No abstract

Context. The location of pure frequencies in the spectrum of an irregularly sampled time series is an important topic in astrophysical data analysis. Especially in the domain of asteroseismology, a highly precise and unambiguous study of frequencies in photometric light or radial velocity curves is required. Aims. Due to sampling irregularities and large observational gaps, the classic methods for frequency estimation (prewhitening techniques, clean, cleanest, etc.) sometimes suffer false detections. We propose a new framework for this problem that allows a more precise and unambiguous frequency location. Methods. Multisine fitting is addressed as the sparse representation of the data in an overcomplete dictionary of frequencies, hence the name SparSpec for the method. We model the data as the sum of an arbitrarily large number of pure frequencies, discretised on a fixed grid. Among all the many representations fitting the data, we seek the one with the fewest non-zero amplitudes. This solution can be computed by minimising a convex criterion with no local minima. A computationally efficient and convergent optimisation strategy is derived and a user-friendly software implementing SparSpec is provided online at http://www.ast.obs-mip.fr/Softwares. Results. The method is first illustrated on a simple test example where SparSpec correctly locates the frequencies while classic methods fail. Then, simulations on more realistic artificial time series reveal the interest of this new methodology in terms of robustness toward sampling aliases. An application to the radial velocity curve of the pre-main sequence Herbig Ae star HD 104237 is finally presented, where the method is able to determine oscillation frequencies even in the presence of strong low-frequency perturbations such as orbital movements. While SparSpec mainly confirms previously published studies for the four more important frequencies, it suggests some ambiguity about the position of a fifth frequency. Additional simulations show that the SparSpec results are more plausible.

Abstract-Sparse approximation addresses the problem of approximately fitting a linear model with a solution having as few non-zero components as possible. While most sparse estimation algorithms rely on suboptimal formulations, this work studies the performance of exact optimization of 0-norm-based problems through Mixed-Integer Programs (MIPs). Nine different sparse optimization problems are formulated based on 1, 2 or ∞ data misfit measures, and involving whether constrained or penalized formulations. For each problem, MIP reformulations allow exact optimization, with optimality proof, for moderate-size yet difficult sparse estimation problems. Algorithmic efficiency of all formulations is evaluated on sparse deconvolution problems. This study promotes error-constrained minimization of the 0 norm as the most efficient choice when associated with 1 and ∞ misfits, while the 2 misfit is more efficiently optimized with sparsity-constrained and sparsity-penalized problems. Then, exact 0-norm optimization is shown to outperform classical methods in terms of solution quality, both for over-and underdetermined problems. Finally, numerical simulations emphasize the relevance of the different p fitting possibilities as a function of the noise statistical distribution. Such exact approaches are shown to be an efficient alternative, in moderate dimension, to classical (suboptimal) sparse approximation algorithms with 2 data misfit. They also provide an algorithmic solution to less common sparse optimization problems based on 1 and ∞ misfits. For each formulation, simulated test problems are proposed where optima have been successfully computed. Data and optimal solutions are made available as potential benchmarks for evaluating other sparse approximation methods.

International audienceDetermining the physical characteristics of a star is an inverse problem consisting of estimating the parameters of models for the stellar structure and evolution, and knowing certain observable quantities. We use a Bayesian approach to solve this problem for α Cen A, which allows us to incorporate prior information on the parameters to be estimated, in order to better constrain the problem. Our strategy is based on the use of a Markov chain Monte Carlo (MCMC) algorithm to estimate the posterior probability densities of the stellar parameters: mass, age, initial chemical composition, etc. We use the stellar evolutionary code ASTEC to model the star. To constrain this model both seismic and non-seismic observations were considered. Several different strategies were tested to fit these values, using either two free parameters or five free parameters in ASTEC. We are thus able to show evidence that MCMC methods become efficient with respect to more classical grid-based strategies when the number of parameters increases. The results of our MCMC algorithm allow us to derive estimates for the stellar parameters and robust uncertainties thanks to the statistical analysis of the posterior probability densities. We are also able to compute odds for the presence of a convective core in α Cen A. When using core-sensitive seismic observational constraints, these can rise above ˜40 per cent. The comparison of results to previous studies also indicates that these seismic constraints are of critical importance for our knowledge of the structure of this star