A reduced basis representation for chirp and ringdown gravitational wave templates

Transcription

1 A reduced basis representation for chirp and ringdown gravitational wave templates 1 Sarah Caudill 2 Chad Galley 3 Frank Herrmann 1 Jan Hesthaven 4 Evan Ochsner 5 Manuel Tiglio 1 1 University of Maryland, Department of Physics 2 California Institute of Technology, Jet Propulsion Laboratory 3 Brown University, Division of Applied Mathematics 4 University of Wisconsin Milwaukee, Center for Gravitation and Cosmology 5 Louisiana State University, Department of Physics and Astronomy PRL 106, (2011) September 16, 2011

2 Outline Greedy construction of a reduced basis catalog

3 The reduced basis space Outline Greedy construction of a reduced basis catalog

4 The reduced basis space Matched filtering: gravitational waveform templates from a catalog are correlated with detector data Templates chosen so that few signals are missed Minimal match (MM) characterizes the event loss for a given catalog Many templates needed, scales like (1 MM) (Parameter dimension)/2 Desire for... Faster and cheaper searches, generate alerts for EM counterparts (LLOID 1 ), use higher dimensional waveforms, reduced burden of overlap computation for parameter estimation 1 K. Cannon et al., Toward Early-Warning Detection of Gravitational Waves from Compact Binary Coalescence, arxiv: v1.

5 The reduced basis space Reduced basis method Highlights include... Generation of an accurate and compact reduced basis space of waveforms Significantly fewer templates (basis) for a given MM Selection of nearly optimal parameter points Non linear space of waveforms can be represented as a small linear space with arbitrarily high accuracy

6 The reduced basis space The reduced basis space and an algorithm to find it

7 The reduced basis space Problem statement Given: P parameters µ = {µ 1,...,µ P } and the space of all normalized waveforms H Chirp, ringdown, reduced models, analytic waveforms, etc. Each waveform is denoted h µ H GOAL: Find an N dimensional linear space W N to accurately represent H Ansatz: Basis vectors of WN are waveforms chosen from H W N is called the reduced basis space

8 The reduced basis space When should we look for a reduced basis space? Projection operator P N : H W N for an orthonormal basis e i of W N P N h N < e i,h > e i < g,f > i=1 b a gf S n (f) df Remark: P N h is the best approximation to h Kolmogorov N width specifies error for an optimal W N d N (H) = min W N max µ H P Nh µ h µ If solutions have smooth dependence on parameters one expects d N (H) Ae bn

9 The reduced basis space A practical approach to finding W N Finding the best space is computationally challenging, so Sample H at a finite set of (training space) points T Now we seek to approximate HT = {h µ H : µ T } by W N 2. Build W N by solving N easy problems (Greedy approach) Suppose we have Wi. The algorithm optimally chooses W i+1 and continues to W N Sequence of hierarchical spaces are constructed W1 W 2... W N W N nearly optimal 2 compared to WN Kol. If N-width decays exponentially so does the approximation error for W N. 2 P. Binev et al., Convergence rates for greedy algorithms in reduced basis methods.

10 The reduced basis space (setup) Choose a parameter and waveform space (continuous and discrete) Initialize reduced basis with choice of µ 1 and thus W 1 = span({h 1 }) To go from W i to W i+1...

11 The reduced basis space The greedy error ε i max µ T h µ P i h µ

12 The reduced basis space The greedy error ε i max µ T h µ P i h µ While ε i Tol i i +1

13 The reduced basis space The greedy error ε i max µ T h µ P i h µ While ε i Tol i i For all µ T compute h µ P i h µ

14 The reduced basis space The greedy error ε i max µ T h µ P i h µ While ε i Tol i i For all µ T compute h µ P i h µ 2. Find the parameter µ i+1 which maximizes the error of step 1

15 The reduced basis space The greedy error ε i max µ T h µ P i h µ While ε i Tol i i For all µ T compute h µ P i h µ 2. Find the parameter µ i+1 which maximizes the error of step 1 3. Let h i+1 = h µi+1 and W i+1 = span({h 1,...,h i+1 })

16 The reduced basis space The greedy error ε i max µ T h µ P i h µ While ε i Tol i i For all µ T compute h µ P i h µ 2. Find the parameter µ i+1 which maximizes the error of step 1 3. Let h i+1 = h µi+1 and W i+1 = span({h 1,...,h i+1 }) W N approximates H T with an error of better than Tol Outputs a collection of points { µ i } N i=1 and corresponding waveforms {h i } N i=1 such that W N = span ( {h i } N ) i=1

17 The reduced basis space Greedy construction

18 The reduced basis space Remarks Straightforward to implement Constructing W N is O(N) Resulting reduced basis is an application specific spectral basis Simplest matched filtering search between the signal s and every member of the training space < s,p N h j >= N s,e i e i,h j i=1 Since e i,h j has been precomputed offline, filtering now involves significantly fewer integrals

19 Outline Greedy construction of a reduced basis catalog 2PN chirp waveforms Ringdown waveforms Greedy construction of a reduced basis catalog

20 2PN chirp waveforms Ringdown waveforms Chirp Waveforms 2PN stationary-phase-approximation chirp waveforms 2 dimensional parameter space described by the compact objects masses, m 1 and m 2 Recall total mass M = m 1 +m 2 and symmetric mass ratio η = m 1 m 2 /M 2 Dense training space where points are uniformly spaced in m 1 and m 2

21 2PN chirp waveforms Ringdown waveforms for [1-3]M with Initial LIGO Figure: Expected exponential convergence of algorithm. Same generic feature for all mass ranges considered.

22 ) Greedy construction of a reduced basis catalog 2PN chirp waveforms Ringdown waveforms Distribution of selected parameters [1-3]M with Initial LIGO (ChirpM = η 3/5 M) Chirp mass (M Symmetric mass ratio Figure: Reduced basis with MM = Figure: Metric placement with MM =.97

23 Greedy construction of a reduced basis catalog 2PN chirp waveforms Ringdown waveforms Table: Number of reduced basis (N RB ) and metric placed (N metric ) waveforms for different minimal matches MM. Detector 1 MM BBH BNS N RB N metric N RB N metric

24 Greedy construction of a reduced basis catalog 2PN chirp waveforms Ringdown waveforms Table: Number of reduced basis (N RB ) and metric placed (N metric ) waveforms for different minimal matches MM. Detector AdvLIGO AdvVirgo 1 MM BBH BNS N RB N metric N RB N metric ,058 19,336 5,395 72, , , , , ,395 42,496 7, , , , , ,

25 2PN chirp waveforms Ringdown waveforms for [1-3]M with Initial LIGO Asymptotic result dim(w N ) = 921. Confirms expectation that we are converging to finite dimensional space for fixed error tolerance.

26 2PN chirp waveforms Ringdown waveforms Ringdown Waveforms Superposition of quasi-normal (i.e. exponentially damped) modes Dimension of parameter space is... 2 for dominate l = m = 2 one-mode waveform: central frequency f22 and quality factor Q 22 5 for two-mode waveform: central frequencies f22 and f 33, quality factors Q 22 and Q 33, and relative amplitude A 3 for two-mode waveform constrained by general relativity as f 33 (f 22,Q 22 ) and Q 33 (f 22,Q 22 ) Training space given by metric placement algorithm and MM =.99

27 2PN chirp waveforms Ringdown waveforms Single mode: convergence and initialization Figure: Greedy error as a function of the number of reduced basis waveforms for all possible initializations. The dark line shows the average and the shaded area the maximum dispersion around it. No fine tuning is needed!

28 2PN chirp waveforms Ringdown waveforms Constrained two-mode: Selected parameters Waveform given by h = C[(1 A)h 220 +Ah 330 ] Figure: The metric-based training space (points) and parameter values selected by the greedy algorithm (bars), with (2,2,0) mode (red) and (3,3,0) mode (blue). Figure: Values for the relative amplitude parameter A selected by the greedy algorithm. 1, 000 samples were used for A [0,1].

29 2PN chirp waveforms Ringdown waveforms Constrained two-mode: Monte Carlo study of errors Figure: Error in representing any template (i.e. outside of the training space) using reduced basis waveforms. Axes defined by A [0,1] f 22 [10,4000]Hz Q 22 [2.1187,20]

30 2PN chirp waveforms Ringdown waveforms 1 MM 2-mode, GR 2-mode N metric N RB N metric N RB , , , , , , ,590 Table: Number of reduced basis waveforms (N RB ) needed to represent 2-mode training spaces with (l,m,n) = (2,2,0) and (3,3,0) for different minimal matches MM. Values of N metric in the second column courtesy of V. Cardoso.

31 Outline Greedy construction of a reduced basis catalog Greedy construction of a reduced basis catalog

32 Greedy construction of a reduced basis catalog Demonstrated the space of waveforms can be accurately represented by a compact set of basis function Proposed an efficient algorithm for finding this space Applied the algorithm to chirp and multi-mode ringdown waveforms Selects most relevant parameter points Significant compressions, especially high dimensional ones like multi-mode ringdown Current work includes... Interpolation Waveforms with spin (anti-)aligned with orbital momentum

33 QUESTIONS?