Towards beating the curse of dimensionality for gravitational waves using reduced basis

Using the reduced basis approach, we efficiently compress and accurately represent the space of waveforms for nonprecessing binary black hole inspirals, which constitutes a four-dimensional parameter space (two masses, two spin magnitudes). Compared to the nonspinning case, we find that only a marginal increase in the (already relatively small) number of reduced basis elements is required to represent any nonprecessing waveform to nearly numerical round-off precision. Most parameters selected by the algorithm are near the boundary of the parameter space, leaving the bulk of its volume sparse. Our results suggest that the full eight-dimensional space (two masses, two spin magnitudes, four spin orientation angles on the unit sphere) may be highly compressible and represented with very high accuracy by a remarkably small number of waveforms, thus providing some hope that the number of numerical relativity simulations of binary black hole coalescences needed to represent the entire space of configurations is not intractable. Finally, we find that the distribution of selected parameters is robust to different choices of seed values starting the algorithm, a property which should be useful for indicating parameters for numerical relativity simulations of binary black holes. In particular, we find that the mass ratios $m_1/m_2$ of nonspinning binaries selected by the algorithm are mostly in the interval $[1,3]$ and that the median of the distribution follows a power-law behavior $\sim (m_1/m_2{)}^{-5.25}$.

Figure 1

The reduced basis representation error as a function of the number of reduced basis for different dimensionalities. The greedy error shows the rapid exponential convergence to round-off. The number of reduced basis elements increases very mildly from the 2D ($\approx 1725\mathrm{RB}$) to the 3D ($\approx 1824\mathrm{RB}$) case and to the 4D ($\approx 1839\mathrm{RB}$) case. This shows that the increasing dimensionality may not be a major obstacle for the reduced basis approach.

Figure 2

Histogram of the selected parameter choices for the mass ratio $m_1/m_2$ in the 4D case. Note the large spike near equal masses and the rapid falloff to small counts for increasing disparity in the component masses.

Figure 3

The reduced basis parameter choices in the $m_1–m_2$ plane. The overall structure is similar for all three (2D, 3D, 4D) nonprecessing cases. The gray scale indicates the number of selected points in a bin on a logarithmic scaling.

Figure 4

The reduced basis parameter choices for the 3D case $(m_1,m_2,\chi )$. Comparing to Fig. 3 one can also see the selection of primarily low-mass systems. For the lowest mass systems a large number of spin values are selected. Few systems from the bulk volume of the parameter space are chosen.

Figure 5

Figure 5a shows a histogram of spin values in the 3D case. Spins antialigned with the orbital angular momentum are predominantly chosen. Figure 5b shows a two-dimensional histogram of spin values in the 4D case. Similarly to Fig. 5a, antialigned systems ${\chi }_i=-1$ are predominantly chosen. The gray scale indicates the number of selected points in a bin on a logarithmic scaling.

Figure 6

Monte Carlo reconstruction error study for the 4D case. The histogram shows the distribution of the representation error $\sigma =\Vert \tilde{h}-P_N[\tilde{h}]{\Vert }^2$ for ${10}^7$ randomly selected parameter values. Notice the logarithmic scale on the vertical axis.

Figure 7

Parameters chosen by the greedy algorithm ($m_1$ left and $m_1/m_2$ right) when using 200 seed values corresponding to equal mass, nonspinning binaries with different total masses. The lines are quantile curves representing the fraction of selected parameters with values greater than the value of the quantile. Black corresponds to the median and the bounding black curves correspond to the 0% (top) and 100% (bottom) quantiles. The red line is a fit to the median. Despite the different choice of greedy parameters that are selected, the distribution of points is rather robust to the choice of seed.