Functional Error Models to Accelerate Nested Sampling

Abstract:

The main challenge in groundwater problems is the reliance on large numbers of unknown parameters with wide rage of associated uncertainties. To translate this uncertainty to quantities of interest (for instance the concentration of pollutant in a drinking well), a large number of forward flow simulations is required.

To make the problem computationally tractable, Josset et al. (2013, 2014) introduced the concept of functional error models. It consists in two elements: a proxy model that is cheaper to evaluate than the full physics flow solver and an error model to account for the missing physics. The coupling of the proxy model and the error models provides reliable predictions that approximate the full physics model's responses. The error model is tailored to the problem at hand by building it for the question of interest. It follows a typical approach in machine learning where both the full physics and proxy models are evaluated for a training set (subset of realizations) and the set of responses is used to construct the error model using functional data analysis. Once the error model is devised, a prediction of the full physics response for a new geostatistical realization can be obtained by computing the proxy response and applying the error model.

We propose the use of functional error models in a Bayesian inference context by combining it to the Nested Sampling (Skilling 2006; El Sheikh et al. 2013, 2014). Nested Sampling offers a mean to compute the Bayesian Evidence by transforming the multidimensional integral into a 1D integral. The algorithm is simple: starting with an active set of samples, at each iteration, the sample with the lowest likelihood is kept aside and replaced by a sample of higher likelihood. The main challenge is to find this sample of higher likelihood. We suggest a new approach: first the active set is sampled, both proxy and full physics models are run and the functional error model is build. Then, at each iteration of the Nested Sampling algorithm, the proposed geostatistical realization is first evaluated through the approximate model to decide whether it is useful or not to perform a full physics simulation. This improves the acceptance rate of full physics simulations and opens the door to iteratively test the performance and improve the quality of the error model.