"""
Plot Smolyak multi-indices
==========================
"""
# %%
# 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`.
# %%
import openturns as ot
import openturns.experimental as otexp
import openturns.viewer as otv
from matplotlib import pylab as plt
# %%
# In the first example, we print the indices involved in
# Smolyak-Legendre quadrature of level 3.
# The multi-indices are computed using the
# :py:meth:`openturns.experimental.SmolyakExperiment.computeCombination` method.
# Actually, the multi-indices do not actually depend on the
# underlying univariate quadratures, but this is required for
# the :class:`openturns.experimental.SmolyakExperiment` class.
collection = [ot.GaussProductExperiment()] * 2
level = 3
print("level = ", level)
experiment = otexp.SmolyakExperiment(collection, level)
indices = experiment.computeCombination()
print(indices)
# %%
# 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.
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
# %%
# 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.
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()
# %%
# 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.
# %%
plt.show()