Quantifying Uncertainties on Excursion Sets Under a Gaussian Random Field Prior

Azzimonti, Dario; Bect, Julien; Chevalier, Clément; Ginsbourger, David

doi:10.1137/141000749

Cited by 39 publications

(34 citation statements)

References 38 publications

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“…In what follows, the points G are called pilot points, borrowing the term from geostatistics (see,e.g. Scheidt, 2006, Chapter 4.2), and here they are selected with Algorihtm B from Azzimonti et al (2016). The number of pilot points can be empirically chosen by stopping when the optimum of Algorithm B's objective function stabilizes around a value.…”

Section: Approximation For Posterior Field Realizationsmentioning

confidence: 99%

Profile Extrema for Visualizing and Quantifying Uncertainties on Excursion Regions: Application to Coastal Flooding

et al. 2019

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We consider the problem of describing excursion sets of a real-valued function f , i.e. the set of inputs where f is above a fixed threshold. Such regions are hard to visualize if the input space dimension, d, is higher than 2. For a given projection matrix from the input space to a lower dimensional (usually 1, 2) subspace, we introduce profile sup (inf) functions that associate to each point in the projection's image the sup (inf) of the function constrained over the pre-image of this point by the considered projection. Plots of profile extrema functions convey a simple, although intrinsically partial, visualization of the set. We consider expensive to evaluate functions where only a very limited number of evaluations, n, is available, e.g. n < 100d, and we surrogate f with a posterior quantity of a Gaussian process (GP) model. We first compute profile extrema functions for the posterior mean given n evaluations of f . We quantify the uncertainty on such estimates by studying the distribution of GP profile extrema with posterior quasi-realizations obtained from an approximating process. We control such approximation with a bound inherited from the Borell-TIS inequality. The technique is applied to analytical functions (d = 2, 3) and to a 5dimensional coastal flooding test case for a site located on the Atlantic French coast.Here f is a numerical model returning the area of flooded surface in the coastal region given some offshore conditions. Profile extrema functions allowed us to better understand which offshore conditions impact large flooding events.

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Section: Approximation For Posterior Field Realizationsmentioning

confidence: 99%

Profile Extrema for Visualizing and Quantifying Uncertainties on Excursion Regions: Application to Coastal Flooding

et al. 2019

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“…Here again we contrast the function-view strategy of emulation, which quantifies the learning of h(·), with the root-view strategy that quantifies learning of X * . For the former, despite some progress on building EI measures for the level-sets and graph of h(·) [2,7], these metrics remain complex. In our view this is a fundamental conceptual hurdle arising from the mismatch between the large model space for h, and the much simpler derived quantity, i.e., the root x * , to be learned.…”

Section: Introductionmentioning

confidence: 99%

Generalized Probabilistic Bisection for Stochastic Root Finding

Rodríguez

Ludkovski

2020

ACM Trans. Model. Comput. Simul.

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We consider numerical schemes for root finding of noisy responses through generalizing the Probabilistic Bisection Algorithm (PBA) to the more practical context where the sampling distribution is unknown and location-dependent. As in standard PBA, we rely on a knowledge state for the approximate posterior of the root location. To implement the corresponding Bayesian updating, we also carry out inference of oracle accuracy, namely learning the probability of correct response. To this end we utilize batched querying in combination with a variety of frequentist and Bayesian estimators based on majority vote, as well as the underlying functional responses, if available. For guiding sampling selection we investigate both Information Directed sampling, as well as Quantile sampling. Our numerical experiments show that these strategies perform quite differently; in particular we demonstrate the efficiency of randomized quantile sampling which is reminiscent of Thompson sampling. Our work is motivated by the root-finding sub-routine in pricing of Bermudan financial derivatives, illustrated in the last section of the paper.

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“…Carlo with conditional realizations of the field, see, e.g. Azzimonti et al (2016) for fast approximations of conditional realizations.…”

Section: Vorob'ev Expectation and Conservative Estimatesmentioning

confidence: 99%

Adaptive Design of Experiments for Conservative Estimation of Excursion Sets

et al. 2019

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We consider the problem of estimating the set of all inputs that leads a system to some particular behavior. The system is modeled with an expensive-to-evaluate function, such as a computer experiment, and we are interested in its excursion set, i.e. the set of points where the function takes values above or below some prescribed threshold. The objective function is emulated with Gaussian Process (GP) models based on an initial design of experiments enriched with evaluation results at (batch-) sequentially determined input points. The GP model provides conservative estimates for the excursion set, which control false positives while minimizing false negatives. We introduce adaptive strategies that sequentially select new evaluations of the function by reducing the uncertainty on conservative estimates. Following the Stepwise Uncertainty Reduction approach we obtain new evaluations by minimizing adapted criteria. Tractable formulae for the conservative criteria are derived which allow more convenient optimization. The method is benchmarked on random functions generated under the model assumptions in two and five dimensions and applied to a reliability engineering test case. Overall, the proposed strategy of minimizing false negatives in conservative estimation achieves competitive performance both in terms of model-based and model-free indicators.

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Quantifying Uncertainties on Excursion Sets Under a Gaussian Random Field Prior

Cited by 39 publications

References 38 publications

Profile Extrema for Visualizing and Quantifying Uncertainties on Excursion Regions: Application to Coastal Flooding

Profile Extrema for Visualizing and Quantifying Uncertainties on Excursion Regions: Application to Coastal Flooding

Generalized Probabilistic Bisection for Stochastic Root Finding

Adaptive Design of Experiments for Conservative Estimation of Excursion Sets

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