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#####################
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.



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Model Analysis
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Below are tutorials on various model analysis techniques



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    <div class="sphx-glr-thumbcontainer" tooltip="Sobol Indices ------------- Any function f with finite variance parameterized by a set of indep...">

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  .. image:: /auto_tutorials/analysis/images/thumb/sphx_glr_plot_sensitivity_analysis_thumb.png
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  :ref:`sphx_glr_auto_tutorials_analysis_plot_sensitivity_analysis.py`

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      <div class="sphx-glr-thumbnail-title">Sensitivity Analysis</div>
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.. toctree::
   :hidden:

   /auto_tutorials/analysis/plot_sensitivity_analysis

*********
Inference
*********
Below is a gallery of foundational tutorials probabilistic inversion



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    <div class="sphx-glr-thumbcontainer" tooltip="Background ---------- When observational data are available, that data should be used to inform...">

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  .. image:: /auto_tutorials/inference/images/thumb/sphx_glr_plot_bayesian_inference_thumb.png
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  :ref:`sphx_glr_auto_tutorials_inference_plot_bayesian_inference.py`

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      <div class="sphx-glr-thumbnail-title">Bayesian Inference</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Directed Acyclic Graphs ----------------------- A Bayesian network (BN) structure is a directed...">

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  .. image:: /auto_tutorials/inference/images/thumb/sphx_glr_plot_bayesian_networks_thumb.png
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  :ref:`sphx_glr_auto_tutorials_inference_plot_bayesian_networks.py`

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      <div class="sphx-glr-thumbnail-title">Gaussian Networks</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="PFI solves the inverse problem of inferring parameters \rv of a deterministic model f(\rv) from...">

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  .. image:: /auto_tutorials/inference/images/thumb/sphx_glr_plot_push_forward_based_inference_thumb.png
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  :ref:`sphx_glr_auto_tutorials_inference_plot_push_forward_based_inference.py`

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      <div class="sphx-glr-thumbnail-title">Push Forward Based Inference</div>
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.. toctree::
   :hidden:

   /auto_tutorials/inference/plot_bayesian_inference
   /auto_tutorials/inference/plot_bayesian_networks
   /auto_tutorials/inference/plot_push_forward_based_inference

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



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**********
Surrogates
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    <div class="sphx-glr-thumbcontainer" tooltip="This tutorial will present methods to approximate univariate functions \hat{f}_{\alpha}:\reals\...">

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  .. image:: /auto_tutorials/surrogates/images/thumb/sphx_glr_plot_univariate_interpolation_thumb.png
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  :ref:`sphx_glr_auto_tutorials_surrogates_plot_univariate_interpolation.py`

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      <div class="sphx-glr-thumbnail-title">Univariate Interpolation</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Generally speaking surrogates are built using a &quot;small&quot; number of model simulations and are the...">

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  .. image:: /auto_tutorials/surrogates/images/thumb/sphx_glr_plot_tensor_product_interpolation_thumb.png
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  :ref:`sphx_glr_auto_tutorials_surrogates_plot_tensor_product_interpolation.py`

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      <div class="sphx-glr-thumbnail-title">Tensor-product Interpolation</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Sparse grids approximate a model (function) f_\alpha with D inputs z=[z_1,\ldots,z_D]^\top as a...">

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  .. image:: /auto_tutorials/surrogates/images/thumb/sphx_glr_plot_sparse_grids_thumb.png
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  :ref:`sphx_glr_auto_tutorials_surrogates_plot_sparse_grids.py`

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      <div class="sphx-glr-thumbnail-title">Sparse Grids</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Adaptive Leja Sequences">

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  .. image:: /auto_tutorials/surrogates/images/thumb/sphx_glr_plot_adaptive_leja_interpolation_thumb.png
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  :ref:`sphx_glr_auto_tutorials_surrogates_plot_adaptive_leja_interpolation.py`

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      <div class="sphx-glr-thumbnail-title">Adaptive Leja Sequences</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Constructing a GP requires specifying a prior mean m(\rv) and covariance kernel C(\rv, \rv^\sta...">

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  .. image:: /auto_tutorials/surrogates/images/thumb/sphx_glr_plot_gaussian_processes_thumb.png
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  :ref:`sphx_glr_auto_tutorials_surrogates_plot_gaussian_processes.py`

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      <div class="sphx-glr-thumbnail-title">Gaussian processes</div>
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.. toctree::
   :hidden:

   /auto_tutorials/surrogates/plot_univariate_interpolation
   /auto_tutorials/surrogates/plot_tensor_product_interpolation
   /auto_tutorials/surrogates/plot_sparse_grids
   /auto_tutorials/surrogates/plot_adaptive_leja_interpolation
   /auto_tutorials/surrogates/plot_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.

.. [NGXSISC2014] `Narayan, A. and Gittelson, C. and Xiu, D. A Stochastic Collocation Algorithm with Multifidelity Models. SIAM Journal on Scientific Computing 36(2), A495-A521, 2014. <https://doi.org/10.1137/130929461>`_
		 







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    <div class="sphx-glr-thumbcontainer" tooltip="using Monte Carlo quadrature applied to an approximation f_\alpha of the function f, e.g. a rep...">

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  .. image:: /auto_tutorials/multi_fidelity/images/thumb/sphx_glr_plot_monte_carlo_thumb.png
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  :ref:`sphx_glr_auto_tutorials_multi_fidelity_plot_monte_carlo.py`

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      <div class="sphx-glr-thumbnail-title">Monte Carlo Quadrature</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="In this tutorial we will discuss using MC to estimate the mean \mat{\mu} and covariance \mat{\S...">

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  .. image:: /auto_tutorials/multi_fidelity/images/thumb/sphx_glr_plot_multioutput_monte_carlo_thumb.png
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  :ref:`sphx_glr_auto_tutorials_multi_fidelity_plot_multioutput_monte_carlo.py`

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      <div class="sphx-glr-thumbnail-title">Monte Carlo Quadrature: Beyond Mean Estimation</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Let us introduce a model f_\kappa with known statistic Q_{\kappa}. We can use this model to est...">

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  .. image:: /auto_tutorials/multi_fidelity/images/thumb/sphx_glr_plot_control_variate_monte_carlo_thumb.png
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  :ref:`sphx_glr_auto_tutorials_multi_fidelity_plot_control_variate_monte_carlo.py`

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      <div class="sphx-glr-thumbnail-title">Two Model Control Variate Monte Carlo</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="CVMC is often not useful for practical analysis of numerical models because typically the stati...">

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  .. image:: /auto_tutorials/multi_fidelity/images/thumb/sphx_glr_plot_approximate_control_variates_thumb.png
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  :ref:`sphx_glr_auto_tutorials_multi_fidelity_plot_approximate_control_variates.py`

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      <div class="sphx-glr-thumbnail-title">Two model Approximate Control Variate Monte Carlo</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="The approximate control variate estimator of a vector-valued statistic \mat{Q}_0\in\reals^{S} t...">

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  .. image:: /auto_tutorials/multi_fidelity/images/thumb/sphx_glr_plot_many_model_acv_thumb.png
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  :ref:`sphx_glr_auto_tutorials_multi_fidelity_plot_many_model_acv.py`

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      <div class="sphx-glr-thumbnail-title">Approximate Control Variate Monte Carlo</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="These covariancesdepend on how samples are allocated to the sets \rvset_\alpha,\rvset_\alpha^, ...">

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  .. image:: /auto_tutorials/multi_fidelity/images/thumb/sphx_glr_acv_covariances_thumb.png
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  :ref:`sphx_glr_auto_tutorials_multi_fidelity_acv_covariances.py`

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      <div class="sphx-glr-thumbnail-title">Delta-Based Covariance Formulas For Approximate Control Variates</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Optimizing and constructing PACV estimators requires estimating \covar{Q_0}{\Delta} and \covar{...">

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  .. image:: /auto_tutorials/multi_fidelity/images/thumb/sphx_glr_plot_allocation_matrices_thumb.png
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  :ref:`sphx_glr_auto_tutorials_multi_fidelity_plot_allocation_matrices.py`

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      <div class="sphx-glr-thumbnail-title">Approximate Control Variate Allocation Matrices</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Two Model MLMC -------------- The two model MLMC estimator is based upon the observations that ...">

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  .. image:: /auto_tutorials/multi_fidelity/images/thumb/sphx_glr_plot_multi_level_monte_carlo_thumb.png
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  :ref:`sphx_glr_auto_tutorials_multi_fidelity_plot_multi_level_monte_carlo.py`

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      <div class="sphx-glr-thumbnail-title">Multi-level Monte Carlo</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="To derive the MFMC estimator first recall the two model ACV estimator">

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  .. image:: /auto_tutorials/multi_fidelity/images/thumb/sphx_glr_plot_multi_fidelity_monte_carlo_thumb.png
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  :ref:`sphx_glr_auto_tutorials_multi_fidelity_plot_multi_fidelity_monte_carlo.py`

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      <div class="sphx-glr-thumbnail-title">Multi-fidelity Monte Carlo</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="sphx_glr_auto_tutorials_multi_fidelity_plot_multi_fidelity_monte_carlo.py) demonstrates that MF...">

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  .. image:: /auto_tutorials/multi_fidelity/images/thumb/sphx_glr_plot_pacv_thumb.png
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  :ref:`sphx_glr_auto_tutorials_multi_fidelity_plot_pacv.py`

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      <div class="sphx-glr-thumbnail-title">Parametrically Defined Approximate Control Variates</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="This tutorial demonstrates how computing statistics for multiple outputs simultaneoulsy can imp...">

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  .. image:: /auto_tutorials/multi_fidelity/images/thumb/sphx_glr_plot_multioutput_acv_thumb.png
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  :ref:`sphx_glr_auto_tutorials_multi_fidelity_plot_multioutput_acv.py`

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      <div class="sphx-glr-thumbnail-title">Multioutput Approximate Control Variates</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="This tutorial presents how to use a pilot-study to compute the statistics needed to compute a M...">

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  .. image:: /auto_tutorials/multi_fidelity/images/thumb/sphx_glr_plot_pilot_studies_thumb.png
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  :ref:`sphx_glr_auto_tutorials_multi_fidelity_plot_pilot_studies.py`

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      <div class="sphx-glr-thumbnail-title">Pilot Studies</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="The following tutorial shows how to choose the best subset of models.">

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  .. image:: /auto_tutorials/multi_fidelity/images/thumb/sphx_glr_plot_ensemble_selection_thumb.png
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  :ref:`sphx_glr_auto_tutorials_multi_fidelity_plot_ensemble_selection.py`

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      <div class="sphx-glr-thumbnail-title">Model Ensemble Selection</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="   * - .. _MLBLUE-sample-allocation:">

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  .. image:: /auto_tutorials/multi_fidelity/images/thumb/sphx_glr_plot_multilevel_blue_thumb.png
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  :ref:`sphx_glr_auto_tutorials_multi_fidelity_plot_multilevel_blue.py`

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      <div class="sphx-glr-thumbnail-title">Multilevel Best Linear Unbiased estimators (MLBLUE)</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Models often utilize numerical discretizations to solve the equations governing the system dyna...">

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  .. image:: /auto_tutorials/multi_fidelity/images/thumb/sphx_glr_plot_multiindex_collocation_thumb.png
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  :ref:`sphx_glr_auto_tutorials_multi_fidelity_plot_multiindex_collocation.py`

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      <div class="sphx-glr-thumbnail-title">Multi-level and Multi-index Collocation</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Multilevel GPs assume that all the available models \{f_m\}_{m=1}^M can be ordered into a hiera...">

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  .. image:: /auto_tutorials/multi_fidelity/images/thumb/sphx_glr_plot_multifidelity_gp_thumb.png
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  :ref:`sphx_glr_auto_tutorials_multi_fidelity_plot_multifidelity_gp.py`

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      <div class="sphx-glr-thumbnail-title">Multifidelity Gaussian processes</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="In the sphx_glr_auto_tutorials_multi_fidelity_plot_multiindex_collocation.py tutorial we showed...">

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  .. image:: /auto_tutorials/multi_fidelity/images/thumb/sphx_glr_plot_gaussian_mfnets_thumb.png
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  :ref:`sphx_glr_auto_tutorials_multi_fidelity_plot_gaussian_mfnets.py`

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      <div class="sphx-glr-thumbnail-title">MFNets: Multi-fidelity networks</div>
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.. toctree::
   :hidden:

   /auto_tutorials/multi_fidelity/plot_monte_carlo
   /auto_tutorials/multi_fidelity/plot_multioutput_monte_carlo
   /auto_tutorials/multi_fidelity/plot_control_variate_monte_carlo
   /auto_tutorials/multi_fidelity/plot_approximate_control_variates
   /auto_tutorials/multi_fidelity/plot_many_model_acv
   /auto_tutorials/multi_fidelity/acv_covariances
   /auto_tutorials/multi_fidelity/plot_allocation_matrices
   /auto_tutorials/multi_fidelity/plot_multi_level_monte_carlo
   /auto_tutorials/multi_fidelity/plot_multi_fidelity_monte_carlo
   /auto_tutorials/multi_fidelity/plot_pacv
   /auto_tutorials/multi_fidelity/plot_multioutput_acv
   /auto_tutorials/multi_fidelity/plot_pilot_studies
   /auto_tutorials/multi_fidelity/plot_ensemble_selection
   /auto_tutorials/multi_fidelity/plot_multilevel_blue
   /auto_tutorials/multi_fidelity/plot_multiindex_collocation
   /auto_tutorials/multi_fidelity/plot_multifidelity_gp
   /auto_tutorials/multi_fidelity/plot_gaussian_mfnets


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    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download all examples in Python source code: auto_tutorials_python.zip </auto_tutorials/auto_tutorials_python.zip>`

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download all examples in Jupyter notebooks: auto_tutorials_jupyter.zip </auto_tutorials/auto_tutorials_jupyter.zip>`


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    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_