Extremes of vector-valued processes by finite dimensional models

TL;DR

This paper introduces finite dimensional models for vector-valued random processes that enable efficient Monte Carlo sampling and accurate estimation of extreme values and other functionals, supported by theoretical guarantees and numerical examples.

Contribution

It develops a framework for constructing finite dimensional models with provable properties for estimating extremes of vector-valued processes.

Findings

FD models enable accurate estimation of process extremes.

Monte Carlo sampling of FD models is computationally feasible.

Numerical examples demonstrate the models' effectiveness.

Abstract

Finite dimensional (FD) models, i.e., deterministic functions of time/space and finite sets of random variables, are constructed for target vector-valued random processes/fields. They are required to have two properties. First, standard Monte Carlo algorithms can be used to generate their samples, referred to as FD samples. Second, under some conditions specified by several theorems, FD samples can be used to estimate distributions of extremes and other functionals of target random functions. Numerical illustrations involving two-dimensional random processes and apparent properties of random microstructures are presented to illustrate the implementation of FD models for these stochastic problems and show that they are accurate if the conditions of our theorems are satisfied.