  The likelihood function for cosmological parameters, given by e.g. weak
lensing shear measurements, depends on contributions to the covariance induced
by the nonlinear evolution of the cosmic web. As nonlinear clustering to date
has only been described by numerical $N$-body simulations in a reliable and
sufficiently precise way, the necessary computational costs for estimating
those covariances at different points in parameter space are tremendous. In
this work we describe the change of the matter covariance and of the weak
lensing covariance matrix as a function of cosmological parameters by
constructing a suitable basis, where we model the contribution to the
covariance from nonlinear structure formation using Eulerian perturbation
theory at third order. We show that our formalism is capable of dealing with
large matrices and reproduces expected degeneracies and scaling with
cosmological parameters in a reliable way. Comparing our analytical results to
numerical simulations we find that the method describes the variation of the
covariance matrix found in the SUNGLASS weak lensing simulation pipeline within
the errors at one-loop and tree-level for the spectrum and the trispectrum,
respectively, for multipoles up to $\ell\leq 1300$. We show that it is possible
to optimize the sampling of parameter space where numerical simulations should
be carried out by minimising interpolation errors and propose a corresponding
method to distribute points in parameter space in an economical way.
