Rare-event Simulation and Efficient Discretization for the Supremum of Gaussian Random Fields

TL;DR

This paper introduces efficient Monte Carlo algorithms for estimating high excursion probabilities and conditional expectations of Gaussian random fields, achieving constant-time complexity for large thresholds and applicable to a broad class of fields.

Contribution

It develops novel, efficient computational methods with constant-time Monte Carlo algorithms for tail probability estimation of Gaussian fields as thresholds grow large.

Findings

Algorithms run in constant time for large thresholds

Applicable to a broad class of H"older continuous Gaussian fields

Provides theoretical insights into the asymptotic analysis of Gaussian extremes

Abstract

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.