.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/plot_multiindex_collocation_ex.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_auto_examples_plot_multiindex_collocation_ex.py>`
        to download the full example code

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_auto_examples_plot_multiindex_collocation_ex.py:


Multi-index Collocation
-----------------------
The following provides an example of how to use multivariate quadrature, e.g. multilevel Monte Carlo, control variates to estimate the mean of a high-fidelity model from an ensemble of related models of varying cost and accuracy. Refer to :ref:`sphx_glr_auto_tutorials_multi_fidelity_plot_multiindex_collocation.py` for a detailed tutorial on the theory behind multi-index collocation.

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Load the necessary modules

.. GENERATED FROM PYTHON SOURCE LINES 8-17

.. code-block:: default

    from scipy import stats
    import numpy as np
    import matplotlib.pyplot as plt
    from functools import partial
    from pyapprox.benchmarks.benchmarks import setup_benchmark
    from pyapprox import interface, variables, surrogates
    # set seed for reproducibility
    np.random.seed(1)








.. GENERATED FROM PYTHON SOURCE LINES 18-19

Set up a :py:class:`~pyapprox.interface.wrappers.MultiIndexModel` that takes samples :math:`x=[z_1,\ldots,z_D,v_1,\ldots, v_C]` which is the concatenation of a random sample `z` and configuration values specifying the discretization parameters of the numerical model.

.. GENERATED FROM PYTHON SOURCE LINES 19-35

.. code-block:: default


    marginals = [stats.uniform(-2, 3)]
    variable = variables.IndependentMarginalsVariable(marginals)


    def fun(eps, zz):
        return np.cos(np.pi*(zz[0]+1)+eps)[:, None]


    def setup_fun(eps):
        return partial(fun, eps)


    config_values = [np.asarray([0.25, 0.125, 0])]
    model = interface.MultiIndexModel(setup_fun, config_values)








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Now set up and run the multi-index algorithm

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.. code-block:: default

    config_var_trans = variables.ConfigureVariableTransformation(config_values)
    mi_result = surrogates.adaptive_approximate(
        model, variable, "sparse_grid",
        {"max_nsamples": 20, "config_var_trans": config_var_trans,
         "max_level_1d": ([np.inf for ii in range(variable.num_vars())] +
                          [len(cv)-1 for cv in config_var_trans.config_values]),
         "config_variables_idx": variable.num_vars(),
         "cost_function": lambda config_sample: 2**config_sample[0],
         "tol": 1e-3, "verbose": 1})
    mi_approx = mi_result.approx





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    Cannot add subspace [0 3]
    Max level of 2 reached in variable 1
    Max num evaluations (20) reached
    Error estimate 0.5336307283210873
    Max num evaluations (20) reached
    Error estimate 0.19831215919512413
    Max num evaluations (20) reached
    Error estimate 0.0831164889209195




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The following can be used to plot the approximation and high-fidelity target function if there is only one random variable z and one configuration variable

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.. code-block:: default

    ax = plt.subplots()[1]
    zz = np.linspace(*variable.get_statistics("interval", 1.0)[0], 101)[None, :]
    ax.plot(zz[0], mi_approx(zz), label="MF")
    ax.plot(zz[0], fun(0, zz), label='HF')
    mi_zz_samples = mi_approx.var_trans.map_from_canonical(
        mi_approx.samples[:variable.num_vars()])
    for ii in range(config_var_trans.config_values[0].shape[0]):
        II = np.where(mi_approx.samples[-1, :] == ii)[0]
        ax.plot(mi_zz_samples[0, II], mi_approx.values[II, 0], 'o',
                label=r"$f_{0}$".format(ii+1))
    ax.legend()
    plt.show()



.. image-sg:: /auto_examples/images/sphx_glr_plot_multiindex_collocation_ex_001.png
   :alt: plot multiindex collocation ex
   :srcset: /auto_examples/images/sphx_glr_plot_multiindex_collocation_ex_001.png
   :class: sphx-glr-single-img






.. rst-class:: sphx-glr-timing

   **Total running time of the script:** ( 0 minutes  0.065 seconds)


.. _sphx_glr_download_auto_examples_plot_multiindex_collocation_ex.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example




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

      :download:`Download Python source code: plot_multiindex_collocation_ex.py <plot_multiindex_collocation_ex.py>`

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

      :download:`Download Jupyter notebook: plot_multiindex_collocation_ex.ipynb <plot_multiindex_collocation_ex.ipynb>`


.. only:: html

 .. rst-class:: sphx-glr-signature

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