Post on 08-Jul-2015
IntroductionParallelization
Variable ResolutionSummary
Improving Parameter Estimation Efficiencyfor Gravitational Wave Data Analysis
J. M. Bell1 2 J. Veitch2
1Departments of Physics and MathematicsMillsaps College
2Gravitational PhysicsNIKHEF
University of Florida IREU in Gravitational Physics
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 1 / 22
IntroductionParallelization
Variable ResolutionSummary
Gravitational WavesParameter EstimationNested SamplingMotivation
General RelativityGravity Curvature
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 2 / 22
IntroductionParallelization
Variable ResolutionSummary
Gravitational WavesParameter EstimationNested SamplingMotivation
Gravitational WavesRipples in Spacetime
Inspiral Merger Ringdown
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 3 / 22
inspiral.mpegMedia File (video/mpeg)
IntroductionParallelization
Variable ResolutionSummary
Gravitational WavesParameter EstimationNested SamplingMotivation
Parameter EstimationExtracting the Physics
I Chirp Mass
I Mc = (m1m2)3/5
m1 +m2I TimeI Sky PositionI DistanceI 2 Orientation AnglesI 6 Spin Components
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 4 / 22
IntroductionParallelization
Variable ResolutionSummary
Gravitational WavesParameter EstimationNested SamplingMotivation
Nested SamplingA Numerical Parameter Estimation Algorithm
The Procedure
1 Begin with Nlive samples2 Remove least likely sample3 Add a more likely sample4 Repeat until satisfied
The Algorithm In Action
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 5 / 22
nestedsampling.mpegMedia File (video/mpeg)
IntroductionParallelization
Variable ResolutionSummary
Gravitational WavesParameter EstimationNested SamplingMotivation
Motivation
I Problem:Nested Sampling requiresLOTS of time
I Solution:Speed up the Algorithm
I ParallelizationI Variable Resolution
One month later...
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 6 / 22
IntroductionParallelization
Variable ResolutionSummary
OverviewMethodResultsConclusions
Parallelization
I Nested sampling converges on the maximum likelihoodmore
I Quickly for small NliveI Accurately for large Nlive
I Goals:I Reduce overall computational timeI Maintain sufficient accuracyI Determine optimal range of live points
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 7 / 22
IntroductionParallelization
Variable ResolutionSummary
OverviewMethodResultsConclusions
ParallelizationMethod
1 Run multiple instances in parallel with different Nlive.
Instances 1 2 4 8 ... 64Nlive 1024 512 256 128 ... 16
2 Draw samples from each instance by weighting andresampling within each.
3 Merge parallel runs by weighting each run according to itsown result.
4 Draw samples from all runs by drawing from the weightedparallel samples.
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 8 / 22
IntroductionParallelization
Variable ResolutionSummary
OverviewMethodResultsConclusions
Parallelization ResultsChirp 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
IntroductionParallelization
Variable ResolutionSummary
OverviewMethodResultsConclusions
Parallelization ResultsAccuracy and Efficiency
Samples vs. Nlive Computational Time vs. Nlive
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 10 / 22
IntroductionParallelization
Variable ResolutionSummary
OverviewMethodResultsConclusions
ParallelizationConclusions
I Reducing Nlive by 50% returns 75% of the samples
I The optimal range for Nlive is 200 to 300I The total Nlive should be 1000
I Parallelization effectively reduces computational time
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 11 / 22
IntroductionParallelization
Variable ResolutionSummary
Sampling TheoremMethodResultsConclusions
Switching GearsTime and Frequency Domains of Inspiral Signals
Fourier Transform
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 12 / 22
IntroductionParallelization
Variable ResolutionSummary
Sampling TheoremMethodResultsConclusions
Waveform Reconstruction
The Sampling Theorem
A frequency domain waveformcontaining no amplitudes
greater than F is completelydetermined by giving itsordinates at a series of
abscissas spaced1
2F = f Nyquist Hz apart.
I f Sampling > f NyquistI Redundant and Slow
I f Sampling < f NyquistI Inaccurate but Fast
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 13 / 22
IntroductionParallelization
Variable ResolutionSummary
Sampling TheoremMethodResultsConclusions
Variable ResolutionWe can rebuild him...
I Its possible to reconstruct a frequency domain waveform
I Optimum accuracy/efficiency is obtained by sampling at
f nyq((fgw )) =12
1(fgw )
I But why sample a decreasing function at a constant rate?
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 14 / 22
IntroductionParallelization
Variable ResolutionSummary
Sampling TheoremMethodResultsConclusions
Variable Resolution Algorithm
1 Choose a number of frequency domain breaks, M2 Determine their location by minimizing the number of
samples required to reconstruct the waveform
N(f1, f2, ..., fM) =f1 fminfnyq(fmin)
+f2 f1fnyq(f1)
+ ...+fmax fMfnyq(fM)
where fmin < f1 < f2 < ... < fM < fmax3 Sample at the Nyquist frequency in each band4 Compose separate bands to generate waveform5 Test match against single band case using interpolation
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 15 / 22
IntroductionParallelization
Variable ResolutionSummary
Sampling TheoremMethodResultsConclusions
Variable Resolution MethodA Broken Waveform
4 Band Broken Waveform
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 16 / 22
IntroductionParallelization
Variable ResolutionSummary
Sampling TheoremMethodResultsConclusions
Variable Resolution MethodThe Composed Waveform
4 Band Composed Waveform
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 17 / 22
IntroductionParallelization
Variable ResolutionSummary
Sampling TheoremMethodResultsConclusions
Variable Resolution ResultsAccuracy
Bands % Match1 99.99972 99.95413 99.75894 99.53515 99.3865
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 18 / 22
IntroductionParallelization
Variable ResolutionSummary
Sampling TheoremMethodResultsConclusions
Variable Resolution ResultsEfficiency
Bands d+hh:mm:ss1 4+09:00:002 3+16:28:543 2+23:22:124 2+18:19:395 2+17:30:19
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 19 / 22
IntroductionParallelization
Variable ResolutionSummary
Sampling TheoremMethodResultsConclusions
Variable Resolution Conclusions
I Variable Resolution is feasible for parameter estimation
I Over 99% accuracy retained with 50% reduction incomputational time
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 20 / 22
IntroductionParallelization
Variable ResolutionSummary
Summary and OutlookAcknowledgments and Questions
Summary and Outlook
I Parallelization and Variable Resolution are viable means ofreducing computational time
I What lies ahead?I Optimization of the multiband algorithmI Simultaneous testing of both approachesI Implementation in the time domain
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 21 / 22
IntroductionParallelization
Variable ResolutionSummary
Summary and OutlookAcknowledgments and Questions
J.M. Bell, J. Veitch Improving Efficiency of CBC Parameter Estimation 22 / 22
IntroductionGravitational WavesParameter EstimationNested SamplingMotivation
ParallelizationOverviewMethodResultsConclusions
Variable ResolutionSampling TheoremMethodResultsConclusions
SummarySummary and OutlookAcknowledgments and Questions