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

.. only:: html

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

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

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

.. _sphx_glr_auto_reliability_sensitivity_design_of_experiments_plot_smolyak_indices.py:


Plot Smolyak multi-indices
==========================

.. GENERATED FROM PYTHON SOURCE LINES 6-25

The goal of this example is to plot the multi-indices used in Smolyak's
quadrature.

For a given dimension :math:`d_x \in \mathbb{N}` and a given level
:math:`\ell \in \mathbb{N}`
Smolyak's quadrature is the combination of tensorized univariate quadratures.
These quadrature are defined by the set of multi-indices:

.. math::

    \mathcal{S}_{\ell, d_x} = \left\{\|\boldsymbol{k}\|_1 \leq \ell
                                     + d_x - 1\right\}

where :math:`\|\boldsymbol{k}\|_1 = k_1 + ... + k_{d_x}` is the 1-norm of the
multi-index :math:`\boldsymbol{k} \in \mathbb{N}^{d_x}`.

The goal of this script is to plot the multi-indices involved in Smolyak's
quadrature for different values of the level :math:`\ell` in
dimension :math:`d_x = 2`.

.. GENERATED FROM PYTHON SOURCE LINES 27-32

.. code-block:: Python

    import openturns as ot
    import openturns.experimental as otexp
    import openturns.viewer as otv
    from matplotlib import pylab as plt








.. GENERATED FROM PYTHON SOURCE LINES 33-40

In the first example, we print the indices involved in
Smolyak-Legendre quadrature of level 3.
The multi-indices are computed using the
:meth:`~openturns.SmolyakExperiment.computeCombination` method.
Actually, the multi-indices do not actually depend on the
underlying univariate quadratures, but this is required for
the :class:`~openturns.SmolyakExperiment` class.

.. GENERATED FROM PYTHON SOURCE LINES 40-48

.. code-block:: Python


    collection = [ot.GaussProductExperiment()] * 2
    level = 3
    print("level = ", level)
    experiment = otexp.SmolyakExperiment(collection, level)
    indices = experiment.computeCombination()
    print(indices)





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

 .. code-block:: none

    level =  3
    [[2,1],[1,2],[3,1],[2,2],[1,3]]




.. GENERATED FROM PYTHON SOURCE LINES 49-55

We see that the multi-indices have a sum which is equal to either 3 or 4.
In other words, these multi-indices belong to two different layers of
constant 1-norms.

In order to see how this evolves depending on the level of the quadrature,
the following function creates a 2D plot of the set of multi-indices.

.. GENERATED FROM PYTHON SOURCE LINES 55-70

.. code-block:: Python



    def drawSmolyakIndices(level):
        # Plot Smolyak indices of given level in 2 dimensions
        collection = [ot.GaussProductExperiment()] * 2
        experiment = otexp.SmolyakExperiment(collection, level)
        indices = experiment.computeCombination()
        sample = indices
        graph = ot.Graph("L = %d" % (level), "k1", "k2", True)
        cloud = ot.Cloud(sample)
        cloud.setPointStyle("bullet")
        graph.add(cloud)
        return graph









.. GENERATED FROM PYTHON SOURCE LINES 71-75

In the following script, we create a grid of plots, where
each graph corresponds to a given quadrature level.
The bounding box of each graph is set to a constant value, so that all graphs
have the same X and Y bounds.

.. GENERATED FROM PYTHON SOURCE LINES 75-96

.. code-block:: Python


    levelMax = 8.0
    boundingBox = ot.Interval([0.0] * 2, [levelMax] * 2)
    nbrows = 2
    nbcols = 3
    grid = ot.GridLayout(nbrows, nbcols)
    level = 1
    for i in range(nbrows):
        for j in range(nbcols):
            graph = drawSmolyakIndices(level)
            if i < nbrows - 1:
                graph.setXTitle("")
            if j > 0:
                graph.setYTitle("")
            graph.setBoundingBox(boundingBox)
            grid.setGraph(i, j, graph)
            level += 1
    view = otv.View(grid, figure_kw={"figsize": (5.0, 3.0)})
    plt.subplots_adjust(wspace=0.5, hspace=0.7)
    plt.tight_layout()




.. image-sg:: /auto_reliability_sensitivity/design_of_experiments/images/sphx_glr_plot_smolyak_indices_001.png
   :alt: , L = 1, L = 2, L = 3, L = 4, L = 5, L = 6
   :srcset: /auto_reliability_sensitivity/design_of_experiments/images/sphx_glr_plot_smolyak_indices_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 97-101

We see that when the level increases, the set of Smolyak multi-indices
correspond to two different layers of constant 1-norm.
This is a consequence of Smolyak's quadrature, which is based on
tensorization of univariate difference quadratures.

.. GENERATED FROM PYTHON SOURCE LINES 103-104

.. code-block:: Python

    plt.show()








.. _sphx_glr_download_auto_reliability_sensitivity_design_of_experiments_plot_smolyak_indices.py:

.. only:: html

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

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

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

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

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