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Computing the inverse covariance matrix (or precision matrix) of large datavectors is crucial in weak lensing (and multi-probe) analyses of the largescale structure of the universe. Analytically computed covariances arenoise-free and hence straightforward to invert, however the modelapproximations might be insufficient for the statistical precision of futurecosmological data. Estimating covariances from numerical simulations improveson these approximations, but the sample covariance estimator is inherentlynoisy, which introduces uncertainties in the error bars on cosmologicalparameters and also additional scatter in their best fit values. For futuresurveys, reducing both effects to an acceptable level requires an unfeasiblylarge number of simulations.  In this paper we describe a way to expand the true precision matrix around acovariance model and show how to estimate the leading order terms of thisexpansion from simulations. This is especially powerful if the covariancematrix is the sum of two contributions, $\smash{\mathbf{C} =\mathbf{A}+\mathbf{B}}$, where $\smash{\mathbf{A}}$ is well understoodanalytically and can be turned off in simulations (e.g. shape-noise for cosmicshear) to yield a direct estimate of $\smash{\mathbf{B}}$. We test our methodin mock experiments resembling tomographic weak lensing data vectors from theDark Energy Survey (DES) and the Large Synoptic Survey Telecope (LSST). For DESwe find that $400$ N-body simulations are sufficient to achive negligiblestatistical uncertainties on parameter constraints. For LSST this is achievedwith $2400$ simulations. The standard covariance estimator would require>$10^5$ simulations to reach a similar precision. We extend our analysis to aDES multi-probe case finding a similar performance.

Precision matrix expansion - efficient use of numerical simulations in estimating errors on cosmological parameters