Abstract
Abstract. Contaminant source localization problems require efficient and robust methods that can account for geological heterogeneities and accommodate relatively small data sets of noisy observations. As realism commands hi-fidelity simulations, computation costs call for global optimization algorithms under parsimonious evaluation budgets. Bayesian optimization approaches are well adapted to such settings as they allow the exploration of parameter spaces in a principled way so as to iteratively locate the point(s) of global optimum while maintaining an approximation of the objective function with an instrumental quantification of prediction uncertainty. Here, we adapt a Bayesian optimization approach to localize a contaminant source in a discretized spatial domain. We thus demonstrate the potential of such a method for hydrogeological applications and also provide test cases for the optimization community. The localization problem is illustrated for cases where the geology is assumed to be perfectly known. Two 2-D synthetic cases that display sharp hydraulic conductivity contrasts and specific connectivity patterns are investigated. These cases generate highly nonlinear objective functions that present multiple local minima. A derivative-free global optimization algorithm relying on a Gaussian process model and on the expected improvement criterion is used to efficiently localize the point of minimum of the objective functions, which corresponds to the contaminant source location. Even though concentration measurements contain a significant level of proportional noise, the algorithm efficiently localizes the contaminant source location. The variations of the objective function are essentially driven by the geology, followed by the design of the monitoring well network. The data and scripts used to generate objective functions are shared to favor reproducible research. This contribution is important because the functions present multiple local minima and are inspired from a practical field application. Sharing these complex objective functions provides a source of test cases for global optimization benchmarks and should help with designing new and efficient methods to solve this type of problem.
Highlights
Many hydrogeological processes are governed by nonlinear equations
Using information from the 25 observation wells, the optimization algorithm is applied over four configurations that depend on the retained geology and on the contaminant source location as described in Table 2, where the noise level κ is set to 0 and the parameter p of the objective function f (x) is set to 2
Comparison of the different scenarios reveals that the geology controls the main features of the objective functions, which reinforce the importance of realistic geological structures in contaminant source localization problems
Summary
Many hydrogeological processes are governed by nonlinear equations (e.g., unsaturated flow problems, heat and transport problems; De Marsily, 1986). Derivative-free global optimization methods such as evolutionary algorithms, simulated annealing and others have become commonplace in the last decades These are typically regarded with caution as they do not systematically come with much guarantee and can potentially require large numbers of function evaluations, a situation that is to be avoided in the case. One of the greatest strengths of common Bayesian optimization algorithms is that they do guide evaluations towards the global optimum and maintain an approximate representation of the objective function together with a quantification of prediction uncertainty. This enables space exploration with a memory so as to prevent or mitigate evaluations at redundant locations. Recent adaptations of popular Bayesian optimization approaches allow the accommodation of evaluation noise (Picheny and Ginsbourger, 2014), parallel evaluations (Marmin et al, 2015), high dimensions (Wang et al, 2018), nonstationarity (Snoek et al, 2014), gradient observations (Wu et al, 2017) and many more features (see for instance Ginsbourger, 2018, for a broader overview of sequential design algorithms for computer experiments)
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