Convergence and round-off errors in a two-dimensional eigenvalue problem using spectral methods and Arnoldi-Chebyshev algorithm

An efficient way of solving 2D stability problems in fluid mechanics is to use, after discretization of the equations that cast the problem in the form of a generalized eigenvalue problem, the incomplete Arnoldi-Chebyshev method. This method preserves the banded structure sparsity of matrices of the algebraic eigenvalue problem and thus decreases memory use and CPU-time consumption. The errors that affect computed eigenvalues and eigenvectors are due to the truncation in the discretization and to finite precision in the computation of the discretized problem. In this paper we analyze those two errors and the interplay between them. We use as a test case the two-dimensional eigenvalue problem yielded by the computation of inertial modes in a spherical shell. This problem contains many difficulties that make it a very good test case. It turns out that that single modes (especially most-damped modes i.e. with high spatial frequency) can be very sensitive to round-off errors, even when apparently good spectral convergence is achieved. The influence of round-off errors is analyzed using the spectral portrait technique and by comparison of double precision and extended precision computations. Through the analysis we give practical recipes to control the truncation and round-off errors on eigenvalues and eigenvectors.Comment: 15 pages, 9 figure

Axisymmetric inertial modes in a spherical shell at low Ekman numbers

We investigate the asymptotic properties of axisymmetric inertial modes propagating in a spherical shell when viscosity tends to zero. We identify three kinds of eigenmodes whose eigenvalues follow very different laws as the Ekman number $E$ becomes very small. First are modes associated with attractors of characteristics that are made of thin shear layers closely following the periodic orbit traced by the characteristic attractor. Second are modes made of shear layers that connect the critical latitude singularities of the two hemispheres of the inner boundary of the spherical shell. Third are quasi-regular modes associated with the frequency of neutral periodic orbits of characteristics. We thoroughly analyse a subset of attractor modes for which numerical solutions point to an asymptotic law governing the eigenvalues. We show that three length scales proportional to $E^{1/6}$, $E^{1/4}$ and $E^{1/3}$ control the shape of the shear layers that are associated with these modes. These scales point out the key role of the small parameter $E^{1/12}$ in these oscillatory flows. With a simplified model of the viscous Poincar\'e equation, we can give an approximate analytical formula that reproduces the velocity field in such shear layers. Finally, we also present an analysis of the quasi-regular modes whose frequencies are close to $\sin(\pi/4)$ and explain why a fluid inside a spherical shell cannot respond to any periodic forcing at this frequency when viscosity vanishes.Comment: 38 pages, 25 figures, to appear in J. Fluid Mechanic

CAR: A MATLAB Package to Compute Correspondence Analysis with Rotations

Correspondence analysis (CA) is a popular method that can be used to analyse relationships between categorical variables. Like principal component analysis, CA solutions can be rotated both orthogonally and obliquely to simple structure without affecting the total amount of explained inertia. We describe a MATLAB package for computing CA. The package includes orthogonal and oblique rotation of axes. It is designed not only for advanced users of MATLAB but also for beginners. Analysis can be done using a user-friendly interface, or by using command lines. We illustrate the use of CAR with one example.

MultipleCar: A Graphical User Interface MATLAB Toolbox to Compute Multiple Correspondence Analysis

In this paper we present the toolbox MultipleCar, which is a general program for computing multiple correspondence analysis and which was designed using a graphical user interface. The procedures implemented in MultipleCar are the usual ones that are already available in other applications, plus some additional procedures. MultipleCar makes it possible to compute (1) joint correspondence analysis, and (2) orthogonal and oblique rotation of coordinates. Although MultipleCar was developed in MATLAB, we compiled it as a standalone application for Windows operative systems based on graphical user interfaces. The users can decide whether to use the advanced MATLAB version of MultipleCar, or the standalone version (which does not require any programming skills)

Exploiting Partial Symmetries for Markov Chain Aggregation

International audience; The technique presented in this paper allows the automatic construction of a lumped Markov chain for almost symmetrical Stochastic Well-formed Net (SWN) models. The starting point is the Extended Symbolic Reachability Graph (ESRG), which is a reduced representation of a SWN model reachability graph (RG), based on the aggregation of states into classes. These classes may be used as aggregates for lumping the Continuous Time Markov Chain (CTMC) isomorphic to the model RG: however it is not always true that the lumpability condition is verified by this partition of states. In the paper we propose an algorithm that progressively refines the ESRG classes until a lumped Markov chain is obtained

Geology of the Zicavo Metamorphic Complex, southern Corsica (France)

In this study, we investigated the Zicavo Metamorphic Complex (southern Corsica), which ispart of the innermost Axial Zone of the Corsica-Sardinia Variscan belt. To better evaluate itsgeological and structural outline, a 1:5000 geological map, coupled with new structural/microstructural and petrographic data, is presented. The complex is formed by threetectonic units, from bottom to top: (i) an Orthogneiss Unit, (ii) a Leptyno-Amphibolite Unit,and (iii) a Micaschist Unit. They are separated by ductile shear zones with a top-to-the-SEsense of shear. They underwent a polyphase deformation and polymetamorphic history,with a shortening stage in the amphibolite facies, responsible for the main structures andshearing, followed by an exhumation phase