Uncertainty Quantification

A comprehensive collection of projects covering computational methods for uncertainty quantification in mathematical models

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Overview

This repository contains implementations in both MATLAB and Python covering fundamental and advanced topics in uncertainty quantification:

• Sensitivity analysis (local and global)
• Parameter estimation and identifiability
• Bayesian inference and MCMC methods
• Uncertainty propagation
• Surrogate modeling techniques
• Model discrepancy quantification

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Course Topics

Topic	Description
Sensitivity Analysis	Local (finite difference, complex-step) and global (Morris, Sobol) methods
Parameter Estimation	OLS, MLE, constrained optimization
Bayesian Inference	Prior specification, posterior computation, MCMC algorithms
Uncertainty Propagation	Confidence intervals, credible intervals, prediction intervals
Surrogate Models	Polynomial regression, Gaussian processes, sparse grids
Model Discrepancy	Quantifying differences between physical and computational models

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Requirements

MATLAB

• MATLAB R2021a or later
• Optimization Toolbox
• Global Optimization Toolbox
• MCMC for MATLAB (for Project 3)

Python

pip install -r requirements.txt

Required packages: numpy, scipy, matplotlib, statsmodels

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Project 1: Sensitivity Analysis

Topics: Local and global sensitivity analysis, Fisher information matrix, parameter identifiability

Problem	Description	MATLAB	Python
1	Compute sensitivities of spring model	UQ_8_5.m	UQ_8_5.py
2	SIR model sensitivities and identifiability	UQ_8_8.m	UQ_8_8.py
3	Heat equation parameter identifiability	UQ_8_9.m	UQ_8_9.py
4	Global sensitivity (Morris, Sobol) for Helmholtz model	UQ_9_6.m	UQ_9_6.py

Project Writeup (PDF)

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Project 2: Parameter Estimation

Topics: OLS estimation, constrained optimization, covariance estimation

Problem	Description	MATLAB	Python
1	Heat model parameter estimation (copper rod)	Problem1.m	Problem1.py
2	OLS for Helmholtz energy model	Problem2.m	Problem2.py
3	SIR model parameter distributions	Problem3.m	Problem3.py

Project Writeup (PDF)

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Project 3: Bayesian Inference and MCMC

Topics: Metropolis-Hastings, DRAM, posterior distributions, convergence diagnostics

Problem	Description	MATLAB	Python
1	Posterior comparison for heat equation	Problem1.m	Problem1.py
2	DRAM vs Metropolis for SIR model	Problem2.m	Problem2.py
3	MCMC for Helmholtz model	Problem3.m	Problem3.py

Project Writeup (PDF)

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Project 4: Uncertainty Propagation

Topics: Confidence intervals, credible intervals, prediction intervals

Problem	Description	MATLAB	Python
1	Frequentist intervals for height-weight model	Problem1.m	Problem1.py
2	Bayesian intervals for aluminum rod	Problem2.m	Problem2.py
3	SIR model credible intervals	Problem3a.m	Problem3.py
4	Frequentist vs Bayesian comparison	Problem4.m	Problem4.py

Project Writeup (PDF)

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Project 5: Surrogate Models

Topics: Polynomial surrogates, Latin hypercube sampling, Gaussian processes

Problem	Description	MATLAB	Python
1	Polynomial surrogate with LHS	Problem1.m	Problem1.py
2	Legendre surrogate model	Problem2.m	Problem2.py
3	Gaussian process regression	Problem3.m	Problem3.py

Project Writeup (PDF)

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Project 6: Model Discrepancy

Topics: Physical vs surrogate model comparison, Dittus-Boelter equation

Problem	Description	MATLAB	Python
1	Dittus-Boelter equation analysis	Final.m	Final.py

Project Writeup (PDF)

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Key Concepts Implemented

Sensitivity Analysis Methods

• Finite Difference: First-order approximation of derivatives
• Complex-Step: Machine-precision derivative approximation
• Morris Screening: Efficient global sensitivity screening
• Sobol Indices: Variance-based sensitivity measures

MCMC Algorithms

• Metropolis-Hastings: Standard random-walk sampler
• Adaptive Metropolis: Self-tuning proposal distribution
• DRAM: Delayed Rejection Adaptive Metropolis

Surrogate Modeling

• Polynomial Regression: Least squares fitting
• Latin Hypercube Sampling: Space-filling experimental design
• Gaussian Process: Non-parametric probabilistic model

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Getting Started

MATLAB

% Navigate to project folder
cd 'Project 1'
% Run a script
UQ_8_5

Python

# Install dependencies
pip install -r requirements.txt

# Run a script
cd "Project 1"
python UQ_8_5.py

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Learning Outcomes

After studying these materials, you will be able to:

• Compute model sensitivities using analytical and numerical methods
• Identify estimable parameter subsets using Fisher information
• Perform global sensitivity analysis with Morris and Sobol methods
• Estimate parameters using frequentist and Bayesian approaches
• Implement MCMC algorithms for posterior sampling
• Construct confidence, credible, and prediction intervals
• Build surrogate models for computational efficiency
• Quantify model discrepancy and prediction uncertainty

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License

This project is licensed under the MIT License - see the LICENSE file for details.