  The Karhunen-Lo\`{e}ve (KL) expansion is a popular method for approximating
random fields by transforming an infinite-dimensional stochastic domain into a
finite-dimensional parameter space. Its numerical approximation is of central
importance to the study of PDEs with random coefficients. In this work, we
analyze the approximation error of the Karhunen-Lo\`eve expansion for lognormal
random fields. We derive error estimates that allow the optimal balancing of
the truncation error of the expansion, the Quasi Monte-Carlo error for sampling
in the stochastic domain and the numerical approximation error in the physical
domain. The estimate is given in the number $M$ of terms maintained in the KL
expansion, in the number of sampling points $N$, and in the discretization mesh
size $h$ in the physical domain employed in the numerical solution of the
eigenvalue problems during the expansion. The result is used to quantify the
error in PDEs with random coefficients. We complete the theoretical analysis
with numerical experiments in one and multiple stochastic dimensions.
