 # Introduction To Uncertainty Quantification This is the note for the summer short course Introduction To Uncertainty Quantification, advised by Prof. Xiaoliang wan

## Introduction To Uncertainty Quantification

First we display the brief history of hurricane forecast modeling.

• 1950s:statistical model
• 1973:Statistical/dynamical hybrid model
• 1990:resolution is fine enough for "accurate" prediction
• 2001:ensemble models

#### hurricane: Science & society

It is impossible to definitely predict the future state of the atmosphere because of the chaotic nature. Furthermore, existing observation have limited resolution in both space and time, especially over large bodies of water such as pacific ocean. Lack of observations introduces uncertainty into the limit state of the atmosphere. To account for this uncertainty ensemble forecasting is used.

Example.$$-\nabla \cdot (K\nabla u)=f$$

K为土壤的渗透系数，带有不确定性，同时边值条件因为观测也有不确定性。

##### 动力系统模型－>UQ

UQ:Physical Model+Uncertainty

Where uncertainty come from: example: Boundary condition, initial condition.

• Probability Theroy:Markov Chain
• Statistical Modeling:uncertainty data (Statistical data=Model data+verification data)
##### Uncertainty Quantification

physical model+UQ: 加大物理模型的能力

Model oriented not Data oriented

##### Typical Objection Of UQ

X:inputs,Y:outputs,F:Problem

Y=F(x)

$Y:F_*(\mu)(B)=Pr(F(x)\in B)$

1)Statistic of Y e.g. mean,

2)Failure Probality $Pr(Y\in B)<<1$

3) Interval Estimation $Pr(Y\in B)>>1-eps$

• Confidence Interval
• Credible Interval

Baysian Inverse Problem:Ill posed->well posed.

4) Model Reduction F_h ->F

Curse of dimensionality: Example:Polynomial Approx

Example:K-nearest algorithm.

#### Aleatoric and Epistemic Uncertaincy

• Aleatoric:内在的，不可控的
• Epistemic:随着知识增加会减小的不确定性

• X:input
• \theta:parameter
• Y:quantity of interest

Assumption:Model Error Gaussian

#### Measure of information and uncertainty

• variance and correlation
• Entorpy
• 'distance' between probatility space

Example. Uncertainty Principle in Quantum Mechanics

Example. Total Variation Distance; Kullback-Leibler Divergence(Relative Entoropy);

-> Dinsker's Inequality: $dTV(\mu,\miu)^2 \le 2 DKL(\mu||\miu)$