Finite Element Maximum Entropy Method for Approximating Absolutely Continuous Invariant Measures

In a chaotic dynamical system, the eventual behavior of iterates of initial points of a map is unpredictable even though the map is deterministic. A system which is chaotic in a deterministic point of view may be regular in a statistical viewpoint. The statistical viewpoint requires the study of absolutely continuous invariant measure (ACIM) of a map with respect to the Lebesgue measure. An invariant density of the Frobenius-Perron (F-P) operator associated with a nonsingular map is employed to evaluate an ACIM of the map. The ACIM is a key factor for studying the eventual behavior of iterates of almost all initial points of the map. It is difficult to obtain an invariant density of the F-P operator in an exact mathematical form except for some simple maps. Different numerical schemes have been developed to approximate such densities. The maximum entropy principle gives a criterion to select a least-biased density among all densities satisfying a system of moment equations. In this principle, a least-biased density maximizes the Boltzmann entropy. In this dissertation, piecewise quadratic functions and quadratic splines are used in the maximum entropy method to calculate the L1 errors between the exact and the approximate invariant densities of the F-P operator associated with nonsingular maps defined from [0;1] to itself. The numerical results are supported by rigorous mathematical proofs. The L1 errors between the exact and approximate invariant densities of the Markov operator associated with Markov type position dependent random maps, defined from [0;1] to itself, are calculated by using the piecewise linear polynomials maximum entropy method

Sunyaev-Zel'dovich clusters reconstruction in multiband bolometer camera surveys

We present a new method for the reconstruction of Sunyaev-Zel'dovich (SZ) galaxy clusters in future SZ-survey experiments using multiband bolometer cameras such as Olimpo, APEX, or Planck. Our goal is to optimise SZ-Cluster extraction from our observed noisy maps. We wish to emphasize that none of the algorithms used in the detection chain is tuned on prior knowledge on the SZ -Cluster signal, or other astrophysical sources (Optical Spectrum, Noise Covariance Matrix, or covariance of SZ Cluster wavelet coefficients). First, a blind separation of the different astrophysical components which contribute to the observations is conducted using an Independent Component Analysis (ICA) method. Then, a recent non linear filtering technique in the wavelet domain, based on multiscale entropy and the False Discovery Rate (FDR) method, is used to detect and reconstruct the galaxy clusters. Finally, we use the Source Extractor software to identify the detected clusters. The proposed method was applied on realistic simulations of observations. As for global detection efficiency, this new method is impressive as it provides comparable results to Pierpaoli et al. method being however a blind algorithm. Preprint with full resolution figures is available at the URL: w10-dapnia.saclay.cea.fr/Phocea/Vie_des_labos/Ast/ast_visu.php?id_ast=728Comment: Submitted to A&A. 32 Pages, text onl

Entropy computing via integration over fractal measures

We discuss the properties of invariant measures corresponding to iterated function systems (IFSs) with place-dependent probabilities and compute their Renyi entropies, generalized dimensions, and multifractal spectra. It is shown that with certain dynamical systems one can associate the corresponding IFSs in such a way that their generalized entropies are equal. This provides a new method of computing entropy for some classical and quantum dynamical systems. Numerical techniques are based on integration over the fractal measures.Comment: 14 pages in Latex, Revtex + 4 figures in .ps attached (revised version, new title, several changes, to appear in CHAOS

An Independent Component Analysis Based Tool for Exploring Functional Connections in the Brain

This thesis describes the use of independent component analysis (ICA) as a measure of voxel similarity, which allows the user to find and view statistically independent maps of correlated voxel activity. The tool developed in this work uses a specialized clustering technique, designed to find and characterize clusters of activated voxels, to compare the independent component spatial maps across patients. This same method is also used to compare SPM results across patients

Extending emission line Doppler tomography ; mapping modulated line flux

Emission line Doppler tomography is a powerful tool that resolves the accretion flow in binaries on micro-arcsecond scales using time-resolved spectroscopy. I present an extension to Doppler tomography that relaxes one of its fundamental axioms and permits the mapping of time-dependent emission sources. Significant variability on the orbital period is a common characteristic of the emission sources that are observed in the accretion flows of cataclysmic variables and X-ray binaries. Modulation Doppler tomography maps sources varying harmonically as a function of the orbital period through the simultaneous reconstruction of three Doppler tomograms. One image describes the average flux distribution like in standard tomography, while the two additional images describe the variable component in terms of its sine and cosine amplitudes. I describe the implementation of such an extension in the form of the maximum entropy based fitting code MODMAP. Test reconstructions of synthetic data illustrate that the technique is robust and well constrained. Artifact free reconstructions of complex emission distributions can be achieved under a wide range of signal to noise levels. An application of the technique is illustrated by mapping the orbital modulations of the asymmetric accretion disc emission in the dwarf nova IP Pegasi.Comment: 8 pages, 4 figures; accepted for publication in MNRA

Weak Lensing Mass Reconstruction using Wavelets

This paper presents a new method for the reconstruction of weak lensing mass maps. It uses the multiscale entropy concept, which is based on wavelets, and the False Discovery Rate which allows us to derive robust detection levels in wavelet space. We show that this new restoration approach outperforms several standard techniques currently used for weak shear mass reconstruction. This method can also be used to separate E and B modes in the shear field, and thus test for the presence of residual systematic effects. We concentrate on large blind cosmic shear surveys, and illustrate our results using simulated shear maps derived from N-Body Lambda-CDM simulations with added noise corresponding to both ground-based and space-based observations.Comment: Accepted manuscript with all figures can be downloaded at: http://jstarck.free.fr/aa_wlens05.pdf and software can be downloaded at http://jstarck.free.fr/mrlens.htm

On the concept of complexity in random dynamical systems

We introduce a measure of complexity in terms of the average number of bits per time unit necessary to specify the sequence generated by the system. In random dynamical system, this indicator coincides with the rate K of divergence of nearby trajectories evolving under two different noise realizations. The meaning of K is discussed in the context of the information theory, and it is shown that it can be determined from real experimental data. In presence of strong dynamical intermittency, the value of K is very different from the standard Lyapunov exponent computed considering two nearby trajectories evolving under the same randomness. However, the former is much more relevant than the latter from a physical point of view as illustrated by some numerical computations for noisy maps and sandpile models.Comment: 35 pages, LaTe