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Register a new userCARPool: fast, accurate computation of large-scale structure statistics by pairing costly and cheap cosmological simulations
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To exploit the power of next-generation large-scale structure surveys, ensembles of numerical simulations are necessary to give accurate theoretical predictions of the statistics of observables. High-fidelity simulations come at a towering computational cost. Therefore, approximate but fast simulations, surrogates, are widely used to gain speed at the price of introducing model error. We propose a general method that exploits the correlation between simulations and surrogates to compute fast, reduced-variance statistics of large-scale structure observables without model error at the cost of only a few simulations. We call this approach Convergence Acceleration by Regression and Pooling (CARPool). In numerical experiments with intentionally minimal tuning, we apply CARPool to a handful of GADGET-III $N$-body simulations paired with surrogates computed using COmoving Lagrangian Acceleration (COLA). We find $sim 100$-fold variance reduction even in the non-linear regime, up to $k_mathrm{max} approx 1.2$ $h {rm Mpc^{-1}}$ for the matter power spectrum. CARPool realises similar improvements for the matter bispectrum. In the nearly linear regime CARPool attains far larger sample variance reductions. By comparing to the 15,000 simulations from the Quijote suite, we verify that the CARPool estimates are unbiased, as guaranteed by construction, even though the surrogate misses the simulation truth by up to $60%$ at high $k$. Furthermore, even with a fully configuration-space statistic like the non-linear matter density probability density function, CARPool achieves unbiased variance reduction factors of up to $sim 10$, without any further tuning. Conversely, CARPool can be used to remove model error from ensembles of fast surrogates by combining them with a few high-accuracy simulations.
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The covariance matrix $boldsymbol{Sigma}$ of non-linear clustering statistics that are measured in current and upcoming surveys is of fundamental interest for comparing cosmological theory and data and a crucial ingredient for the likelihood approximations underlying widely used parameter inference and forecasting methods. The extreme number of simulations needed to estimate $boldsymbol{Sigma}$ to sufficient accuracy poses a severe challenge. Approximating $boldsymbol{Sigma}$ using inexpensive but biased surrogates introduces model error with respect to full simulations, especially in the non-linear regime of structure growth. To address this problem we develop a matrix generalization of Convergence Acceleration by Regression and Pooling (CARPool) to combine a small number of simulations with fast surrogates and obtain low-noise estimates of $boldsymbol{Sigma}$ that are unbiased by construction. Our numerical examples use CARPool to combine GADGET-III $N$-body simulations with fast surrogates computed using COmoving Lagrangian Acceleration (COLA). Even at the challenging redshift $z=0.5$, we find variance reductions of at least $mathcal{O}(10^1)$ and up to $mathcal{O}(10^4)$ for the elements of the matter power spectrum covariance matrix on scales $8.9times 10^{-3}<k_mathrm{max} <1.0$ $h {rm Mpc^{-1}}$. We demonstrate comparable performance for the covariance of the matter bispectrum, the matter correlation function and probability density function of the matter density field. We compare eigenvalues, likelihoods, and Fisher matrices computed using the CARPool covariance estimate with the standard sample covariance estimators and generally find considerable improvement except in cases where $Sigma$ is severely ill-conditioned.
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