Approximate Interval Method for Epistemic Uncertainty Propagation using Polynomial Chaos and Evidence Theory

TL;DR

This paper introduces an approximate method combining Polynomial Chaos and Evidence Theory to efficiently estimate bounds of a function's output under epistemic uncertainty, useful for complex nonlinear systems.

Contribution

It presents a novel approach that transforms Polynomial Chaos expansions into Bernstein form to compute bounds, integrating evidence theory for uncertainty propagation.

Findings

Accurately bounds the output of nonlinear functions under uncertainty.

Efficiently propagates Dempster-Shafer structures through complex functions.

Demonstrates effectiveness on an algebraic challenge problem.

Abstract

The paper builds upon a recent approach to find the approximate bounds of a real function using Polynomial Chaos expansions. Given a function of random variables with compact support probability distributions, the intuition is to quantify the uncertainty in the response using Polynomial Chaos expansion and discard all the information provided about the randomness of the output and extract only the bounds of its compact support. To solve for the bounding range of polynomials, we transform the Polynomial Chaos expansion in the Bernstein form, and use the range enclosure property of Bernstein polynomials to find the minimum and maximum value of the response. This procedure is used to propagate Dempster-Shafer structures on closed intervals through nonlinear functions and it is applied on an algebraic challenge problem.