. Typically, such data can be represented using one or more directions, and directional statistics is the branch of statistics that deals with their analysis. In this paper we provide a review of the many recent developments in the field since the publication of Mardia and Jupp (1999), still the most comprehensive text on directional statistics. Many of those developments have been stimulated by interesting applications in fields as diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics, image analysis, text mining, environmetrics, and machine learning. We begin by considering developments for the exploratory analysis of directional data before progressing to distributional models, general approaches to inference, hypothesis testing, regression, nonparametric curve estimation, methods for dimension reduction, classification and clustering, and the modelling of time series, spatial and spatio-temporal data. An overview of currently available software for analysing directional data is also provided, and potential future developments discussed.Comment: 61 page

A New Spherical Harmonics Scheme for Multi-Dimensional Radiation Transport I: Static Matter Configurations

Recent work by McClarren & Hauck [29] suggests that the filtered spherical harmonics method represents an efficient, robust, and accurate method for radiation transport, at least in the two-dimensional (2D) case. We extend their work to the three-dimensional (3D) case and find that all of the advantages of the filtering approach identified in 2D are present also in the 3D case. We reformulate the filter operation in a way that is independent of the timestep and of the spatial discretization. We also explore different second- and fourth-order filters and find that the second-order ones yield significantly better results. Overall, our findings suggest that the filtered spherical harmonics approach represents a very promising method for 3D radiation transport calculations.Comment: 29 pages, 13 figures. Version matching the one in Journal of Computational Physic

A linear and regularized ODF estimation algorithm to recover multiple fibers in Q-Ball imaging

Due the well-known limitations of diffusion tensor imaging (DTI), high angular resolution diffusion imaging is currently of great interest to characterize voxels containing multiple fiber crossings. In particular, Q-ball imaging (QBI) is now a popular reconstruction method to obtain the orientation distribution function (ODF) of these multiple fiber distributions. The latter captures all important angular contrast by expressing the probability that a water molecule will diffuse into any given solid angle. However, QBI and other high order spin displacement estimation methods involve non-trivial numerical computations and lack a straightforward regularization process. In this paper, we propose a simple linear and regularized analytic solution for the Q-ball reconstruction of the ODF. First, the signal is modeled with a physically meaningful high order spherical harmonic series by incorporating the Laplace-Beltrami operator in the solution. This leads to an elegant mathematical simplification of the Funk-Radon transform using the Funk-Hecke formula. In doing so, we obtain a fast and robust model-free ODF approximation. We validate the accuracy of the ODF estimation quantitatively using the multi-tensor synthetic model where the exact ODF can be computed. We also demonstrate that the estimated ODF can recover known multiple fiber regions in a biological phantom and in the human brain. Another important contribution of the paper is the development of ODF sharpening methods. We show that sharpening the measured ODF enhances each underlying fiber compartment and considerably improves the extraction of fibers. The proposed techniques are simple linear transformations of the ODF and can easily be computed using our spherical harmonics machinery