Adaptive reconstruction of imperfectly-observed monotone functions, with applications to uncertainty quantification

Reconstruction adaptative de fonctions monotones partiellement observées, et applications à la quantification d'incertitude

Luc Bonnet (1, 2) , Jean-Luc Akian (3) , Eric Savin (4) , Tim J. Sullivan (5, 6)

1 DAAA, ONERA, Université Paris-Saclay [Châtillon]
2 MSSMat - Laboratoire de mécanique des sols, structures et matériaux
3 DMAS, ONERA, Université Paris Saclay [Châtillon]
4 DTIS, ONERA, Université Paris Saclay [Palaiseau]
5 Department of Mathematics, University of Warwick
6 ZIB - Zuse Institute Berlin

Motivated by the desire to numerically calculate rigorous upper and lower bounds on deviation probabilities over large classes of probability distributions, we present an adap-tive algorithm for the reconstruction of increasing real-valued functions. While this problem is similar to the classical statistical problem of isotonic regression, the optimisation setting alters several characteristics of the problem and opens natural algorithmic possibilities. We present our algorithm, establish sufficient conditions for convergence of the reconstruction to the ground truth, and apply the method to synthetic test cases and a real-world example of uncertainty quantification for aerodynamic design.

Mots clés: adaptive approximation; isotonic regression; optimisation under uncertainty; uncertainty quantification; Aerodynamic design

Mots clés: APPROXIMATION ADAPTATIVE; RÉGRESSION ISOTONE; OPTIMISATION SOUS INCERTITUDE; QUANTIFICATION INCERTITUDE; AERODYNAMIQUE NUMERIQUE

Domaines: Autre; Optimisation et contrôle [math.OC]

Luc Bonnet, Jean-Luc Akian, Eric Savin, Tim J. Sullivan. Adaptive reconstruction of imperfectly-observed monotone functions, with applications to uncertainty quantification. Algorithms, 2020, 13 (8), pp.196. ⟨10.3390/a13080196⟩. ⟨hal-02908957v2⟩