Optimal Bounds on Nonlinear Partial Differential Equations in Model Certification, Validation, and Experimental Design

TL;DR

This paper applies the Optimal Uncertainty Quantification (OUQ) theory to PDE-governed systems, demonstrating rigorous model validation and optimal design with improved accuracy and efficiency over traditional methods.

Contribution

It introduces a novel application of OUQ to complex PDE systems, enabling rigorous bounds and efficient validation for physical models.

Findings

OUQ provides tighter bounds than Monte Carlo methods.

The approach reduces computational cost for PDE uncertainty quantification.

Demonstrated on Burgers' equation for shock location prediction.

Abstract

We demonstrate that the recently developed Optimal Uncertainty Quantification (OUQ) theory, combined with recent software enabling fast global solutions of constrained non-convex optimization problems, provides a methodology for rigorous model certification, validation, and optimal design under uncertainty. In particular, we show the utility of the OUQ approach to understanding the behavior of a system that is governed by a partial differential equation -- Burgers' equation. We solve the problem of predicting shock location when we only know bounds on viscosity and on the initial conditions. Through this example, we demonstrate the potential to apply OUQ to complex physical systems, such as systems governed by coupled partial differential equations. We compare our results to those obtained using a standard Monte Carlo approach, and show that OUQ provides more accurate bounds at a lower computational cost. We discuss briefly about how to extend this approach to more complex systems, and how to integrate our approach into a more ambitious program of optimal experimental design.