Presentation Template

Trappler Victor

Created: 2021-02-28 dim. 18:06

\( \DeclareMathOperator*{\argmin}{arg\,min} \)

1 Introduction

2 Deterministical calibration

2.1 Numerical models

Numerical models are used \(\mathcal{M}(x)\)

2.2 Inverse Problems

Given some observations \(y\), how to find \(\theta\) such that "\(y = \mathcal{M}(x)\)", or at least as close as possible ?

2.3 Deterministic Calibration of numerical models

Let us define \(J(x) = \| \mathcal{M}(x) - y\|^2_{\Sigma}\) and

\begin{equation} \hat{x} = \min_{x \in \mathbb{X}} J(x) \end{equation}

3 Calibration under uncertainty

3.1 Different types of uncertainties

A distinction can be made for the uncertainties:

  • Epistemic: That can be reduced with more studies, more explorations
  • Aleatoric: intrinsic variability

3.2 Introducing the aleatoric uncertainty

Input variable becomes \(x := (\theta, U)\) with

  • Epistemic: real parameter \(\theta\)
  • Aleatoric: real random variable \(U\)

and

\begin{equation} J(\theta, U) = \| \mathcal{M}(\theta, U) - y\|^2_{\Sigma} \end{equation}

is now a random variable as well, that we want to minimise in some sense

3.3 Many criteria of optimisation

Minimisation of the mean:

\begin{equation} \hat{\theta}_{\mathrm{mean}} = \argmin_{\theta\in\Theta} \mathbb{E}_{U}\left[J(\theta, U)\right] \end{equation}

4 Regret-based estimates

4.1 Conditional minimisers

Each sample \(u \sim U\) introduce a new situation of calibration, we introduce then

\begin{align} \theta^*(u) \end{align}

4.2 Region of acceptability

\((\theta, u) \quad \alpha-\text{acceptable } \iff J(\theta, u) \leq \alpha J^*(u)\)

5 SUR methods using GP

\begin{equation} Z(\theta, u) \sim \mathcal{N}(m_Z(\theta,u), \sigma^2_Z(\theta, u) \end{equation}

6 Conclusion