# Fast random field generation with $H$-matrices

## TL;DR

This paper introduces a linear-cost method using $H$-matrices to efficiently generate normal and log-normal random fields on arbitrary point sets, with proven error bounds and broad applicability.

## Contribution

It presents a novel $H$-matrix based approach for approximate square root computation of covariance matrices, enabling fast random field generation on general point sets.

## Key findings

- Linear computational cost for random field generation
- Rigorous error estimates demonstrating convergence
- Applicable to non-stationary covariance functions and irregular point sets

## Abstract

We use the $H$-matrix technology to compute the approximate square root of a covariance matrix in linear cost. This allows us to generate normal and log-normal random fields on general point sets with optimal cost. We derive rigorous error estimates which show convergence of the method. Our approach requires only mild assumptions on the covariance function and on the point set. Therefore, it might be also a nice alternative to the circulant embedding approach which applies only to regular grids and stationary covariance functions.