  In this paper, we consider a classic problem concerning the high excursion
probabilities of a Gaussian random field $f$ living on a compact set $T$. We
develop efficient computational methods for the tail probabilities $P(\sup_T
f(t) > b)$ and the conditional expectations $E(\Gamma(f) | \sup_T f(t) > b)$ as
$b\rightarrow \infty$. For each $\varepsilon$ positive, we present Monte Carlo
algorithms that run in \emph{constant} time and compute the interesting
quantities with $\varepsilon$ relative error for arbitrarily large $b$. The
efficiency results are applicable to a large class of H\"older continuous
Gaussian random fields. Besides computations, the proposed change of measure
and its analysis techniques have several theoretical and practical indications
in the asymptotic analysis of extremes of Gaussian random fields.
