- [4.2.1 **What do we want to know?**](https://ndcbe.github.io/cbe67701-uncertainty-quantification/04.02-Simple-ADR-Example.html#4.2.1-**What-do-we-want-to-know?**)
- [4.2.2 **Why do we want to know that?**](https://ndcbe.github.io/cbe67701-uncertainty-quantification/04.02-Simple-ADR-Example.html#4.2.2-**Why-do-we-want-to-know-that?**)
- [4.2.3 **4.3.1 Simple ADR Example**](https://ndcbe.github.io/cbe67701-uncertainty-quantification/04.02-Simple-ADR-Example.html#4.2.3-**4.3.1-Simple-ADR-Example**)

Chapter 5.0 Regression Approximations to Estimate Sensitivities ¶

5.1 Ridge Regression ¶

5.1.1 Sensitivities with Least-Squares Regression
5.1.2 Ridge Regression
5.1.3 Cross Validation

5.1.4 Example Problem

  - [5.1.4.1 I < J](https://ndcbe.github.io/cbe67701-uncertainty-quantification/05.01-Contributed-Example.html#5.1.4.1-I-<-J)

5.1.5 Ridge Regression with SKLearn

5.2 Lasso Regression ¶

5.3 Elastic Net Regression ¶

Chapter 6.0 Adjoint-Based Local Sensitivity Analysis ¶

6.1 Nonlinear Diffusion-Reaction Equation ¶

6.2 A Simple Example of Adjoint Sensitivity Analysis ¶

6.3 Sensitivity Analysis with Adjoint Operators ¶

6.3.1 Review of inner products
6.3.2 Sensitivity Analysis of Linear equations
6.3.3 Case 1. Perturbations in the RHS
- 6.3.3.1 Example: Multi-product plant
- 6.3.3.2 Solution
6.3.4 Case 2. Sensitivity to changes in equation coefficients
- 6.3.4.1 Example
- 6.3.4.2 Solution
6.3.5 Case 3. Generic Case
6.3.6 Systems of first-order linear differential equations
6.3.7 Application: Pharmacokinetics
```
  - [6.3.7.1 Linear pharmacokinetics](https://ndcbe.github.io/cbe67701-uncertainty-quantification/06.03-Sensitivity-Analysis-with-Adjoint-Operators.html#6.3.7.1-Linear-pharmacokinetics)
```
6.3.8 Systems of first-order nonlinear differential equations
- 6.3.8.1 Example: Over-damped oscillator
6.3.9 Systems of first-order differential equations
Markdown Links

6.4 Adjoint Sensitivity Notes on Numerical Computation ¶

Chapter 7.0 Sampling-Based Uncertainty Quantification: Monte Carlo and Beyond ¶

7.1 Latin Hypercube and Quasi-Monte Carlo Sampling ¶

7.1.1 Latin hypercube sampling
7.1.2 Quasi MC: Halton’s low discrepency sequences
7.1.3 Comparison (1 dimensional))
Markdown Figures
- ./figures/LHS.png
Markdown Links
- low discrepancy (or quasi-random) sequence
- Halton sequence

7.2 Latin Hypercube Sampling ¶

7.3 Meaningful Title Goes Here ¶

Chapter 8.0 Reliability Methods for Estimating the Probability of Failure ¶

Chapter 9.0 Stochastic Projection and Collocation ¶

9.1 Hermite Expansion for Normally Distributed Parameters ¶

9.2 Uniform Random Variables: Legendre Polynomials ¶

9.2.1 Generalized Polynomial Chaos
9.2.2 Example: $G \sim g(x) = cos(x)$ and $x \sim \mathcal{U}(0,2\pi)$-=-cos(x)$-and-$x-\sim-\mathcal{U}(0,2\pi)$)
9.2.3 Importing libraries
9.2.4 Functions to calculate exact and $n^{\mathrm{th}}$-order Legendre polynomial approximation of $g(x)$$)
9.2.5 Function to calculate $\mathrm{Var}(g(x))$, $E[G]$, and $\mathrm{Var}(G)$)$,-$E[G]$,-and-$\mathrm{Var}(G)$)
9.2.6 Generating samples from $\mathcal{U}[0,2\pi]$
9.2.7 Convergence of variance for $g(x) = cos(x)$, where $ x \sim \mathcal{U} (0, 2\pi )$-=-cos(x)$,-where-$-x-\sim-\mathcal{U}-(0,-2\pi-)$)
9.2.8 Approximate $g(x)$ using different orders of Legendre polynomials$-using-different-orders-of-Legendre-polynomials)
9.2.9 Plot PDF of the random variable $g(x) = \cos(x)$, where $x \sim \mathcal{U}(0, 2\pi)$, and various approximations-=-\cos(x)$,-where-$x-\sim-\mathcal{U}(0,-2\pi)$,-and-various-approximations)
Markdown Figures
- Figure1
Markdown Links
- https://link.springer.com/chapter/10.1007/978-3-319-99525-0_9
- https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.eval_legendre.html

Chapter 10.0 Gaussian Process Emulators and Surrogate Models ¶

10.1 Using GPflow package for Gaussian Process Regression ¶

10.2 A simple example of Bayesian quadrature ¶

- [10.2.1 Bayesian quadrature uses Gaussian process regression as the approximation to the integrand](https://ndcbe.github.io/cbe67701-uncertainty-quantification/10.02-Bayesian-quadrature.html#10.2.1-Bayesian-quadrature-uses-Gaussian-process-regression-as-the-approximation-to-the-integrand)
- [10.2.2 The procedures of Bayesian Quadrature](https://ndcbe.github.io/cbe67701-uncertainty-quantification/10.02-Bayesian-quadrature.html#10.2.2-The-procedures-of-Bayesian-Quadrature)
- [10.2.3 Implementation](https://ndcbe.github.io/cbe67701-uncertainty-quantification/10.02-Bayesian-quadrature.html#10.2.3-Implementation)
    - [10.2.3.1 Take the GPR model as the surrogate](https://ndcbe.github.io/cbe67701-uncertainty-quantification/10.02-Bayesian-quadrature.html#10.2.3.1-Take-the-GPR-model-as-the-surrogate)
- [10.2.4 Result](https://ndcbe.github.io/cbe67701-uncertainty-quantification/10.02-Bayesian-quadrature.html#10.2.4-Result)
    - [10.2.4.1 Find the next quadrature point by minimizing the posterior variance](https://ndcbe.github.io/cbe67701-uncertainty-quantification/10.02-Bayesian-quadrature.html#10.2.4.1-Find-the-next-quadrature-point-by-minimizing-the-posterior-variance)

10.3 Using scikit-learn for Gaussian Process Regression¶

Chapter 11.0 Predictive Models Informed by Simulation, Measurement, and Surrogates ¶

11.1 Gibbs Sampling to Approximate Bayes' Integral ¶

- [11.1.1 Example: Multivariate distribution](https://ndcbe.github.io/cbe67701-uncertainty-quantification/11.01-Gibbs-Sampling.html#11.1.1-Example:-Multivariate-distribution)
- [11.1.2 When will the method break?](https://ndcbe.github.io/cbe67701-uncertainty-quantification/11.01-Gibbs-Sampling.html#11.1.2-When-will-the-method-break?)