Reconstructing initial data using observers: error analysis of the semi-discrete and fully discrete approximations

A new iterative algorithm for solving initial data inverse problems from partial observations has been recently proposed in Ramdani et al. (Automatica 46(10), 1616-1625, 2010 ). Based on the concept of observers (also called Luenberger observers), this algorithm covers a large class of abstract evolution PDE's. In this paper, we are concerned with the convergence analysis of this algorithm. More precisely, we provide a complete numerical analysis for semi-discrete (in space) and fully discrete approximations derived using finite elements in space and an implicit Euler method in time. The analysis is carried out for abstract Schrödinger and wave conservative systems with bounded observation (locally distributed)

Reconstructing initial data using observers : error analysis of the semi-discrete and fully discrete approximations

A new iterative algorithm for solving initial data inverse problems from partial observations has been recently proposed in Ramdani, Tucsnak and Weiss [15]. Based on the concept of observers (also called Luenberger observers), this algorithm covers a large class of abstract evolution PDE's. In this paper, we are concerned with the convergence analysis of this algorithm. More precisely, we provide a complete numerical analysis for semi-discrete (in space) and fully discrete approximations derived using finite elements in space and finite differences in time. The analysis is carried out for abstract Schr\"odinger and wave conservative systems with bounded observation (locally distributed).Comment: 38 pages, 1 figure

QPOs from Random X-ray Bursts around Rotating Black Holes

We continue our earlier studies of quasi-periodic oscillations (QPOs) in the power spectra of accreting, rapidly-rotating black holes that originate from the geometric "light echoes" of X-ray flares occurring within the black hole ergosphere. Our present work extends our previous treatment to three-dimensional photon emission and orbits to allow for arbitrary latitudes in the positions of the distant observers and the X-ray sources in place of the mainly equatorial positions and photon orbits of the earlier consideration. Following the trajectories of a large number of photons we calculate the response functions of a given geometry and use them to produce model light curves which we subsequently analyze to compute their power spectra and autocorrelation functions. In the case of an optically-thin environment, relevant to advection-dominated accretion flows, we consistently find QPOs at frequencies of order of ~kHz for stellar-mass black hole candidates while order of ~mHz for typical active galactic nuclei (~10^7 Msun) for a wide range of viewing angles (30 to 80deg) from X-ray sources predominantly concentrated toward the equator within the ergosphere. As in our previous treatment, here too, the QPO signal is produced by the frame-dragging of the photons by the rapidly-rotating black hole, which results in photon "bunches" separated by constant time-lags, the result of multiple photon orbits around the hole. Our model predicts for various source/observer configurations the robust presence of a new class of QPOs, which is inevitably generic to curved spacetime structure in rotating black hole systems.Comment: 26 pages, 8 b/w figs, accepted to Ap