The Influence of Numerical Error on an Inverse Problem Methodology in PDE Models

TL;DR

This paper investigates how numerical errors in PDE-based inverse problems affect parameter estimation and uncertainty quantification, providing analytical, computational insights, and practical guidelines for practitioners.

Contribution

It introduces a combined analytical and simulation approach to understand the impact of numerical errors in inverse PDE problems and offers guidelines for error management.

Findings

Numerical errors significantly influence the least squares cost function.

Residual patterns can be used to improve parameter estimation.

Guidelines help practitioners identify dominant error sources.

Abstract

The inverse problem methodology is a commonly-used framework in the sciences for parameter estimation and inference. It is typically performed by fitting a mathematical model to noisy experimental data. There are two significant sources of error in the process: 1.\ Noise from the measurement and collection of experimental data and 2.\ numerical error in approximating the true solution to the mathematical model. Little attention has been paid to how this second source of error alters the results of an inverse problem. As a first step towards a better understanding of this problem, we present a modeling and simulation study using a simple advection-driven PDE model. We present both analytical and computational results concerning how the different sources of error impact the least squares cost function as well as parameter estimation and uncertainty quantification. We investigate residual patterns to derive an autocorrelative statistical model that can improve parameter estimation and confidence interval computation for first order methods. Building on the results of our investigation, we provide guidelines for practitioners to determine when numerical or experimental error is the main source of error in their inference, along with suggestions of how to efficiently improve their results.