The purpose of this project is to develop practical rigorous methods for estimating the error in computed waveforms from gravitational wave simulation with reliable accuracy, in support of the NSF-funded Laser Interferometer Gravitational Observatory (LIGO). The project brings together a team of applied and computational mathematicians with expertise in constructing error estimates for solutions of partial differential equations and physicists with expertise in numerical solutions of the Einstein equation and gravitational wave data analysis. The primary technical goal is to develop and analyze new mathematical and computational methods that can be used by the gravitational physics community to compute rigorous and reliably accurate estimates for the errors of numerical solutions of the Einstein equations and the gravitational waveforms that are determined from them. In particular, this research explores the following issues: (1) Error quantification and a posteriori analysis using adjoint sensitivity techniques, and their associated numerical implementation; (2) Adaptive algorithms that are driven by goal-oriented error control, and their associated theoretical convergence analysis; and (3) The role of covariance symmetry and associated geometric structures in error analysis and the construction of numerical methods. As part of the a posteriori analysis, the project team will develop the basic theory of adjoint operators and duality for the Einstein equations. This will provide the foundation for future investigations into sensitivity analysis, data assimilation and uncertainty quantification for using LIGO data. It should be emphasized that the main thrusts of the proposed research are discretization-neutral, and therefore have broad applicability to the breadth of numerical relativity codes in existence.
The NSF-supported Laser Interferometer Gravitational Observatory (LIGO) can be successful only if highly accurate gravitational waveform models are available for use as part of the data analysis process, both for detecting gravitational waves and also for measuring the physical properties of any detected signals. The strongest sources of gravitational waves are expected to be collisions between heavy, dense stars or black holes, which can only be modeled accurately using complex numerical simulations to calculate the anticipated gravitational waveforms. Such waveforms are needed to construct the filters that allow detection of the weak gravitational-wave signals in the noisy detector, and such waveforms are also needed to measure the physical properties of the sources of any detected signals. The waveform accuracy needed to accomplish the required data analysis tasks is quite high. However, the numerical relativity community has yet to develop the analytic and computational tools needed to evaluate rigorously the accuracy of the numerical waveform models. If the qualitative accuracy measures currently used by the numerical relativity community are too optimistic, the rigorous new methods developed by this project could make the difference between success and failure of LIGO. If the current numerical waveforms are in fact accurate enough, the methods developed by this project could improve the computational efficiency of determining waveforms with a specified accuracy level, and thus reduce the cost of producing them.
One objective of the research supported by this grant was to explore new ways of characterizing and measuring the errors made by computer simulations of the gravitational waves produced by possible gravitational wave sources, like pairs of black holes that orbit one another. These computer model generated waves are needed to analyze the data from the large NSF supported gravitational wave experimental effort called LIGO. After a lot of interaction and discussion with our applied mathematics collaborators in this project, better methods (than those presently being used) of estimating the particular error quantities needed in gravitational wave data analysis were not identified. Funds from this grant were also used to make advances on a number of fundamental problems related to constructing accurate computer simulations of solutions of Einstein's equation, i.e. the equations used to predict gravitational wave signals. Methods were developed, for example, for solving these equations on manifolds with arbitrary topologies. Computer simulations of the differential rotation caused by the graviational radiation driven instability in the r-modes of rotating neutron stars were undertaken. New methods and computer software were developed for constructing accurate measurements of the neutron star equation of state from gravitational wave measurements of the properties of neutron star orbits. New gravitational wave accuracy requirements were derived for one of the more recently introduced methods of testing data for potential gravitational wave signals.
