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We consider the random field M(t)=\sup_{n\geq 1}\big\{-\logA_{n}+X_{n}(t)\big\}\,,\qquad t\in T\, for a set $T\subset \mathbb{R}^{m}$,where $(X_{n})$ is an iid sequence of centered Gaussian random fields on $T$and $0<A_{1}<A_{2}<\cdots $ are the arrivals of a general renewal process on$(0,\infty )$, independent of $(X_{n})$. In particular, a large class ofmax-stable random fields with Gumbel marginals have such a representation.Assume that one needs $c\left( d\right) =c(\{t_{1},\ldots,t_{d}\})$ functionevaluations to sample $X_{n}$ at $d$ locations $t_{1},\ldots ,t_{d}\in T$. Weprovide an algorithm which, for any $\epsilon >0$, samples $M(t_{1}),\ldots,M(t_{d})$ with complexity $o(c(d)\,d^{\epsilon })$. Moreover, if $X_{n}$ hasan a.s. converging series representation, then $M$ can be a.s. approximatedwith error $\delta $ uniformly over $T$ and with complexity $O(1/(\delta \log(1/\delta ))^{1/\alpha })$, where $\alpha $ relates to the H\"{o}ldercontinuity exponent of the process $X_{n}$ (so, if $X_{n}$ is Brownian motion,$\alpha =1/2$).

On Optimal Exact Simulation of Max-Stable and Related Random Fields