Integration based profile likelihood calculation for PDE constrained parameter estimation problems

TL;DR

This paper introduces an integration-based method for calculating profile likelihoods in PDE-constrained parameter estimation, offering improved speed and accuracy over traditional optimization-based approaches.

Contribution

The novel approach exploits a dynamical system along the likelihood profile, reducing computational effort compared to existing methods.

Findings

Good accuracy demonstrated on reaction-diffusion model

Significant speed-up over state-of-the-art algorithms

Facilitates rigorous uncertainty analysis in PDE-constrained problems

Abstract

Partial differential equation (PDE) models are widely used in engineering and natural sciences to describe spatio-temporal processes. The parameters of the considered processes are often unknown and have to be estimated from experimental data. Due to partial observations and measurement noise, these parameter estimates are subject to uncertainty. This uncertainty can be assessed using profile likelihoods, a reliable but computationally intensive approach. In this paper, we introduce an integration based approach for the profile likelihood calculation for inverse problems with PDE constraints. While existing approaches rely on repeated optimization, the proposed approach exploits a dynamical system evolving along the likelihood profile. We derive the dynamical system for the reduced and the full estimation problem and study its properties. To evaluate the proposed method, we compare it with state-of-the-art algorithms for a simple reaction-diffusion model for a cellular patterning process. We observe a good accuracy of the method as well as a significant speed up as compared to established methods. Integration based profile calculation facilitates rigorous uncertainty analysis for computationally demanding parameter estimation problems with PDE constraints.