1. Introduction
Parallelization
Variable Resolution
Summary
Improving Parameter Estimation Efficiency
for Gravitational Wave Data Analysis
J. M. Bell1 2 J. Veitch2
1Departments of Physics and Mathematics
Millsaps College
2Gravitational Physics
NIKHEF
University of Florida IREU in Gravitational Physics
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 1 / 22

2. Introduction
Parallelization
Variable Resolution
Summary
Gravitational Waves
Parameter Estimation
Nested Sampling
Motivation
General Relativity
Gravity ≡ Curvature
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 2 / 22

3. Introduction
Parallelization
Variable Resolution
Summary
Gravitational Waves
Parameter Estimation
Nested Sampling
Motivation
Gravitational Waves
Ripples in Spacetime
Inspiral Merger Ringdown
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 3 / 22

4. Introduction
Parallelization
Variable Resolution
Summary
Gravitational Waves
Parameter Estimation
Nested Sampling
Motivation
Parameter Estimation
Extracting the Physics
Chirp Mass
Mc =
(m1m2)3/5
m1 + m2
Time
Sky Position
Distance
2 Orientation Angles
6 Spin Components
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 4 / 22

5. Introduction
Parallelization
Variable Resolution
Summary
Gravitational Waves
Parameter Estimation
Nested Sampling
Motivation
Nested Sampling
A Numerical Parameter Estimation Algorithm
The Procedure
1 Begin with Nlive samples
2 Remove least likely sample
3 Add a more likely sample
4 Repeat until satisfied
The Algorithm In Action
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 5 / 22

6. Introduction
Parallelization
Variable Resolution
Summary
Gravitational Waves
Parameter Estimation
Nested Sampling
Motivation
Motivation
Problem:
Nested Sampling requires
LOTS of time
Solution:
Speed up the Algorithm
Parallelization
Variable Resolution
One month later...
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 6 / 22

7. Introduction
Parallelization
Variable Resolution
Summary
Overview
Method
Results
Conclusions
Parallelization
Nested sampling converges on the maximum likelihood
more
Quickly for small Nlive
Accurately for large Nlive
Goals:
Reduce overall computational time
Maintain sufficient accuracy
Determine optimal range of live points
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 7 / 22

8. Introduction
Parallelization
Variable Resolution
Summary
Overview
Method
Results
Conclusions
Parallelization
Method
1 Run multiple instances in parallel with different Nlive.
Instances 1 2 4 8 ... 64
Nlive 1024 512 256 128 ... 16
2 Draw samples from each instance by weighting and
resampling within each.
3 Merge parallel runs by weighting each run according to its
own result.
4 Draw samples from all runs by drawing from the weighted
parallel samples.
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 8 / 22

9. Introduction
Parallelization
Variable Resolution
Summary
Overview
Method
Results
Conclusions
Parallelization Results
Chirp Mass Cumulative Distributions for 1024 Total Live Points
Nlive from 1024 to 16 Nlive from 256 to 64
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 9 / 22

10. Introduction
Parallelization
Variable Resolution
Summary
Overview
Method
Results
Conclusions
Parallelization Results
Accuracy and Efficiency
Samples vs. Nlive Computational Time vs. Nlive
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 10 / 22

11. Introduction
Parallelization
Variable Resolution
Summary
Overview
Method
Results
Conclusions
Parallelization
Conclusions
Reducing Nlive by 50% returns 75% of the samples
The optimal range for Nlive is 200 to 300
The total Nlive should be ≥ 1000
Parallelization effectively reduces computational time
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 11 / 22

12. Introduction
Parallelization
Variable Resolution
Summary
Sampling Theorem
Method
Results
Conclusions
Switching Gears
Time and Frequency Domains of Inspiral Signals
Fourier Transform ⇒
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 12 / 22

13. Introduction
Parallelization
Variable Resolution
Summary
Sampling Theorem
Method
Results
Conclusions
Waveform Reconstruction
The Sampling Theorem
A frequency domain waveform
containing no amplitudes
greater than F is completely
determined by giving its
ordinates at a series of
abscissas spaced
1
2F = f Nyquist Hz apart.
f Sampling > f Nyquist
Redundant and Slow
f Sampling < f Nyquist
Inaccurate but Fast
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 13 / 22

14. Introduction
Parallelization
Variable Resolution
Summary
Sampling Theorem
Method
Results
Conclusions
Variable Resolution
We can rebuild him...
It’s possible to reconstruct a frequency domain waveform
Optimum accuracy/efficiency is obtained by sampling at
f nyq(τ(fgw )) =
1
2
1
τ(fgw )
But why sample a decreasing function at a constant rate?
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 14 / 22

15. Introduction
Parallelization
Variable Resolution
Summary
Sampling Theorem
Method
Results
Conclusions
Variable Resolution Algorithm
1 Choose a number of frequency domain breaks, M
2 Determine their location by minimizing the number of
samples required to reconstruct the waveform
N(f1, f2, ..., fM) =
f1 − fmin
fnyq(fmin)
+
f2 − f1
fnyq(f1)
+ ... +
fmax − fM
fnyq(fM)
where fmin < f1 < f2 < ... < fM < fmax
3 Sample at the Nyquist frequency in each band
4 Compose separate bands to generate waveform
5 Test match against single band case using interpolation
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 15 / 22

16. Introduction
Parallelization
Variable Resolution
Summary
Sampling Theorem
Method
Results
Conclusions
Variable Resolution Method
A Broken Waveform
4 Band Broken Waveform
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 16 / 22

17. Introduction
Parallelization
Variable Resolution
Summary
Sampling Theorem
Method
Results
Conclusions
Variable Resolution Method
The Composed Waveform
4 Band Composed Waveform
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 17 / 22

18. Introduction
Parallelization
Variable Resolution
Summary
Sampling Theorem
Method
Results
Conclusions
Variable Resolution Results
Accuracy
Bands % Match
1 99.9997
2 99.9541
3 99.7589
4 99.5351
5 99.3865
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 18 / 22

19. Introduction
Parallelization
Variable Resolution
Summary
Sampling Theorem
Method
Results
Conclusions
Variable Resolution Results
Efficiency
Bands d+hh:mm:ss
1 ≈ 4+09:00:00
2 3+16:28:54
3 2+23:22:12
4 2+18:19:39
5 2+17:30:19
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 19 / 22

20. Introduction
Parallelization
Variable Resolution
Summary
Sampling Theorem
Method
Results
Conclusions
Variable Resolution Conclusions
Variable Resolution is feasible for parameter estimation
Over 99% accuracy retained with 50% reduction in
computational time
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 20 / 22

21. Introduction
Parallelization
Variable Resolution
Summary
Summary and Outlook
Acknowledgments and Questions
Summary and Outlook
Parallelization and Variable Resolution are viable means of
reducing computational time
What lies ahead?
Optimization of the multiband algorithm
Simultaneous testing of both approaches
Implementation in the time domain
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 21 / 22

22. Introduction
Parallelization
Variable Resolution
Summary
Summary and Outlook
Acknowledgments and Questions
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 22 / 22