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High-Precision Numerical Simulations on a CUDA GPU: Kerr Black Hole Tails

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Abstract

Computational science has advanced significantly over the past decade and has impacted almost every area of science and engineering. Most numerical scientific computation today is performed with double-precision floating-point accuracy (64-bit or \(\sim \)15 decimal digits); however, there are a number of applications that benefit from a higher level of numerical precision. In this paper, we describe such an application in the research area of black hole physics: studying the late-time behavior of decaying fields in Kerr black hole space-time. More specifically, this application involves a hyperbolic partial-differential-equation solver that uses high-order finite-differencing and quadruple (128-bit or \(\sim \)30 decimal digits) or octal (256-bit or \(\sim \)60 decimal digits) floating-point precision. Given the computational demands of this high-order and high-precision solver, in addition to the rather long evolutions required for these studies, we accelerate the solver using a many-core Nvidia graphics-processing-unit and obtain an order-of-magnitude speed-up over a high-end multi-core processor. We thus demonstrate a practical solution for demanding problems that utilize high-precision numerics today.

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Notes

  1. Perhaps the most significant and tragic event that occurred due a limited numerical precision issue in recent history was the failure of the Patriot Missle during the Persian Gulf War in the early 1990s. This was attributed to the limited precision capabilities of the control computer that was unable to operate accurately over long periods of time due to the accumulation of round-off error [3].

  2. A pseudo-spectral approach is better suited to address the challenges mentioned in this context; however, our higher-order finite-difference implementation requires relatively modest changes to our original second-order code, therefore we simply proceed with that approach. It is worth pointing out that other work in the same context using a pseudo-spectral approach posed similar precision issues [22], and thus made use of quadruple precision numerics on a CPU.

  3. The Cell BE port of the QD library performed quite well. However, the GPU implementation performed poorly, simply because we did not make any GPU-specific optimizations at the time and worked with much older, relatively low-performing hardware.

  4. It is worth pointing out that (surprisingly) the performance of the Intel C++ compiler supported long double datatype is much lower than that of double-double (DD) from the QD library.

  5. Note a slight variation to this trend in the CPU-DD plot. This is more clearly seen in Fig. 4.

References

  1. Bailey, D.H.: High-precision arithmetic in scientific computation. Comput. Sci. Eng. 7, 54 (2005)

    Article  Google Scholar 

  2. Bailey, D.H., Barrio, R., Borwein, J.M.: High-precision computation: Mathematical physics and dynamics. Appl. Math. Comput. 218, 10106 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Higham, N..: Accuracy and Stability of Numerical Analysis, pp 506–507. SIAM, (1996).

  4. Valdettaro, L., Rieutord, M., Braconnier, T., Fraysse, V.: Convergence and round-off errors in a two-dimensional eigenvalue problem using spectral methods and ArnoldiChebyshev algorithm. J. Comput. Appl. Math. 205, 382 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Sarra, S.: Radial basis function approximation methods with extended precision floating point arithmetic. Eng. Anal. Boundary Elements 35, 68 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gottieb, S., Fischer, Paul F.: Modified Conjugate Gradient Method for the Solution of \(Ax = b\), J. Sci. Comp. 13, 173 (1998)

    Article  Google Scholar 

  7. Nvidia’s CUDA, http://www.nvidia.com/cuda/. Accessed 13 Aug 2012

  8. Burko, L., Khanna, G.: Radiative falloff in the background of rotating black hole. Phys. Rev. D 67, 081502 (2003)

    Article  MathSciNet  Google Scholar 

  9. Burko, L., Khanna, G.: Universality of massive scalar field late-time tails in black-hole spacetimes. Phys. Rev. D 70, 044018 (2004)

    Article  MathSciNet  Google Scholar 

  10. Burko, L., Khanna, G.: Late-time Kerr tails revisited. Class. Quant. Grav. 26, 015014 (2009)

    Article  MathSciNet  Google Scholar 

  11. Burko, L., Khanna, G.: Late-time Kerr tails: generic and non-generic initial data sets, “up” modes, and superposition. Class. Quant. Grav. 28, 025012 (2011)

    Article  MathSciNet  Google Scholar 

  12. Zenginoglu, A., Khanna, G., and Burko, L., Mode coupling and intermediate behavior of Kerr tails, Class. Quant. Grav., 2012.

  13. Baumgarte, T., Shapiro, S.: Numerical Relativity. Cambridge University Press,   (2010)

  14. Alcubierre, M.: Introduction to 3+1 Numerical Relativity. Oxford University Press,   (2008)

  15. Cactus Code, http://www.cactuscode.org/. Accessed 13 Aug 2012

  16. Gundlach, C.: Critical Phenomena in Gravitational Collapse. Living Rev. Relat. 2, 4 (1999)

    MathSciNet  Google Scholar 

  17. Amaro-Seoane, P., et al.: Intermediate and Extreme Mass-Ratio Inspirals - Astrophysics, Science Applications and Detection using LISA. Class. Quant. Grav. 24, R113 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Teukolsky, S.: Perturbations of a rotating black hole. Astrophys. J. 185, 635 (1973)

    Article  MathSciNet  Google Scholar 

  19. Lax, P., Richtmyer, R.: Survey of the stability of linear finite difference equations. Comm. Pure Appl. Math. 9, 267 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  20. Krivan, W., Laguna, P., Papadopoulos, P., Andersson, N.: Dynamics of perturbations of rotating black holes. Phys. Rev. D 56, 3395 (1997)

    Article  MathSciNet  Google Scholar 

  21. Price, R.: Nonspherical Perturbations of Relativistic Gravitational Collapse. Phys. Rev. D 5, 2419 (1972)

    Article  MathSciNet  Google Scholar 

  22. Tiglio, M., Kidder, L., Teukolsky, S.: High accuracy simulations of Kerr tails: coordinate dependence and higher multipoles. Class. Quant. Grav. 25, 105022 (2008)

    Article  Google Scholar 

  23. Khanna, G., McKennon, J.: Numerical modeling of gravitational wave sources accelerated by OpenCL. Comput. Phys. Commun. 181, 1605 (2010)

    Article  MATH  Google Scholar 

  24. McKennon, J., Forrester, G. and Khanna, G.,: High accuracy gravitational waveforms from black hole binary inspirals using OpenCL. Proceedings of the NSF XSEDE12 Conference, Chicago IL, 2012.

  25. LBNL QD Library, http://crd.lbl.gov/~dhbailey/mpdist/. Accessed 13 Aug 2012

  26. GQD Library, http://code.google.com/p/gpuprec/. Accessed 13 Aug 2012

  27. Ginjupalli, R., Khanna, G.: High-precision numerical simulations of rotating black holes accelerated by CUDA, Proceedings of the International Conference on High Performance Computing Systems (HPCS). Orlando, FL 2010.

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  1. Physics Department, University of Massachusetts Dartmouth, 285 Old Westport Rd., North Dartmouth, MA, 02747, USA

    Gaurav Khanna

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Khanna, G. High-Precision Numerical Simulations on a CUDA GPU: Kerr Black Hole Tails. J Sci Comput 56, 366–380 (2013). https://doi.org/10.1007/s10915-012-9679-3

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