Fast prediction and evaluation of gravitational waveforms using surrogate models (1308.3565v2)

Abstract: [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.

• The paper develops surrogate models that achieve over three orders of magnitude reduction in computation time for gravitational waveform prediction.
• The methodology employs a greedy algorithm for reduced basis selection and an empirical interpolation method to capture key waveform dynamics.
• Fitting waveform amplitude and phase at strategic time nodes ensures high-fidelity predictions, enabling real-time gravitational wave analysis.

Overview of Predictive Models for Gravitational Waveforms

The paper under review presents a comprehensive approach to addressing the computational challenges associated with generating gravitational waveforms, particularly when dealing with binary black hole coalescences. These waveforms, which carry essential information about highly gravitating objects, are crucial for gravitational wave physics but notoriously expensive to compute via traditional numerical methods.

The authors introduce surrogate models as a solution, enabling rapid and accurate predictions without compromising on the underlying physics. This method is structured into three offline steps that build upon reduced order modeling techniques: selecting appropriate parameter values with a greedy algorithm, employing empirical interpolation to determine crucial time values, and fitting waveform values at these times. The surrogate models dramatically reduce the computation cost, demonstrating speedups of more than three orders of magnitude compared to traditional methods.

Key Methodological Components

1. Reduced Basis Selection: The authors employ a greedy algorithm to establish a reduced basis that captures the essential waveform characteristics across a range of parameters. This basis proves to be efficient as it requires significantly fewer waveforms than standard parameter explorations would suggest. The exponential decay of the greedy error solidifies the efficacy of the reduced basis in constraining the problem's complexity.
2. Empirical Interpolation Method (EIM): The EIM selects specific time nodes critical for waveform reconstruction, leveraging the reduced basis to build a highly accurate temporal interpolant. This step ensures minimal interpolation errors while maintaining well-conditioned computations. The empirical nodes are carefully chosen to represent the waveform family accurately, reflecting its structure through minimal points and enabling rapid evaluations.
3. Fitting Parameter Dependence: By fitting the waveform's amplitude and phase at the empirical nodes, the model retains high fidelity in parameter space. This allows for accurate predictions at arbitrary parameter values, addressing one of the principal computational bottlenecks in gravitational waveform modeling.

Numerical Results and Implications

The paper provides numerical evidence from surrogate models built for Effective One Body (EOB) waveforms of non-spinning binary black hole systems, showcasing substantial speedups and error reductions. For astrophysical waveforms relevant to gravitational wave detection, these surrogates present an efficient alternative to traditional methods, offering a promising pathway for real-time data analysis and parameter estimation. Notably, this approach is anticipated to extend to spinning waveforms and more complex models, such as those derived from numerical relativity simulations.

Future Prospects

The surrogate modeling framework outlined in this paper represents a significant advancement in the computational handling of gravitational waveforms. As gravitational wave detectors continue to evolve, these models will play a pivotal role in facilitating rapid analyses, allowing researchers to explore vast parameter spaces more efficiently. Continued development could integrate advanced fitting techniques or expand the models to multi-dimensional parameter spaces, further enhancing their utility in gravitational wave physics.

In conclusion, the methodology and results of this paper underscore the potential of surrogate models as a practical tool for gravitational waveform predictions, setting a foundation for further research and application in this critical field of paper.