General Bayesian L2 calibration of mathematical models

TL;DR

This paper introduces and compares Bayesian and general Bayesian methods for calibrating mathematical models to physical systems by estimating unknown parameters to minimize the squared L2 difference.

Contribution

It develops a unified framework for Bayesian calibration of models using L2 norm minimization, including new methods and comparisons.

Findings

Proposes new Bayesian calibration methods based on L2 norm.

Analyzes the theoretical properties of these calibration methods.

Demonstrates effectiveness through numerical experiments.

Abstract

A mathematical model is a function taking certain arguments and returning a theoretical prediction of a feature of a physical system. The arguments to the mathematical model can be split into two groups; (a) controllable variables of the system; and (b) calibration parameters: unknown characteristics of the physical system that cannot be controlled or directly measured. Of interest is the estimation of the calibration parameter using physical observations. Since the mathematical model will be an inexact representation of the physical system: the aim is to estimate values for the calibration parameters to make the mathematical model ``close" to the physical system. Closeness is defined as the squared $L^2$ norm of the difference between the mathematical model and the physical system. Different Bayesian and general Bayesian methods are introduced, developed and compared for this task.