Bayesian sequential design of computer experiments for quantile set inversion

TL;DR

This paper introduces a Bayesian sequential design method using Gaussian processes and the SUR principle to efficiently identify input sets that meet probabilistic output criteria in complex simulations.

Contribution

It proposes a novel Bayesian approach for Quantile Set Inversion using Gaussian process models and the SUR strategy, tailored for complex systems with uncertain inputs.

Findings

Effective in approximating the target set with fewer evaluations

Demonstrates improved efficiency over non-sequential methods

Applicable to reliability-based optimization problems

Abstract

We consider an unknown multivariate function representing a system-such as a complex numerical simulator-taking both deterministic and uncertain inputs. Our objective is to estimate the set of deterministic inputs leading to outputs whose probability (with respect to the distribution of the uncertain inputs) of belonging to a given set is less than a given threshold. This problem, which we call Quantile Set Inversion (QSI), occurs for instance in the context of robust (reliability-based) optimization problems, when looking for the set of solutions that satisfy the constraints with sufficiently large probability. To solve the QSI problem we propose a Bayesian strategy, based on Gaussian process modeling and the Stepwise Uncertainty Reduction (SUR) principle, to sequentially choose the points at which the function should be evaluated to efficiently approximate the set of interest. We illustrate the performance and interest of the proposed SUR strategy through several numerical experiments.