Efficient simulation of Brown-Resnick processes based on variance reduction of Gaussian processes

TL;DR

This paper proposes a variance reduction method for simulating Brown-Resnick processes, improving accuracy and efficiency by optimizing the underlying Gaussian process, with theoretical extensions and practical validation.

Contribution

It extends Matheron's approach to minimize maximal variance for convex and non-convex variograms, enhancing simulation methods for Brown-Resnick processes.

Findings

Significant variance reduction in simulations

Improved accuracy over existing algorithms

Validated through comprehensive simulation studies

Abstract

Brown-Resnick processes are max-stable processes that are associated to Gaussian processes. Their simulation is often based on the corresponding spectral representation which is not unique. We study to what extent simulation accuracy and efficiency can be improved by minimizing the maximal variance of the underlying Gaussian process. Such a minimization is a difficult mathematical problem that also depends on the geometry of the simulation domain. We extend Matheron's (1974) seminal contribution in two aspects: (i) making his description of a minimal maximal variance explicit for convex variograms on symmetric domains and (ii) proving that the same strategy reduces the maximal variance also for a huge class of non-convex variograms representable through a Bernstein function. A simulation study confirms that our non-costly modification can lead to substantial improvements among Gaussian representations. We also compare it with three other established algorithms.