# A Multilevel, Hierarchical Sampling Technique for Spatially Correlated Random Fields

## TL;DR

This paper introduces a scalable, hierarchical sampling method for spatially correlated random fields using a multilevel decomposition of stochastic reaction-diffusion equations, improving efficiency for large-scale uncertainty propagation.

## Contribution

It presents a novel multilevel, hierarchical sampling technique that overcomes computational limitations of traditional methods like KL expansion for large-scale spatial fields.

## Key findings

- Method is scalable and efficient for large problems.
- Numerical experiments demonstrate effective multilevel Monte Carlo simulations.
- Applicable to subsurface porous media flow problems.

## Abstract

We propose an alternative method to generate samples of a spatially correlated random field with applications to large-scale problems for forward propagation of uncertainty. A classical approach for generating these samples is the Karhunen-Lo\`{e}ve (KL) decomposition. However, the KL expansion requires solving a dense eigenvalue problem and is therefore computationally infeasible for large-scale problems. Sampling methods based on stochastic partial differential equations provide a highly scalable way to sample Gaussian fields, but the resulting parametrization is mesh dependent. We propose a multilevel decomposition of the stochastic field to allow for scalable, hierarchical sampling based on solving a mixed finite element formulation of a stochastic reaction-diffusion equation with a random, white noise source function. Numerical experiments are presented to demonstrate the scalability of the sampling method as well as numerical results of multilevel Monte Carlo simulations for a subsurface porous media flow application using the proposed sampling method.