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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[3]{Andreas Hartmann}%
\affil[1]{Technische Universität Dresden}%
\affil[2]{University of Potsdam}%
\affil[3]{Dresden University of Technology}%


\vspace{-1em}



  
  \date{June 13, 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 its application to 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 example inverse
problems, using a rainfall-runoff model and groundwater flow model,
demonstrate the substantially increased computational efficiency of
MLGLUE compared to GLUE as well as the similarity of inversion results.
Findings are furthermore compared to inversion results from Markov-chain
Monte Carlo (MCMC) and multilevel delayed acceptance MCMC, a
corresponding multilevel variant, to compare the effects of the
multilevel extension. All examples demonstrate the wide-range
suitability of the approach and include guidelines for practical
applications.%
\end{abstract}%



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\includegraphics[width=0.70\columnwidth]{figures/EX01-HYMOD-convergence-analysis-time/EX01-HYMOD-convergence-analysis-time}

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