# Uncertainty quantification using periodic random variables

## TL;DR

This paper introduces a novel periodic random variable model for uncertainty quantification in PDEs, achieving higher convergence rates with lattice cubature rules, and compares it to traditional affine models.

## Contribution

The paper proposes a periodic random field model with the same mean and covariance as the affine model, improving convergence in uncertainty quantification.

## Key findings

- Higher order cubature convergence rate of O(n^{-1/p})
- Periodic model performs comparably or better than affine model
- Numerical examples validate the theoretical convergence improvements

## Abstract

Many studies in uncertainty quantification have been carried out under the assumption of an input random field in which a countable number of independent random variables are each uniformly distributed on an interval, with these random variables entering linearly in the input random field (the so-called affine model). In this paper we consider an alternative model of the random field, in which the random variables have the same uniform distribution on an interval, but the random variables enter the input field as periodic functions. The field is constructed in such a way as to have the same mean and covariance function as the affine random field. Higher moments differ from the affine case, but in general the periodic model seems no less desirable. The new model of the random field is used to compute expected values of a quantity of interest arising from an elliptic PDE with random coefficients. The periodicity is shown to yield a higher order cubature convergence rate of $\mathcal{O}(n^{-1/p})$ independently of the dimension when used in conjunction with rank-1 lattice cubature rules constructed using suitably chosen smoothness-driven product and order dependent weights, where $n$ is the number of lattice points and $p$ is the summability exponent of the fluctuations in the series expansion of the random coefficient. We present numerical examples that assess the performance of our method.