improving parameter estimation efficiency for gravitational wave data analysis

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



Gravitational WavesParameter EstimationNested SamplingMotivation

General RelativityGravity Curvature





Gravitational WavesRipples in Spacetime

Inspiral Merger Ringdown


inspiral.mpegMedia File (video/mpeg)




Parameter EstimationExtracting the Physics

I Chirp Mass

I Mc = (m1m2)3/5

m1 +m2I TimeI Sky PositionI DistanceI 2 Orientation AnglesI 6 Spin Components





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


nestedsampling.mpegMedia File (video/mpeg)




Motivation

I Problem:Nested Sampling requiresLOTS of time

I Solution:Speed up the Algorithm

I ParallelizationI Variable Resolution

One month later...




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





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.





Parallelization ResultsChirp Mass Cumulative Distributions for 1024 Total Live Points

Nlive from 1024 to 16 Nlive from 256 to 64





Parallelization ResultsAccuracy and Efficiency

Samples vs. Nlive Computational Time vs. Nlive





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




Sampling TheoremMethodResultsConclusions

Switching GearsTime and Frequency Domains of Inspiral Signals

Fourier Transform





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





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?





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





Variable Resolution MethodA Broken Waveform

4 Band Broken Waveform





Variable Resolution MethodThe Composed Waveform

4 Band Composed Waveform





Variable Resolution ResultsAccuracy

Bands % Match1 99.99972 99.95413 99.75894 99.53515 99.3865





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





Variable Resolution Conclusions

I Variable Resolution is feasible for parameter estimation

I Over 99% accuracy retained with 50% reduction incomputational time




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




Summary and OutlookAcknowledgments and Questions


IntroductionGravitational WavesParameter EstimationNested SamplingMotivation

ParallelizationOverviewMethodResultsConclusions

Variable ResolutionSampling TheoremMethodResultsConclusions

SummarySummary and OutlookAcknowledgments and Questions

improving parameter estimation efficiency for gravitational wave data analysis

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