# Theoretical Tutorials

Below is a gallery of tutorials providing detailed mathematical background on the methods in PyApprox.

This tutorials provide more detail than the set of examples found here which simply show how to use different methods with the least amount of code.

## Model Analysis

Below are tutorials on various model analysis techniques

Sensitivity Analysis

Sensitivity Analysis

## Inference

Below is a gallery of foundational tutorials probabilistic inversion

Bayesian Inference

Bayesian Inference

Gaussian Networks

Gaussian Networks

Push Forward Based Inference

Push Forward Based Inference

## Experimental Design

The next release will contain a gallery of foundational tutorials on experimental design

## Surrogates

Univariate Interpolation

Univariate Interpolation

Tensor-product Interpolation

Tensor-product Interpolation

Sparse Grids

Sparse Grids

Gaussian processes

Gaussian processes

## Multi-Fidelity Methods

Multi-fidelity methods utilize an ensemble of models, enriching a small number of high-fidelity simulations with larger numbers of simulations from models of varying prediction accuracy and reduced cost, to enable greater exploration and resolution of uncertainty while maintaining deterministic prediction accuracy. The effectiveness of multi-fidelity approaches depends on the ability to identify and exploit relationships among models within the ensemble.

The relationships among models within a model ensemble vary greatly, and most existing approaches focus on exploiting a specific type of structure for a presumed model sequence. For example, [KOB2000], [LGIJUQ2014], [NGXSISC2014], [TJWGSIAMUQ2015] build surrogate approximations that leverage a 1D hierarchy of models of increasing fidelity, with varying physics and/or numerical discretizations. While Multi-index collocation [HNTTCMAME2016] leverage a multi-dimensional hierarchy controlled my two or more numerical discretization hyper-parameters. Similary [CGSTCVS2011], [GOR2008] exploit a 1D hierarchy of models to estimate statistics such as mean and variance using Monte Carlo methods.

This gallery of tutorials discusses the most popular multi-fidelity methods for quantifying uncertainty in complex models.

Monte Carlo Quadrature: Beyond Mean Estimation

Monte Carlo Quadrature: Beyond Mean Estimation

Two Model Control Variate Monte Carlo

Two Model Control Variate Monte Carlo

Two model Approximate Control Variate Monte Carlo

Two model Approximate Control Variate Monte Carlo

Approximate Control Variate Monte Carlo

Approximate Control Variate Monte Carlo

Delta-Based Covariance Formulas For Approximate Control Variates

Delta-Based Covariance Formulas For Approximate Control Variates

Approximate Control Variate Allocation Matrices

Approximate Control Variate Allocation Matrices

Multi-level Monte Carlo

Multi-level Monte Carlo

Multi-fidelity Monte Carlo

Multi-fidelity Monte Carlo

Parametrically Defined Approximate Control Variates

Parametrically Defined Approximate Control Variates

Multioutput Approximate Control Variates

Multioutput Approximate Control Variates

Pilot Studies

Pilot Studies

Model Ensemble Selection

Model Ensemble Selection

Multilevel Best Linear Unbiased estimators (MLBLUE)

Multilevel Best Linear Unbiased estimators (MLBLUE)

Multi-level and Multi-index Collocation

Multi-level and Multi-index Collocation

Multifidelity Gaussian processes

Multifidelity Gaussian processes

MFNets: Multi-fidelity networks

MFNets: Multi-fidelity networks

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