77 research outputs found
The Emergence of Gravitational Wave Science: 100 Years of Development of Mathematical Theory, Detectors, Numerical Algorithms, and Data Analysis Tools
On September 14, 2015, the newly upgraded Laser Interferometer
Gravitational-wave Observatory (LIGO) recorded a loud gravitational-wave (GW)
signal, emitted a billion light-years away by a coalescing binary of two
stellar-mass black holes. The detection was announced in February 2016, in time
for the hundredth anniversary of Einstein's prediction of GWs within the theory
of general relativity (GR). The signal represents the first direct detection of
GWs, the first observation of a black-hole binary, and the first test of GR in
its strong-field, high-velocity, nonlinear regime. In the remainder of its
first observing run, LIGO observed two more signals from black-hole binaries,
one moderately loud, another at the boundary of statistical significance. The
detections mark the end of a decades-long quest, and the beginning of GW
astronomy: finally, we are able to probe the unseen, electromagnetically dark
Universe by listening to it. In this article, we present a short historical
overview of GW science: this young discipline combines GR, arguably the
crowning achievement of classical physics, with record-setting, ultra-low-noise
laser interferometry, and with some of the most powerful developments in the
theory of differential geometry, partial differential equations,
high-performance computation, numerical analysis, signal processing,
statistical inference, and data science. Our emphasis is on the synergy between
these disciplines, and how mathematics, broadly understood, has historically
played, and continues to play, a crucial role in the development of GW science.
We focus on black holes, which are very pure mathematical solutions of
Einstein's gravitational-field equations that are nevertheless realized in
Nature, and that provided the first observed signals.Comment: 41 pages, 5 figures. To appear in Bulletin of the American
Mathematical Societ
A multi-block infrastructure for three-dimensional time-dependent numerical relativity
We describe a generic infrastructure for time evolution simulations in
numerical relativity using multiple grid patches. After a motivation of this
approach, we discuss the relative advantages of global and patch-local tensor
bases. We describe both our multi-patch infrastructure and our time evolution
scheme, and comment on adaptive time integrators and parallelisation. We also
describe various patch system topologies that provide spherical outer and/or
multiple inner boundaries.
We employ penalty inter-patch boundary conditions, and we demonstrate the
stability and accuracy of our three-dimensional implementation. We solve both a
scalar wave equation on a stationary rotating black hole background and the
full Einstein equations. For the scalar wave equation, we compare the effects
of global and patch-local tensor bases, different finite differencing
operators, and the effect of artificial dissipation onto stability and
accuracy. We show that multi-patch systems can directly compete with the
so-called fixed mesh refinement approach; however, one can also combine both.
For the Einstein equations, we show that using multiple grid patches with
penalty boundary conditions leads to a robustly stable system. We also show
long-term stable and accurate evolutions of a one-dimensional non-linear gauge
wave. Finally, we evolve weak gravitational waves in three dimensions and
extract accurate waveforms, taking advantage of the spherical shape of our grid
lines.Comment: 18 pages. Some clarifications added, figure layout improve
A sparse representation of gravitational waves from precessing compact binaries
Many relevant applications in gravitational wave physics share a significant
common problem: the seven-dimensional parameter space of gravitational
waveforms from precessing compact binary inspirals and coalescences is large
enough to prohibit covering the space of waveforms with sufficient density. We
find that by using the reduced basis method together with a parametrization of
waveforms based on their phase and precession, we can construct ultra-compact
yet high-accuracy representations of this large space. As a demonstration, we
show that less than judiciously chosen precessing inspiral waveforms are
needed for cycles, mass ratios from to and spin magnitudes . In fact, using only the first reduced basis waveforms yields a
maximum mismatch of over the whole range of considered parameters. We
test whether the parameters selected from the inspiral regime result in an
accurate reduced basis when including merger and ringdown; we find that this is
indeed the case in the context of a non-precessing effective-one-body model.
This evidence suggests that as few as numerical simulations of
binary black hole coalescences may accurately represent the seven-dimensional
parameter space of precession waveforms for the considered ranges.Comment: 5 pages, 3 figures. The parameters selected for the basis of
precessing waveforms can be found in the source file
Reduced Order and Surrogate Models for Gravitational Waves
We present an introduction to some of the state of the art in reduced order
and surrogate modeling in gravitational wave (GW) science. Approaches that we
cover include Principal Component Analysis, Proper Orthogonal Decomposition,
the Reduced Basis approach, the Empirical Interpolation Method, Reduced Order
Quadratures, and Compressed Likelihood evaluations. We divide the review into
three parts: representation/compression of known data, predictive models, and
data analysis. The targeted audience is that one of practitioners in GW
science, a field in which building predictive models and data analysis tools
that are both accurate and fast to evaluate, especially when dealing with large
amounts of data and intensive computations, are necessary yet can be
challenging. As such, practical presentations and, sometimes, heuristic
approaches are here preferred over rigor when the latter is not available. This
review aims to be self-contained, within reasonable page limits, with little
previous knowledge (at the undergraduate level) requirements in mathematics,
scientific computing, and other disciplines. Emphasis is placed on optimality,
as well as the curse of dimensionality and approaches that might have the
promise of beating it. We also review most of the state of the art of GW
surrogates. Some numerical algorithms, conditioning details, scalability,
parallelization and other practical points are discussed. The approaches
presented are to large extent non-intrusive and data-driven and can therefore
be applicable to other disciplines. We close with open challenges in high
dimension surrogates, which are not unique to GW science.Comment: Invited article for Living Reviews in Relativity. 93 page
On ab initio-based, free and closed-form expressions for gravitational waves
We introduce a new approach for fnding high accuracy, free and closed-form expressions for the gravitational waves emitted by binary black hole collisions from ab initio models. More precisely, our expressions are built from numerical surrogate models based on supercomputer simulations of the Einstein equations, which have been shown to be essentially indistinguishable from each other. Distinct aspects of our approach are that: (i) representations of the gravitational waves can be explicitly written in a few lines, (ii) these representations are free-form yet still fast to search for and validate and (iii) there are no underlying physical approximations in the underlying model. The key strategy is combining techniques from Artifcial Intelligence and Reduced Order Modeling for parameterized systems. Namely, symbolic regression through genetic programming combined with sparse representations in parameter space and the time domain using Reduced Basis and the Empirical Interpolation Method enabling fast free-form symbolic searches and large-scale a posteriori validations. As a proof of concept we present our results for the collision of two black holes, initially without spin, and with an initial separation corresponding to 25–31 gravitational wave cycles before merger. The minimum overlap, compared to ground truth solutions, is 99%. That is, 1% diference between our closed-form expressions and supercomputer simulations; this is considered for gravitational (GW) science more than the minimum required due to experimental numerical errors which otherwise dominate. This paper aims to contribute to the feld of GWs in particular and Artifcial Intelligence in general.Fil: Tiglio, Manuel. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física. Sección Ciencias de la Computación; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba; ArgentinaFil: Villanueva, Uziel Aarón. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física. Sección Ciencias de la Computación; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba; Argentin
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