Multilevel Monte Carlo for Polytopic DG Discretisations with Spatially Represented Random Coefficients
Abstract
We develop a multilevel Monte Carlo framework for coercive elliptic problems with random diffusion and reaction coefficients on polytopic meshes, with particular emphasis on the approximation and inter-level coupling of random coefficients generated from spatially represented latent fields. The main analytical contribution is a Strang-type error decomposition in a DG norm that separates the forward discretisation error from the error introduced by approximating the random coefficients. On this basis, we study two coefficient-generation and multilevel-coupling strategies, a spectral sampler based on a stationary Gaussian latent field defined on a fixed Cartesian reference grid, and an SPDE-based Matérn sampler in which levels are coupled through projected white-noise loads on agglomerated hierarchies. Both fit into the same coefficient-aware analytical framework and couple adjacent levels directly through a common latent random input. The theory is supported by numerical experiments on general Voronoi and agglomerated polytopic meshes, demonstrating the predicted forward and coefficient-representation errors, decay of multilevel corrections, strong inter-level coupling, and the resulting work-accuracy behaviour relative to single-level Monte Carlo.
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BibTeX
@misc{chronholm2026multilevel,
title = {{Multilevel Monte Carlo for Polytopic DG Discretisations with Spatially Represented Random Coefficients}},
author = {Veronika Chronholm and Matt Evans and Emmanuil H. Georgoulis and Tristan Pryer},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31861},
eprint = {2610.31861},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31861},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Multilevel Monte Carlo for Polytopic DG Discretisations with Spatially Represented Random Coefficients
AU - Veronika Chronholm
AU - Matt Evans
AU - Emmanuil H. Georgoulis
AU - Tristan Pryer
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31861
UR - https://arxiv.org/abs/2610.31861
AB - We develop a multilevel Monte Carlo framework for coercive elliptic problems with random diffusion and reaction coefficients on polytopic meshes, with particular emphasis on the approximation and inter-level coupling of random coefficients generated from spatially represented latent fields. The main analytical contribution is a Strang-type error decomposition in a DG norm that separates the forward discretisation error from the error introduced by approximating the random coefficients. On this basis, we study two coefficient-generation and multilevel-coupling strategies, a spectral sampler based on a stationary Gaussian latent field defined on a fixed Cartesian reference grid, and an SPDE-based Matérn sampler in which levels are coupled through projected white-noise loads on agglomerated hierarchies. Both fit into the same coefficient-aware analytical framework and couple adjacent levels directly through a common latent random input. The theory is supported by numerical experiments on general Voronoi and agglomerated polytopic meshes, demonstrating the predicted forward and coefficient-representation errors, decay of multilevel corrections, strong inter-level coupling, and the resulting work-accuracy behaviour relative to single-level Monte Carlo.
ER -