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Warping and sampling approaches to non-stationary gaussian process modelling.

Abstract : This work deals with approximating expensive-to-evaluatefunctions exhibiting heterogeneous sensitivity to input perturbationsdepending on regions of the input space. Motivated by real test caseswith high computational costs coming mainly from IRSN nuclear safetystudies, we resort to surrogate models of the numerical simulatorsusing Gaussian processes (GP). GP models are popular for sequentialevaluation strategies in design of experiments under limited evaluationbudget. While it is common to make stationarity assumptions for theprocesses and use sampling criteria based on its variance forexploration, we tackle the problem of accommodating the GP-based designto the heterogeneous behaviour of the function from two angles: firstvia a novel class of covariances (WaMI-GP) that simultaneouslygeneralises existing kernels of Multiple Index and of tensorised warpedGP and second, by introducing derivative-based sampling criteriadedicated to the exploration of high variation regions. The novel GPclass is investigated both through mathematical analysis and numericalexperiments, and it is shown that it allows encoding muchexpressiveness while remaining with a moderate number of parameters tobe inferred. Moreover, exploiting methodological links between waveletsanalysis and non-stationary GP modelling, we propose a new non-stationary GP (Wav-GP) with non-parametric warping. The key point is aniterated estimation of the so-called local scale that approximates thederivative of the warping. Wav-GP is applied to two mechanical casestudies highlighting promising prediction performance. Independently ofnon-stationarity assumptions, we conduct derivations for new variance-based criteria relying on the norm of the GP gradient field. Criteriaand models are compared with state-of-the-art methods on engineeringtest cases. It is found on these applications that some of the proposedgradient-based criteria outperform usual variance-based criteria in thecase of a stationary GP model, but that it is even better to usevariance-based criteria with WaMI-GP, which dominates mostly for smalldesigns and in sequential set up. Other contributions in samplingcriteria address the problem of global optimisation, focusing on theexpected improvement criterion and its multipoint version for parallelbatch evaluations. Closed form formulas and fast approximations areestablished for a generalised version of the criterion and its gradient. Numerical experiments illustrate that the proposed approachesenable substantial computational savings.
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Submitted on : Monday, March 26, 2018 - 4:58:40 PM
Last modification on : Thursday, January 23, 2020 - 6:22:13 PM
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  • HAL Id : tel-01743815, version 1



Sebastien Marmin. Warping and sampling approaches to non-stationary gaussian process modelling.. General Mathematics [math.GM]. Ecole Centrale Marseille; Université de Berne. Collegium generale, 2017. English. ⟨NNT : 2017ECDM0007⟩. ⟨tel-01743815⟩



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