Quantifying uncertainty on Pareto fronts with Gaussian process conditional simulations

Binois, Mickaël; Ginsbourger, David; Roustant, Olivier

doi:10.1016/j.ejor.2014.07.032

Cited by 43 publications

(36 citation statements)

References 34 publications

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“…While GRF conditional simulations constitute a standard and important topic in the literature of geostatistics (Chilès and Delfiner, 2012) with a variety of applications in geosciences and natural resources characterization (Delhomme, 1979;Chilès and Allard, 2005;Deutsch, 2002;Dimitrakopoulos, 2011;Journel and Kyriakidis, 2004), they have been increasingly used in engineering and related areas, where Gaussian random field models have been used as prior distributions on expensive-to-evaluation functions (Hoshiya, 1995;Santner et al, 2003;Villemonteix et al, 2009;Roustant et al, 2012;Binois et al, 2014).…”

Section: Introductionmentioning

confidence: 99%

Fast Update of Conditional Simulation Ensembles

2014

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Gaussian random fields (GRF) conditional simulation is a key ingredient in many spatial statistics problems for computing Monte-Carlo estimators and quantifying uncertainties on non-linear functionals of GRFs conditional on data. Conditional simulations are known to often be computer intensive, especially when appealing to matrix decomposition approaches with a large number of simulation points. Here we study the settings where conditioning observations are assimilated batch-sequentially, i.e. one point or batch of points at each stage. Assuming that conditional simulations have been performed at a previous stage, we aim at taking advantage of already available sample paths and by-products in order to produce updated conditional simulations at minimal cost. We provide explicit formulas allowing to update an ensemble of sample paths conditioned on n ≥ 0 observations to an ensemble conditioned on n + q observations, for arbitrary q ≥ 1. Compared to direct approaches, the proposed formulas prove to substantially reduce computational complexity. Moreover, these formulas enable explicitly exhibiting how the q "new" observations are updating the "old" sample paths. Detailed complexity calculations highlighting the benefits of our approach with respect to state-of-the-art algorithms are provided and are complemented by numerical experiments.

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Section: Introductionmentioning

confidence: 99%

Fast Update of Conditional Simulation Ensembles

2014

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“…(4) and the level lines of F Y are expressed as: Proof. For an Archimedean copula with generator φ, the level curve of level α > 0 is ∂L…”

Section: Parametric Form In the Archimedean Casementioning

confidence: 99%

“…In multi-objective optimization, the connection seems to be new. Uncertainty quantification around the Pareto front has been recently considered by [4], using conditional simulations of Kriging metamodels and concepts from random sets theory. Whereas such approach is relevant in a sequential algorithm, it may be inappropriate in the initial stage that we consider here, due to a potentially large model error in metamodeling.…”

Section: Introductionmentioning

confidence: 99%

On the estimation of Pareto fronts from the point of view of copula theory

Binois

Rullière

Roustant

2015

Information Sciences

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International audienceGiven a first set of observations from a design of experiments sampled randomly in the design space, the corresponding set of non-dominated points usually does not give a good approximation of the Pareto front. We propose here to study this problem from the point of view of multivariate analysis, introducing a probabilistic framework with the use of copulas. This approach enables the expression of level lines in the objective space, giving an estimation of the position of the Pareto front when the level tends to zero. In particular, when it is possible to use Archimedean copulas, analytical expressions for Pareto front estimators are available. Several case studies illustrate the interest of the approach, which can be used at the beginning of the optimization when sampling randomly in the design space

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“…The methodology relies on building probabilistic surrogates of the objectives and uses the EEIHV IAF to quantify the merit of evaluating the expensive stochastic computer code at a new design. We leverage the work done in [2] to quantify our uncertainty about the estimated PF at each stage/iteration. We apply the above methodology to solve a multi-pass steel wire manufacturing problem under uncertainty.…”

Section: Introductionmentioning

confidence: 99%

Stochastic Multiobjective Optimization on a Budget: Application to Multipass Wire Drawing With Quantified Uncertainties

Pandita

Bilionis

Panchal

et al. 2018

Int. J. UncertaintyQuantification

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Design optimization of engineering systems with multiple competing objectives is a painstakingly tedious process especially when the objective functions are expensiveto-evaluate computer codes with parametric uncertainties. The effectiveness of the state-of-the-art techniques is greatly diminished because they require a large number of objective evaluations, which makes them impractical for problems of the above kind. Bayesian global optimization (BGO), has managed to deal with these challenges in solving single-objective optimization problems and has recently been extended to multi-objective optimization (MOO). BGO models the objectives via probabilistic surrogates and uses the epistemic uncertainty to define an information acquisition function (IAF) that quantifies the merit of evaluating the objective at new designs. This iterative data acquisition process continues until a stopping criterion is met. The most commonly used IAF for MOO is the expected improvement over the dominated hypervolume (EIHV) which in its original form is unable to deal with parametric uncertainties or measurement noise. In this work, we provide a systematic reformulation of EIHV to deal with stochastic MOO problems. The primary contribution of this paper lies in being able to filter out the noise and reformulate the EIHV without having to observe or estimate the stochastic parameters. An addendum of the probabilistic nature of our methodology is that it enables us to characterize our confidence about the predicted Pareto front. We verify and validate the proposed methodology by applying it to synthetic test problems with known solutions. We demonstrate our approach on an industrial problem of die pass design for a steel wire drawing process.

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Quantifying uncertainty on Pareto fronts with Gaussian process conditional simulations

Cited by 43 publications

References 34 publications

Fast Update of Conditional Simulation Ensembles

Fast Update of Conditional Simulation Ensembles

On the estimation of Pareto fronts from the point of view of copula theory

Stochastic Multiobjective Optimization on a Budget: Application to Multipass Wire Drawing With Quantified Uncertainties

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