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Source author record

Manuel Tiglio

Manuel Tiglio appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

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gr-qcastro-ph.IMphysics.data-anastro-ph.COastro-ph.HEmath.NAastro-ph.GAComputational Engineering, Finance, and ScienceNumerical Analysisastro-phastro-ph.EPastro-ph.SRComputationComputer VisionDistributed, Parallel, and Cluster ComputingGraphicshep-thMachine Learningmath-phmath.MPPerformancephysics.comp-phphysics.hist-phphysics.ins-det

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Published work

26 published item(s)

preprint2026arXiv

AQMP: Image compression through Adaptive Quadtree Refinement and Matching Pursuit with Hyperparameter Optimization

We present AQMP, a novel image codec combining Adaptive Quadtree Refinement with Matching Pursuit. Unlike conventional Matching Pursuit methods that operate on fixed-size sub-images, AQMP dynamically adapts block sizes to local image structure, allocating finer partitions where the image is complex and coarser ones where it is smooth. This adaptivity yields superior compression ratios compared to fixed-size block Matching Pursuit at equivalent image quality, while offering significant parallelization opportunities at both the tree-leaf level and during compression of individual nodes. The algorithm is governed by user-specified accuracy and sparsity parameters alongside a small set of additional hyperparameters. To navigate the trade-off between compression efficiency and visual quality, we perform multi-objective hyperparameter optimization using the Tree-Structured Parzen Estimator, producing comprehensive Pareto fronts. Experimental results show that AQMP achieves up to $4\times$ higher compression rates than JPEG at comparable SSIM values, while maintaining competitive quality across a broad range of compression regimes. Performance evaluation is provided using a representative set of test images. To ensure reproducibility and promote adoption, we have made our implementation publicly available on GitHub under the MIT license.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

Gravitational wave surrogates through automated machine learning

We analyze a prospect for predicting gravitational waveforms from compact binaries based on automated machine learning (AutoML) from around a hundred different possible regression models, without having to resort to tedious and manual case-by-case analyses and fine-tuning. The particular study of this article is within the context of the gravitational waves emitted by the collision of two spinless black holes in initial quasi-circular orbit. We find, for example, that approaches such as Gaussian process regression with radial bases as kernels do provide a sufficiently accurate solution, an approach which is generalizable to multiple dimensions with low computational evaluation cost. The results here presented suggest that AutoML might provide a framework for regression in the field of surrogates for gravitational waveforms. Our study is within the context of surrogates of numerical relativity simulations based on Reduced Basis and the Empirical Interpolation Method, where we find that for the particular case analyzed AutoML can produce surrogates which are essentially indistinguishable from the NR simulations themselves.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

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.

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preprint2016arXiv

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.

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preprint2015arXiv

Accelerated gravitational-wave parameter estimation with reduced order modeling

Inferring the astrophysical parameters of coalescing compact binaries is a key science goal of the upcoming advanced LIGO-Virgo gravitational-wave detector network and, more generally, gravitational-wave astronomy. However, current parameter estimation approaches for such scenarios can lead to computationally intractable problems in practice. Therefore there is a pressing need for new, fast and accurate Bayesian inference techniques. In this letter we demonstrate that a reduced order modeling approach enables rapid parameter estimation studies. By implementing a reduced order quadrature scheme within the LIGO Algorithm Library, we show that Bayesian inference on the 9-dimensional parameter space of non-spinning binary neutron star inspirals can be sped up by a factor of 30 for the early advanced detectors' configurations. This speed-up will increase to about $150$ as the detectors improve their low-frequency limit to 10Hz, reducing to hours analyses which would otherwise take months to complete. Although these results focus on gravitational detectors, the techniques are broadly applicable to any experiment where fast Bayesian analysis is desirable.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Fast and accurate prediction of numerical relativity waveforms from binary black hole coalescences using surrogate models

Simulating a binary black hole (BBH) coalescence by solving Einstein's equations is computationally expensive, requiring days to months of supercomputing time. Using reduced order modeling techniques, we construct an accurate surrogate model, which is evaluated in a millisecond to a second, for numerical relativity (NR) waveforms from non-spinning BBH coalescences with mass ratios in $[1, 10]$ and durations corresponding to about $15$ orbits before merger. We assess the model's uncertainty and show that our modeling strategy predicts NR waveforms {\em not} used for the surrogate's training with errors nearly as small as the numerical error of the NR code. Our model includes all spherical-harmonic ${}_{-2}Y_{\ell m}$ waveform modes resolved by the NR code up to $\ell=8.$ We compare our surrogate model to Effective One Body waveforms from $50$-$300 M_\odot$ for advanced LIGO detectors and find that the surrogate is always more faithful (by at least an order of magnitude in most cases).

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

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 $100$ judiciously chosen precessing inspiral waveforms are needed for $200$ cycles, mass ratios from $1$ to $10$ and spin magnitudes $\le 0.9$. In fact, using only the first $10$ reduced basis waveforms yields a maximum mismatch of $0.016$ 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 $\sim 100$ numerical simulations of binary black hole coalescences may accurately represent the seven-dimensional parameter space of precession waveforms for the considered ranges.

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preprint2014arXiv

Fast prediction and evaluation of gravitational waveforms using surrogate models

[Abridged] We propose a solution to the problem of quickly and accurately predicting gravitational waveforms within any given physical model. The method is relevant for both real-time applications and in more traditional scenarios where the generation of waveforms using standard methods can be prohibitively expensive. Our approach is based on three offline steps resulting in an accurate reduced-order model that can be used as a surrogate for the true/fiducial waveform family. First, a set of m parameter values is determined using a greedy algorithm from which a reduced basis representation is constructed. Second, these m parameters induce the selection of m time values for interpolating a waveform time series using an empirical interpolant. Third, a fit in the parameter dimension is performed for the waveform's value at each of these m times. The cost of predicting L waveform time samples for a generic parameter choice is of order m L + m c_f online operations where c_f denotes the fitting function operation count and, typically, m << L. We generate accurate surrogate models for Effective One Body (EOB) waveforms of non-spinning binary black hole coalescences with durations as long as 10^5 M, mass ratios from 1 to 10, and for multiple harmonic modes. We find that these surrogates are three orders of magnitude faster to evaluate as compared to the cost of generating EOB waveforms in standard ways. Surrogate model building for other waveform models follow the same steps and have the same low online scaling cost. For expensive numerical simulations of binary black hole coalescences we thus anticipate large speedups in generating new waveforms with a surrogate. As waveform generation is one of the dominant costs in parameter estimation algorithms and parameter space exploration, surrogate models offer a new and practical way to dramatically accelerate such studies without impacting accuracy.

0 citations0 reviewsNo code linkedOpen access
preprint2013arXiv

Gravitational wave parameter estimation with compressed likelihood evaluations

One of the main bottlenecks in gravitational wave (GW) astronomy is the high cost of performing parameter estimation and GW searches on the fly. We propose a novel technique based on Reduced Order Quadratures (ROQs), an application and data-specific quadrature rule, to perform fast and accurate likelihood evaluations. These are the dominant cost in Markov chain Monte Carlo (MCMC) algorithms, which are widely employed in parameter estimation studies, and so ROQs offer a new way to accelerate GW parameter estimation. We illustrate our approach using a four dimensional GW burst model embedded in noise. We build an ROQ for this model, and perform four dimensional MCMC searches with both the standard and ROQs quadrature rules, showing that, for this model, the ROQ approach is around 25 times faster than the standard approach with essentially no loss of accuracy. The speed-up from using ROQs is expected to increase for more complex GW signal models and therefore has significant potential to accelerate parameter estimation of GW sources such as compact binary coalescences.

0 citations0 reviewsNo code linkedOpen access
preprint2013arXiv

Two-step greedy algorithm for reduced order quadratures

We present an algorithm to generate application-specific, global reduced order quadratures (ROQ) for multiple fast evaluations of weighted inner products between parameterized functions. If a reduced basis (RB) or any other projection-based model reduction technique is applied, the dimensionality of integrands is reduced dramatically; however, the cost of approximating the integrands by projection still scales as the size of the original problem. In contrast, using discrete empirical interpolation (DEIM) points as ROQ nodes leads to a computational cost which depends linearly on the dimension of the reduced space. Generation of a reduced basis via a greedy procedure requires a training set, which for products of functions can be very large. Since this direct approach can be impractical in many applications, we propose instead a two-step greedy targeted towards approximation of such products. We present numerical experiments demonstrating the accuracy and the efficiency of the two-step approach. The presented ROQ are expected to display very fast convergence whenever there is regularity with respect to parameter variation. We find that for the particular application here considered, one driven by gravitational wave physics, the two-step approach speeds up the offline computations to build the ROQ by more than two orders of magnitude. Furthermore, the resulting ROQ rule is found to converge exponentially with the number of nodes, and a factor of ~50 savings, without loss of accuracy, is observed in evaluations of inner products when ROQ are used as a downsampling strategy for equidistant samples using the trapezoidal rule. While the primary focus of this paper is on quadrature rules for inner products of parameterized functions, our method can be easily adapted to integrations of single parameterized functions, and some examples of this type are considered.

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preprint2012arXiv

Continuum and Discrete Initial-Boundary-Value Problems and Einstein's Field Equations

Many evolution problems in physics are described by partial differential equations on an infinite domain; therefore, one is interested in the solutions to such problems for a given initial dataset. A prominent example is the binary black hole problem within Einstein's theory of gravitation, in which one computes the gravitational radiation emitted from the inspiral of the two black holes, merger and ringdown. Powerful mathematical tools can be used to establish qualitative statements about the solutions, such as their existence, uniqueness, continuous dependence on the initial data, or their asymptotic behavior over large time scales. However, one is often interested in computing the solution itself, and unless the partial differential equation is very simple, or the initial data possesses a high degree of symmetry, this computation requires approximation by numerical discretization. When solving such discrete problems on a machine, one is faced with a finite limit to computational resources, which leads to the replacement of the infinite continuum domain with a finite computer grid. This, in turn, leads to a discrete initial-boundary value problem. The hope is to recover, with high accuracy, the exact solution in the limit where the grid spacing converges to zero with the boundary being pushed to infinity. The goal of this article is to review some of the theory necessary to understand the continuum and discrete initial-boundary value problems arising from hyperbolic partial differential equations and to discuss its applications to numerical relativity; in particular, we present well-posed initial and initial-boundary value formulations of Einstein's equations, and we discuss multi-domain high-order finite difference and spectral methods to solve them.

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preprint2012arXiv

Numerical simulations with a first order BSSN formulation of Einstein's field equations

We present a new fully first order strongly hyperbolic representation of the BSSN formulation of Einstein's equations with optional constraint damping terms. We describe the characteristic fields of the system, discuss its hyperbolicity properties, and present two numerical implementations and simulations: one using finite differences, adaptive mesh refinement and in particular binary black holes, and another one using the discontinuous Galerkin method in spherical symmetry. The results of this paper constitute a first step in an effort to combine the robustness of BSSN evolutions with very high accuracy numerical techniques, such as spectral collocation multi-domain or discontinuous Galerkin methods.

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preprint2012arXiv

Reduced Basis representations of multi-mode black hole ringdown gravitational waves

We construct compact and high accuracy Reduced Basis (RB) representations of single and multiple quasinormal modes (QNMs). The RB method determines a hierarchical and relatively small set of the most relevant waveforms. We find that the exponential convergence of the method allows for a dramatic compression of template banks used for ringdown searches. Compressing a catalog with a minimal match $\MMm=0.99$, we find that the selected RB waveforms are able to represent {\em any} QNM, including those not in the original bank, with extremely high accuracy, typically less than $10^{-13}$. We then extend our studies to two-mode QNMs. Inclusion of a second mode is expected to help with detection, and might make it possible to infer details of the progenitor of the final black hole. We find that the number of RB waveforms needed to represent any two-mode ringdown waveform with the above high accuracy is {\em smaller} than the number of metric-based, one-mode templates with $\MMm=0.99$. For unconstrained two-modes, which would allow for consistency tests of General Relativity, our high accuracy RB has around $10^4$ {\em fewer} waveforms than the number of metric-based templates for $\MMm=0.99$. The number of RB elements grows only linearly with the number of multipole modes versus exponentially with the standard approach, resulting in very compact representations even for many multiple modes. The results of this paper open the possibility of searches of multi-mode ringdown gravitational waves.

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preprint2012arXiv

Towards beating the curse of dimensionality for gravitational waves using Reduced Basis

Using the Reduced Basis approach, we efficiently compress and accurately represent the space of waveforms for non-precessing binary black hole inspirals, which constitutes a four dimensional parameter space (two masses, two spin magnitudes). Compared to the non-spinning case, we find that only a {\it marginal} increase in the (already relatively small) number of reduced basis elements is required to represent any non-precessing waveform to nearly numerical round-off precision. Most parameters selected by the algorithm are near the boundary of the parameter space, leaving the bulk of its volume sparse. Our results suggest that the full eight dimensional space (two masses, two spin magnitudes, four spin orientation angles on the unit sphere) may be highly compressible and represented with very high accuracy by a remarkably small number of waveforms, thus providing some hope that the number of numerical relativity simulations of binary black hole coalescences needed to represent the entire space of configurations is not intractable. Finally, we find that the {\it distribution} of selected parameters is robust to different choices of seed values starting the algorithm, a property which should be useful for indicating parameters for numerical relativity simulations of binary black holes. In particular, we find that the mass ratios $m_1/m_2$ of non-spinning binaries selected by the algorithm are mostly in the interval $[1,3]$ and that the median of the distribution follows a power-law behavior $\sim (m_1/m_2)^{-5.25}$.

0 citations0 reviewsNo code linkedOpen access
preprint2011arXiv

Reduced basis catalogs for gravitational wave templates

We introduce a reduced basis approach as a new paradigm for modeling, representing and searching for gravitational waves. We construct waveform catalogs for non-spinning compact binary coalescences, and we find that for accuracies of 99% and 99.999% the method generates a factor of about $10-10^5$ fewer templates than standard placement methods. The continuum of gravitational waves can be represented by a finite and comparatively compact basis. The method is robust under variations in the noise of detectors, implying that only a single catalog needs to be generated.

0 citations0 reviewsNo code linkedOpen access
preprint2010arXiv

Mode coupling of Schwarzschild perturbations: Ringdown frequencies

Within linearized perturbation theory, black holes decay to their final stationary state through the well-known spectrum of quasinormal modes. Here we numerically study whether nonlinearities change this picture. For that purpose we study the ringdown frequencies of gauge-invariant second-order gravitational perturbations induced by self-coupling of linearized perturbations of Schwarzschild black holes. We do so through high-accuracy simulations in the time domain of first and second-order Regge-Wheeler-Zerilli type equations, for a variety of initial data sets. We consider first-order even-parity $(\ell=2,m=\pm 2)$ perturbations and odd-parity $(\ell=2,m=0)$ ones, and all the multipoles that they generate through self-coupling. For all of them and all the initial data sets considered we find that ---in contrast to previous predictions in the literature--- the numerical decay frequencies of second-order perturbations are the same ones of linearized theory, and we explain the observed behavior. This would indicate, in particular, that when modeling or searching for ringdown gravitational waves, appropriately including the standard quasinormal modes already takes into account nonlinear effects.

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preprint2010arXiv

Statistical constraints on binary black hole inspiral dynamics

We perform a statistical analysis of the binary black hole problem in the post-Newtonian approximation by systematically sampling and evolving the parameter space of initial configurations for quasi-circular inspirals. Through a principal component analysis of spin and orbital angular momentum variables we systematically look for uncorrelated quantities and find three of them which are highly conserved in a statistical sense, both as functions of time and with respect to variations in initial spin orientations. We also look for and find the variables that account for the largest variations in the problem. We present binary black hole simulations of the full Einstein equations analyzing to what extent these results might carry over to the full theory in the inspiral and merger regimes. Among other applications these results should be useful both in semi-analytical and numerical building of templates of gravitational waves for gravitational wave detectors.

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preprint2009arXiv

Integrating Post-Newtonian Equations on Graphics Processing Units

We report on early results of a numerical and statistical study of binary black hole inspirals. The two black holes are evolved using post-Newtonian approximations starting with initially randomly distributed spin vectors. We characterize certain aspects of the distribution shortly before merger. In particular we note the uniform distribution of black hole spin vector dot products shortly before merger and a high correlation between the initial and final black hole spin vector dot products in the equal-mass, maximally spinning case. These simulations were performed on Graphics Processing Units, and we demonstrate a speed-up of a factor 50 over a more conventional CPU implementation.

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preprint2009arXiv

Orbiting binary black hole evolutions with a multipatch high order finite-difference approach

We present numerical simulations of orbiting black holes for around twelve cycles, using a high-order multipatch approach. Unlike some other approaches, the computational speed scales almost perfectly for thousands of processors. Multipatch methods are an alternative to AMR (adaptive mesh refinement), with benefits of simplicity and better scaling for improving the resolution in the wave zone. The results presented here pave the way for multipatch evolutions of black hole-neutron star and neutron star-neutron star binaries, where high resolution grids are needed to resolve details of the matter flow.

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preprint2009arXiv

Radiation reaction and gravitational waves in the effective field theory approach

We compute the contribution to the Lagrangian from the leading order (2.5 post-Newtonian) radiation reaction and the quadrupolar gravitational waves emitted from a binary system using the effective field theory (EFT) approach of Goldberger and Rothstein. We use an initial value formulation of the underlying (quantum) framework to implement retarded boundary conditions and describe these real-time dissipative processes. We also demonstrate why the usual scattering formalism of quantum field theory inadequately accounts for these. The methods discussed here should be useful for deriving real-time quantities (including radiation reaction forces and gravitational wave emission) and hereditary terms in the post-Newtonian approximation (including memory, tail and other causal, history-dependent integrals) within the EFT approach. We also provide a consistent formulation of the radiation sector in the equivalent effective field theory approach of Kol and Smolkin.

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preprint2009arXiv

Solving the Einstein constraint equations on multi-block triangulations using finite element methods

In order to generate initial data for nonlinear relativistic simulations, one needs to solve the Einstein constraints, which can be cast into a coupled set of nonlinear elliptic equations. Here we present an approach for solving these equations on three-dimensional multi-block domains using finite element methods. We illustrate our approach on a simple example of Brill wave initial data, with the constraints reducing to a single linear elliptic equation for the conformal factor $ψ$. We use quadratic Lagrange elements on semi-structured simplicial meshes, obtained by triangulation of multi-block grids. In the case of uniform refinement the scheme is superconvergent at most mesh vertices, due to local symmetry of the finite element basis with respect to local spatial inversions. We show that in the superconvergent case subsequent unstructured mesh refinements do not improve the quality of our initial data. As proof of concept that this approach is feasible for generating multi-block initial data in three dimensions, after constructing the initial data we evolve them in time using a high order finite-differencing multi-block approach and extract the gravitational waves from the numerical solution.

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preprint2009arXiv

Turduckening black holes: an analytical and computational study

We provide a detailed analysis of several aspects of the turduckening technique for evolving black holes. At the analytical level we study the constraint propagation for a general family of BSSN-type formulation of Einstein's field equations and identify under what conditions the turducken procedure is rigorously justified and under what conditions constraint violations will propagate to the outside of the black holes. We present high-resolution spherically symmetric studies which verify our analytical predictions. Then we present three dimensional simulations of single distorted black holes using different variations of the turduckening method and also the puncture method. We study the effect that these different methods have on the coordinate conditions, constraint violations, and extracted gravitational waves. We find that the waves agree up to small but non-vanishing differences, caused by escaping superluminal gauge modes. These differences become smaller with increasing detector location.

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preprint2007arXiv

Multi-patch methods in general relativistic astrophysics - I. Hydrodynamical flows on fixed backgrounds

Many systems of interest in general relativistic astrophysics, including neutron stars, accreting compact objects in X-ray binaries and active galactic nuclei, core collapse, and collapsars, are assumed to be approximately spherically symmetric or axisymmetric. In Newtonian or fixed-background relativistic approximations it is common practice to use spherical polar coordinates for computational grids; however, these coordinates have singularities and are difficult to use in fully relativistic models. We present, in this series of papers, a numerical technique which is able to use effectively spherical grids by employing multiple patches. We provide detailed instructions on how to implement such a scheme, and present a number of code tests for the fixed background case, including an accretion torus around a black hole.

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preprint2006arXiv

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.

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preprint2004arXiv

Recent analytical and numerical techniques applied to the Einstein equations

Combining deeper insight of Einstein's equations with sophisticated numerical techniques promises the ability to construct accurate numerical implementations of these equations. We illustrate this in two examples, the numerical evolution of ``bubble'' and single black hole spacetimes. The former is chosen to demonstrate how accurate numerical solutions can answer open questions and even reveal unexpected phenomena. The latter illustrates some of the difficulties encountered in three-dimensional black hole simulations, and presents some possible remedies.

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preprint2002arXiv

Convergence and stability in numerical relativity

It is often the case in numerical relativity that schemes that are known to be convergent for well posed systems are used in evolutions of weakly hyperbolic (WH) formulations of Einstein's equations. Here we explicitly show that with several of the discretizations that have been used through out the years, this procedure leads to non-convergent schemes. That is, arbitrarily small initial errors are amplified without bound when resolution is increased, independently of the amount of numerical dissipation introduced. The lack of convergence introduced by this instability can be particularly subtle, in the sense that it can be missed by several convergence tests, especially in 3+1 dimensional codes. We propose tests and methods to analyze convergence that may help detect these situations.

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