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\begin{document}

\title{Extending GLUE with Multilevel Methods to Accelerate Statistical
Inversion of Hydrological Models}



\author[1]{Max Gustav Rudolph}%
\author[1]{Thomas Wöhling}%
\author[2]{Thorsten Wagener}%
\author[1]{Andreas Hartmann}%
\affil[1]{Technische Universität Dresden}%
\affil[2]{University of Potsdam}%


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  \date{June 18, 2024}


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\begin{abstract}
Inverse problems are ubiquitous in hydrological modelling for parameter
estimation, system understanding, sustainable water resources
management, and the operation of digital twins. While statistical
inversion is especially popular, its sampling-based nature often
inhibits the inversion of computationally costly models, which has
compromised the use of the Generalized Likelihood Uncertainty Estimation
(GLUE) methodology, e.g., for spatially distributed (partial)
differential equation based models. In this study we introduce
multilevel GLUE (MLGLUE), which alleviates the computational burden of
statistical inversion by utilizing a hierarchy of model resolutions.
Inspired by multilevel Monte Carlo, most parameter samples are evaluated
on lower levels with computationally cheap low-resolution models and
only samples associated with a likelihood above a certain threshold are
subsequently passed to higher levels with costly high-resolution models
for evaluation. Inferences are made at the level of the
highest-resolution model but substantial computational savings are
achieved by discarding samples with low likelihood already on levels
with low resolution and low computational cost. Two test problems
demonstrate the similarity of inferred parameter posteriors and
uncertainty estimates of MLGLUE and GLUE as well as increased
computational efficiency. Findings are furthermore compared to inversion
results from Markov-chain Monte Carlo (MCMC) and from multilevel delayed
acceptance MCMC. The computation time of inversion of a groundwater flow
model was decreases by [?]45\% and [?]57\% when using MLGLUE instead of
conventional formulations of GLUE and MCMC, respectively.%
\end{abstract}%



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