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Warping and sampling approaches to non-stationary gaussian process modelling

Sébastien Marmin has defended his thesis on 12th December 2017 at École Centrale de Marseille (France).​

Document type > *Mémoire/HDR/Thesis

Keywords >

Research Unit > IRSN/PSN-RES/SEMIA/LIMAR

Authors > MARMIN Sébastien

Publication Date > 12/12/2017

Summary

​This work deals with approximating expensive-to-evaluate functions exhibiting heterogeneous sensitivity to input perturbations depending on regions of the input space. Motivated by real test cases with high computational costs coming mainly from IRSN nuclear safety studies, we resort to surrogate models of the numerical simulators using Gaussian processes (GP). GP models are popular for sequential evaluation strategies in design of experiments under limited evaluation budget. While it is common to make stationarity assumptions for the processes and use sampling criteria based on its variance for exploration, we tackle the problem of accommodating the GP-based design to the heterogeneous behaviour of the function from two angles: first via a novel class of covariances (WaMI-GP) that simultaneously generalises existing kernels of Multiple Index and of tensorised warped GP and second, by introducing derivative-based sampling criteria dedicated to the exploration of high variation regions. The novel GP class is investigated both through mathematical analysis and numerical experiments, and it is shown that it allows encoding much expressiveness while remaining with a moderate number of parameters to be inferred. Moreover, exploiting methodological links between wavelets analysis and non-stationary GP modelling, we propose a new non-stationary GP (Wav-GP) with non-parametric warping. The key point is an iterated estimation of the so-called local scale that approximates the derivative of the warping. Wav-GP is applied to two mechanical case studies highlighting promising prediction performance. Independently of non-stationarity assumptions, we conduct derivations for new variance-based criteria relying on the norm of the GP gradient field. Criteria and models are compared with state-of-the-art methods on engineering test cases. It is found on these applications that some of the proposed gradient-based criteria outperform usual variance-based criteria in the case of a stationary GP model, but that it is even better to use variance-based criteria with WaMI-GP, which dominates mostly for small designs and in sequential set up. Other contributions in sampling criteria address the problem of global optimisation, focusing on the expected improvement criterion and its multipoint version for parallel batch evaluations. Closed form formulas and fast approximations are established for a generalised version of the criterion and its gradient. Numerical experiments illustrate that the proposed approaches enable substantial computational savings.


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