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The covariance matrix $\boldsymbol{\Sigma}$ of non-linear clusteringstatistics that are measured in current and upcoming surveys is of fundamentalinterest for comparing cosmological theory and data and a crucial ingredientfor the likelihood approximations underlying widely used parameter inferenceand 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 surrogatesintroduces model error with respect to full simulations, especially in thenon-linear regime of structure growth. To address this problem we develop amatrix generalization of Convergence Acceleration by Regression and Pooling(CARPool) to combine a small number of simulations with fast surrogates andobtain low-noise estimates of $\boldsymbol{\Sigma}$ that are unbiased byconstruction. Our numerical examples use CARPool to combine GADGET-III $N$-bodysimulations with fast surrogates computed using COmoving LagrangianAcceleration (COLA). Even at the challenging redshift $z=0.5$, we find variancereductions of at least $\mathcal{O}(10^1)$ and up to $\mathcal{O}(10^4)$ forthe elements of the matter power spectrum covariance matrix on scales$8.9\times 10^{-3}<k_\mathrm{max} <1.0$ $h {\rm Mpc^{-1}}$. We demonstratecomparable performance for the covariance of the matter bispectrum, the mattercorrelation function and probability density function of the matter densityfield. We compare eigenvalues, likelihoods, and Fisher matrices computed usingthe CARPool covariance estimate with the standard sample covariance estimatorsand generally find considerable improvement except in cases where $\Sigma$ isseverely ill-conditioned.

CARPool Covariance: Fast, unbiased covariance estimation for large-scale structure observables