3,764 research outputs found
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 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 , and
control the shape of the shear layers that are associated with these
modes. These scales point out the key role of the small parameter 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 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
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