Efficient Monte Carlo for high excursions of Gaussian random fields

TL;DR

This paper develops efficient Monte Carlo methods for estimating high-level excursion probabilities and conditional expectations of Gaussian random fields, reducing computational complexity from exponential to polynomial in the threshold level.

Contribution

It introduces Monte Carlo procedures with polynomial complexity for high excursions of Gaussian fields, improving over naive exponential-cost methods, and discusses tuning based on regularity properties.

Findings

Monte Carlo methods with polynomial complexity for tail probability estimation

Procedures adapt to regularity, enhancing efficiency

Applicable to high thresholds with controlled accuracy

Abstract

Our focus is on the design and analysis of efficient Monte Carlo methods for computing tail probabilities for the suprema of Gaussian random fields, along with conditional expectations of functionals of the fields given the existence of excursions above high levels, b. Na\"{i}ve Monte Carlo takes an exponential, in b, computational cost to estimate these probabilities and conditional expectations for a prescribed relative accuracy. In contrast, our Monte Carlo procedures achieve, at worst, polynomial complexity in b, assuming only that the mean and covariance functions are H\"{o}lder continuous. We also explain how to fine tune the construction of our procedures in the presence of additional regularity, such as homogeneity and smoothness, in order to further improve the efficiency.