Goal-Oriented A-Posteriori Estimation of Model Error as an Aid to Parameter Estimation

TL;DR

This paper introduces a Bayesian calibration method that uses goal-oriented a-posterior error estimates in quantities of interest to efficiently calibrate high-fidelity PDE models based on low-fidelity surrogates, improving parameter estimation accuracy.

Contribution

It develops a computationally efficient Bayesian calibration framework utilizing goal-oriented error estimates for high-fidelity PDE models, enhancing parameter estimation with low-fidelity models.

Findings

Effective calibration of PDE model parameters demonstrated in elliptic BVPs.

Successful application to nonlinear tumor growth PDE models.

Error estimates improve the likelihood function accuracy in Bayesian inversion.

Abstract

In this work, a Bayesian model calibration framework is presented that utilizes goal-oriented a-posterior error estimates in quantities of interest (QoIs) for classes of high-fidelity models characterized by PDEs. It is shown that for a large class of computational models, it is possible to develop a computationally inexpensive procedure for calibrating parameters of high-fidelity models of physical events when the parameters of low-fidelity (surrogate) models are known with acceptable accuracy. The main ingredients in the proposed model calibration scheme are goal-oriented a-posteriori estimates of error in QoIs computed using a so-called lower fidelity model compared to those of an uncalibrated higher fidelity model. The estimates of error in QoIs are used to define likelihood functions in Bayesian inversion analysis. A standard Bayesian approach is employed to compute the posterior distribution of model parameters of high-fidelity models. As applications, parameters in a quasi-linear second-order elliptic boundary-value problem (BVP) are calibrated using a second-order linear elliptic BVP. In a second application, parameters of a tumor growth model involving nonlinear time-dependent PDEs are calibrated using a lower fidelity linear tumor growth model with known parameter values.