Functional error modeling for uncertainty quantification in hydrogeology

Abstract

Approximate models (proxies) can be employed to reduce the computational costs of estimating uncertainty. The price to pay is that the approximations introduced by the proxy model can lead to a biased estimation. To avoid this problem and ensure a reliable uncertainty quantification, we propose to combine functional data analysis and machine learning to build error models that allow us to obtain an accurate prediction of the exact response without solving the exact model for all realizations. We build the relationship between proxy and exact model on a learning set of geostatistical realizations for which both exact and approximate solvers are run. Functional principal components analysis (FPCA) is used to investigate the variability in the two sets of curves and reduce the dimensionality of the problem while maximizing the retained information. Once obtained, the error model can be used to predict the exact response of any realization on the basis of the sole proxy response. This methodology is purpose-oriented as the error model is constructed directly for the quantity of interest, rather than for the state of the system. Also, the dimensionality reduction performed by FPCA allows a diagnostic of the quality of the error model to assess the informativeness of the learning set and the fidelity of the proxy to the exact model. The possibility of obtaining a prediction of the exact response for any newly generated realization suggests that the methodology can be effectively used beyond the context of uncertainty quantification, in particular for Bayesian inference and optimization.

Efficient Monte Carlo With Graph‐Based Subsurface Flow and Transport Models

“…It is similar in concept to multilevel MC methods (Giles, ; Lu et al, ), but does not require the presence of multiple grids with different resolutions. It also resembles other recently developed methods (Josset et al, ; Linde et al, ), but does not require machine learning methods to eliminate biases in the lower‐fidelity models. Instead, it relies on the form of the estimator to eliminate this bias.…”

Stochastic inversion for soil hydraulic parameters in the presence of model error: An example involving ground-penetrating radar monitoring of infiltration

“…Construction of the error model can be done in a number of different ways. This includes simple nearest-neighbour or linear interpolation between model-error realizations (e.g., Cui et al, 2011;O'Sullivan and Christie, 2006), representing the discrepancy as a Gaussian process conditioned to the points in the parameter space where the model error is known (e.g., Kennedy and O'Hagan, 2001;Xu and Valocchi , 2015), or using statistical regression approaches (e.g., Doherty and Christensen, 2011;Josset et al, 2015). In all of this work, the implicit assumption is that the full and approximate model-response surfaces are regular enough such that the model error for a set of parameter values where it is unknown can be effectively predicted through some kind of interpolation between the existing realizations.…”

Oil production uncertainty assessment by predicting reservoir production curves and confidence intervals from arbitrary proxy responses

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