A variational method for quantitative photoacoustic tomography with piecewise constant coefficients

In this article, we consider the inverse problem of determining spatially heterogeneous absorption and diffusion coefficients from a single measurement of the absorbed energy (in the steady-state diffusion approximation of light transfer). This problem, which is central in quantitative photoacoustic tomography, is in general ill-posed since it admits an infinite number of solution pairs. We show that when the coefficients are known to be piecewise constant functions, a unique solution can be obtained. For the numerical determination of the coefficients, we suggest a variational method based based on an Ambrosio-Tortorelli-approximation of a Mumford-Shah-like functional, which we implemented numerically and tested on simulated two-dimensional data

Quantitative photoacoustic tomography with piecewise constant material parameters

The goal of quantitative photoacoustic tomography is to determine optical and acoustical material properties from initial pressure maps as obtained, for instance, from photoacoustic imaging. The most relevant parameters are absorption, diffusion and Grueneisen coefficients, all of which can be heterogeneous. Recent work by Bal and Ren shows that in general, unique reconstruction of all three parameters is impossible, even if multiple measurements of the initial pressure (corresponding to different laser excitation directions at a single wavelength) are available. Here, we propose a restriction to piecewise constant material parameters. We show that in the diffusion approximation of light transfer, piecewise constant absorption, diffusion and Gr\"uneisen coefficients can be recovered uniquely from photoacoustic measurements at a single wavelength. In addition, we implemented our ideas numerically and tested them on simulated three-dimensional data

A Toy Model for Testing Finite Element Methods to Simulate Extreme-Mass-Ratio Binary Systems

Extreme mass ratio binary systems, binaries involving stellar mass objects orbiting massive black holes, are considered to be a primary source of gravitational radiation to be detected by the space-based interferometer LISA. The numerical modelling of these binary systems is extremely challenging because the scales involved expand over several orders of magnitude. One needs to handle large wavelength scales comparable to the size of the massive black hole and, at the same time, to resolve the scales in the vicinity of the small companion where radiation reaction effects play a crucial role. Adaptive finite element methods, in which quantitative control of errors is achieved automatically by finite element mesh adaptivity based on posteriori error estimation, are a natural choice that has great potential for achieving the high level of adaptivity required in these simulations. To demonstrate this, we present the results of simulations of a toy model, consisting of a point-like source orbiting a black hole under the action of a scalar gravitational field.Comment: 29 pages, 37 figures. RevTeX 4.0. Minor changes to match the published versio

A Phase-Space Discontinuous Galerkin Scheme for the Radiative Transfer Equation in Slab Geometry

We derive and analyze a symmetric interior penalty discontinuous Galerkin scheme for the approximation of the second-order form of the radiative transfer equation in slab geometry. Using appropriate trace lemmas, the analysis can be carried out as for more standard elliptic problems. Supporting examples show the accuracy and stability of the method also numerically, for different polynomial degrees. For discretization, we employ quad-tree grids, which allow for local refinement in phase-space, and we show exemplary that adaptive methods can efficiently approximate discontinuous solutions. We investigate the behavior of hierarchical error estimators and error estimators based on local averaging

Numerical Relativity: A review

Computer simulations are enabling researchers to investigate systems which are extremely difficult to handle analytically. In the particular case of General Relativity, numerical models have proved extremely valuable for investigations of strong field scenarios and been crucial to reveal unexpected phenomena. Considerable efforts are being spent to simulate astrophysically relevant simulations, understand different aspects of the theory and even provide insights in the search for a quantum theory of gravity. In the present article I review the present status of the field of Numerical Relativity, describe the techniques most commonly used and discuss open problems and (some) future prospects.Comment: 2 References added; 1 corrected. 67 pages. To appear in Classical and Quantum Gravity. (uses iopart.cls

Compact binary merger simulations in numerical relativity

The era of Gravitational Waves Astronomy was launched after the success of the first observation run of the LIGO Scientific Collaboration and the VIRGO Collaboration. The large laser interferometers incredible achievement prompted the need of extensive studies in the field of compact astrophysical objects, such as Black Holes and Neutron Stars. Today, seven years after this event, the field of study underwent a notable expansion, corroborated by the detection of a signal coming from a Binary Neutron Star merger, together with its electro-magnetic counterpart, and, more recently, by the first detections of signals coming from mixed compact binaries, i.e. Black Hole- Neutron Star binaries. In this thesis work we span our attention across different aspects of compact objects mergers, including the inclusion of new physics into the already performing numerical relativity code BAM and the study of specific systems of compact objects. We first explore the treatment of neutrinos in case of Binary Neutron Star mergers and a tool to identify and further analyze regions containing trapped neutrinos, in the hot remnant of such mergers. Neutrinos, play in fact a key role into the rapid processes that characterize the formation of elements in the dynamical ejecta expelled during these catastrophic events. In the following we explore a variety of configurations of mixed compact binary systems. After the development of the new ID code Elliptica, and the steps taken to verify its accuracy, we make use of its capability to evolve sets of physical system with various properties. Exploring the space of parameters we study different spin configurations and magnitudes of single objects and their effects on the merger dynamics