Swarm Intelligence Methods for Extreme Mass Ratio Inspiral Search: First Application of Particle Swarm Optimization
Abstract
:1. Introduction
2. Data Description
2.1. TDI Combinations
2.2. Noise Model and Signal-to-Noise Ratio
2.3. Signal Model: EMRI Waveform
3. Generalized Likelihood Ratio Test
3.1. LLR
3.2. The Ten-Dimensional Search
4. Particle Swarm Optimization
5. Results
- As noted above, one expects a successful PSO search to find a 10-dimensional fitness value that is larger than the one at the true location. We show the corresponding estimated SNR from the successful run in bold.
- The parameter estimation errors listed in the table are evaluated based on the best-fit locations of the successful PSO search. For the parameters , M, , , , and , we show their estimation errors, defined as the difference between the true and best-fit values, relative to their respective CRLB errors (evaluated at the true location), while the error is shown relative to its true value for D. For the parameters , , , and , we simply show the error itself.
- Having obtained the 10-dimensional best-fit location of the successful PSO search, which uses templates restricted to the 10 loudest harmonics, we rerun the 3-dimensional local maximization at this location using all 25 harmonics to estimate the three initial angles, , , . This is done to reveal the influence of the weak harmonics beyond the loudest 10 on the initial angles. The estimated initial angles are then used in the estimation of the distance D using Equation (27).
- With the 14-dimensional recovered parameters in hand, the estimated TDI A and E signals can be obtained by rerunning Equations (2) and (1). The overlap between the estimated A and E signals and the corresponding true signals, computed separately and in combination as defined in Equation (15), are reported as the quantities , , and .
6. Discussion
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
AK | Analytical Kludge |
CO | Compact Object |
CRLB | Cramer–Rao Lower Bound |
DFT | Discrete Fourier Transform |
EMRI | Extreme Mass Ratio Inspiral |
FIM | Fisher Information Matrix |
GW | Gravitational Wave |
GLRT | Generalized Likelihood Ratio Test |
GPU | Graphics Processing Unit |
LDC | LISA Data Challenge |
LLR | Log-Likelihood Ratio |
LISA | Laser Interferometer Space Antenna |
MLE | Maximum Likelihood Estimation |
MCMC | Markov Chain Monte Carlo |
MLDC | Mock LISA Data Challenge |
MBH | Massive Black Hole |
MPI | Message Passing Interface |
ODE | Ordinary Differential Equation |
PSD | Power Spectral Density |
PSO | Particle Swarm Optimization |
SNR | Signal-to-Noise Ratio |
SSB | Solar System Barycenter |
SI | Swarm Intelligence |
TDI | Time Delay Interferometry |
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SNR Order (Descending) | LDC Parameters | ||||
---|---|---|---|---|---|
1 | 7/7 | 7/7 | 7/7 | 7/7 | 17/12 |
2 | 8/8 | 8/8 | 8/8 | 12/12 | 22/17 |
3 | 12/12 | 12/12 | 12/12 | 8/8 | 18/22 |
SNR fraction | 0.849/0.887 | 0.826/0.876 | 0.906/0.904 | 0.702/0.736 | 0.673/0.746 |
4 | 13/13 | 13/13 | 6/6 | 13/13 | 23/7 |
5 | 6/6 | 17/6 | 9/9 | 17/17 | 12/2 |
6 | 9/9 | 6/9 | 13/13 | 18/2 | 13/12 |
7 | 17/17 | 9/17 | 10/17 | 22/18 | 7/18 |
8 | 18/11 | 18/11 | 17/10 | 23/22 | 16/23 |
9 | 11/14 | 11/14 | 11/14 | 6/6 | 21/8 |
10 | 14/2 | 14/18 | 14/11 | 11/3 | 8/3 |
SNR fraction | 0.985/0.987 | 0.981/0.985 | 0.992/0.991 | 0.945/0.943 | 0.945/0.943 |
Parameters | Values | Search Range |
---|---|---|
29.490000 | 10 | |
1.1349449 | 10 | |
2.1422000 | 10 | |
0.96970000 | 10 | |
0.22865665 | 10 | |
7.3804631 | 200 | |
0.4989445 | ||
2.232797 | ||
1.522100 | ||
3.946698 |
SNR 50 | SNR 40 | SNR 30 | |
---|---|---|---|
Square root of fitness values | |||
True 13-dimensional location | 48.737520 | 38.994759 | 29.251997 |
True 10-dimensional location | 48.794305 | 39.02358 | 29.260434 |
mBest location from PSO | 47.468231 39.266858 48.888190 | 39.176120 | 24.556344 29.525760 23.467734 |
Parameter estimation errors | |||
4.872139 | 1.048 6.090174 | 0.311 8.120229 | |
0.875 3.582834 | 4.478545 | 5.971391 | |
9.471417 | 1.349 1.183927 | 0.368 1.578570 | |
3.153740 | 1.334 3.942175 | 0.363 5.256233 | |
1.534 1.842612 | 0.604 2.303266 | 3.071021 | |
0.117 3.202842 | 4.003554 | 5.338071 | |
0.059 | 0.045 | 0.065 | |
0.037 | 0.093 | 0.137 | |
0.004 | 0.983 | ||
0.044 | 3.426 | ||
Overlap between the estimated and true signals | |||
0.983481 | |||
0.968413 | |||
0.977902 |
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Zou, X.-B.; Mohanty, S.D.; Luo, H.-G.; Liu, Y.-X. Swarm Intelligence Methods for Extreme Mass Ratio Inspiral Search: First Application of Particle Swarm Optimization. Universe 2024, 10, 96. https://doi.org/10.3390/universe10020096
Zou X-B, Mohanty SD, Luo H-G, Liu Y-X. Swarm Intelligence Methods for Extreme Mass Ratio Inspiral Search: First Application of Particle Swarm Optimization. Universe. 2024; 10(2):96. https://doi.org/10.3390/universe10020096
Chicago/Turabian StyleZou, Xiao-Bo, Soumya D. Mohanty, Hong-Gang Luo, and Yu-Xiao Liu. 2024. "Swarm Intelligence Methods for Extreme Mass Ratio Inspiral Search: First Application of Particle Swarm Optimization" Universe 10, no. 2: 96. https://doi.org/10.3390/universe10020096
APA StyleZou, X. -B., Mohanty, S. D., Luo, H. -G., & Liu, Y. -X. (2024). Swarm Intelligence Methods for Extreme Mass Ratio Inspiral Search: First Application of Particle Swarm Optimization. Universe, 10(2), 96. https://doi.org/10.3390/universe10020096