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

TL;DR

This paper introduces an adaptive algorithm for reconstructing monotone functions from imperfect data, enabling rigorous bounds on deviation probabilities, with applications in uncertainty quantification for engineering.

Contribution

It presents a novel adaptive reconstruction method tailored for monotone functions, extending isotonic regression techniques for uncertainty quantification tasks.

Findings

Algorithm converges under certain conditions.

Successfully applied to synthetic and real-world aerodynamic data.

Provides rigorous bounds on deviation probabilities.

Abstract

Motivated by the desire to numerically calculate rigorous upper and lower bounds on deviation probabilities over large classes of probability distributions, we present an adaptive 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.