Uncertainty Quantification

Probability Theory

• Sample Space vs Event
• Random Variable
• Random Variables, Vectors and Fields
• Expected Value
• Normal Distribution and Log-normal Distribution
• Probability Density Function (PDF)
• Cumulative Distribution Function (CDF)
• Relationship between CDF and PDF
• Moment of a Random Variable
• Moment Generation Function
• Covariance
• Correlation
• Variance of a Uniform Distribution
• Quick way to find Eigenvalues
• Quick way to find Eigenvectors
• Shortcuts for finding Eigenvalues & Eigenvectors
• Orthogonality
• Orthogonal Polynomials

Karhunen-Loève Expansion

• Multivariate Gaussian PDF
• Karhunen-Loève Expansion
  • Discrete KLE Example - a
  • Discrete KLE Example - b
  • Continuous KLE Example

First Order Second Moment Method

• FOSM
  • Approximating Mean
  • Approximating Variance
• FOSM Example

Quadrature

• Quadrature to Approximate Expected Value
• Quadrature Example
• Curse of Dimensionality in Quadrature
• Flashcards

Surrogate Modeling

• Definition of Surrogate Modeling
• Polynomial Chaos Expansion (gPC)
• Orthogonal Polynomials in gPC Expansions
• Expected Value and Variance Calculations using PCE
  • Multi-Dimensional Polynomial Chaos Expansion
  • gPC Example
  • Using KLE to prepare Random Variables for GPC
• non-Intrusive Projection

Sensitivity Analysis

• Sobol Indices
• Calculating Sobol Indices
• Flashcards

Monte Carlo Methods

• Monte Carlo Methods
  • Approximating Expectation
  • Approximating Variance
  • Mean-Square Error
• Reducing Errors in Monte Carlo Methods
  • Variance Reduction Method
  • Control Variate Method
  • Multilevel Monte Carlo (MLMC)

Misc

• Overview of Uncertainty Quantification
• UQ Applications in Engineering Scenarios
• Formulary
• Practice Problems
• Exam