plot_stacking.py

"""
Operators concatenation
=======================

This example shows how to use 'stacking' operators such as
:py:class:`pylops.VStack`, :py:class:`pylops.HStack`,
:py:class:`pylops.Block`, :py:class:`pylops.BlockDiag`,
and :py:class:`pylops.Kronecker`.

These operators allow for different combinations of multiple linear operators
in a single operator. Such functionalities are used within PyLops as the basis
for the creation of complex operators as well as in the definition of various
types of optimization problems with regularization or preconditioning.

Some of this operators naturally lend to embarassingly parallel computations.
Within PyLops we leverage the
`Multiprocessing <https://docs.python.org/3/library/multiprocessing.html>`_
module to run multiple processes at the same time evaluating a subset of the
operators involved in one of the stacking operations.
"""
import matplotlib.pyplot as plt
import numpy as np

import pylops

plt.close("all")

###############################################################################
# Let's start by defining two second derivatives :py:class:`pylops.SecondDerivative`
# that we will be using in this example.
D2hop = pylops.SecondDerivative(dims=(11, 21), axis=1, dtype="float32")
D2vop = pylops.SecondDerivative(dims=(11, 21), axis=0, dtype="float32")

###############################################################################
# Chaining of operators represents the simplest concatenation that
# can be performed between two or more linear operators.
# This can be easily achieved using the ``*`` symbol
#
# .. math::
#    \mathbf{D_{cat}}=  \mathbf{D_v} \mathbf{D_h}

Nv, Nh = 11, 21
X = np.zeros((Nv, Nh))
X[int(Nv / 2), int(Nh / 2)] = 1

D2op = D2vop * D2hop
Y = D2op * X

fig, axs = plt.subplots(1, 2, figsize=(10, 3))
fig.suptitle("Chain", fontsize=14, fontweight="bold", y=0.95)
im = axs[0].imshow(X, interpolation="nearest")
axs[0].axis("tight")
axs[0].set_title(r"$x$")
plt.colorbar(im, ax=axs[0])
im = axs[1].imshow(Y, interpolation="nearest")
axs[1].axis("tight")
axs[1].set_title(r"$y=(D_x+D_y) x$")
plt.colorbar(im, ax=axs[1])
plt.tight_layout()
plt.subplots_adjust(top=0.8)

###############################################################################
# We now want to *vertically stack* three operators
#
# .. math::
#    \mathbf{D_{Vstack}} =
#     \begin{bmatrix}
#       \mathbf{D_v}    \\
#       \mathbf{D_h}
#     \end{bmatrix}, \qquad
#    \mathbf{y} =
#     \begin{bmatrix}
#       \mathbf{D_v}\mathbf{x}    \\
#       \mathbf{D_h}\mathbf{x}
#     \end{bmatrix}

Nv, Nh = 11, 21
X = np.zeros((Nv, Nh))
X[int(Nv / 2), int(Nh / 2)] = 1
Dstack = pylops.VStack([D2vop, D2hop])

Y = np.reshape(Dstack * X.ravel(), (Nv * 2, Nh))

fig, axs = plt.subplots(1, 2, figsize=(10, 3))
fig.suptitle("Vertical stacking", fontsize=14, fontweight="bold", y=0.95)
im = axs[0].imshow(X, interpolation="nearest")
axs[0].axis("tight")
axs[0].set_title(r"$x$")
plt.colorbar(im, ax=axs[0])
im = axs[1].imshow(Y, interpolation="nearest")
axs[1].axis("tight")
axs[1].set_title(r"$y$")
plt.colorbar(im, ax=axs[1])
plt.tight_layout()
plt.subplots_adjust(top=0.8)

###############################################################################
# If one now wants to have one or more null operators in the stack, it is
# reccomended to use the :py:class:`pylops.Zero` operator (instead of, for example,
# a :py:class:`pylops.MatrixMult` filled with a zero matrix); under the hood, a
# :py:class:`pylops.Zero` operator will be simply by-passed both in the forward and
# adjoint steps.
#
# .. math::
#    \mathbf{D_{Vstack}} =
#     \begin{bmatrix}
#       \mathbf{D_v}    \\
#       \mathbf{0}      \\
#       \mathbf{D_h}
#     \end{bmatrix}, \qquad
#    \mathbf{y} =
#     \begin{bmatrix}
#       \mathbf{D_v}\mathbf{x}    \\
#       \mathbf{0}                \\
#       \mathbf{D_h}\mathbf{x}
#     \end{bmatrix}
#
# Note that this feature will become particularly handy when defining
# :py:class:`pylops.Block` operators with large zero blocks.

# With MatrixMult operator filled with zeros
Zop = pylops.MatrixMult(np.zeros((Nv * Nh, Nv * Nh)))
Dstack = pylops.VStack([D2vop, Zop, D2hop])
Y = np.reshape(Dstack * X.ravel(), (Nv * 3, Nh))

# With Zero operator
Zop = pylops.Zero(Nv * Nh)
D1stack = pylops.VStack([D2vop, Zop, D2hop])
Y1 = np.reshape(D1stack * X.ravel(), (Nv * 3, Nh))

print("Y == Y1:", np.allclose(Y, Y1))

###############################################################################
# Similarly we can now *horizontally stack* three operators
#
# .. math::
#    \mathbf{D_{Hstack}} =
#     \begin{bmatrix}
#        \mathbf{D_v}  & 0.5*\mathbf{D_v} & -1*\mathbf{D_h}
#     \end{bmatrix}, \qquad
#    \mathbf{y} =
#     \mathbf{D_v}\mathbf{x}_1 + 0.5*\mathbf{D_v}\mathbf{x}_2 -
#     \mathbf{D_h}\mathbf{x}_3

Nv, Nh = 11, 21
X = np.zeros((Nv * 3, Nh))
X[int(Nv / 2), int(Nh / 2)] = 1
X[int(Nv / 2) + Nv, int(Nh / 2)] = 1
X[int(Nv / 2) + 2 * Nv, int(Nh / 2)] = 1

Hstackop = pylops.HStack([D2vop, 0.5 * D2vop, -1 * D2hop])
Y = np.reshape(Hstackop * X.ravel(), (Nv, Nh))

fig, axs = plt.subplots(1, 2, figsize=(10, 3))
fig.suptitle("Horizontal stacking", fontsize=14, fontweight="bold", y=0.95)
im = axs[0].imshow(X, interpolation="nearest")
axs[0].axis("tight")
axs[0].set_title(r"$x$")
plt.colorbar(im, ax=axs[0])
im = axs[1].imshow(Y, interpolation="nearest")
axs[1].axis("tight")
axs[1].set_title(r"$y$")
plt.colorbar(im, ax=axs[1])
plt.tight_layout()
plt.subplots_adjust(top=0.8)

###############################################################################
# We can even stack them both *horizontally* and *vertically* such that we
# create a *block* operator
#
# .. math::
#    \mathbf{D_{Block}} =
#     \begin{bmatrix}
#        \mathbf{D_v} & 0.5*\mathbf{D_v} & -1*\mathbf{D_h} \\
#        \mathbf{D_h} & 2*\mathbf{D_h}   & \mathbf{D_v} \\
#     \end{bmatrix}, \qquad
#    \mathbf{y} =
#     \begin{bmatrix}
#        \mathbf{D_v} \mathbf{x_1} + 0.5*\mathbf{D_v} \mathbf{x_2} -
#        \mathbf{D_h} \mathbf{x_3} \\
#        \mathbf{D_h} \mathbf{x_1} + 2*\mathbf{D_h} \mathbf{x_2} +
#        \mathbf{D_v} \mathbf{x_3}
#     \end{bmatrix}

Bop = pylops.Block([[D2vop, 0.5 * D2vop, -1 * D2hop], [D2hop, 2 * D2hop, D2vop]])
Y = np.reshape(Bop * X.ravel(), (2 * Nv, Nh))

fig, axs = plt.subplots(1, 2, figsize=(10, 3))
fig.suptitle("Block", fontsize=14, fontweight="bold", y=0.95)
im = axs[0].imshow(X, interpolation="nearest")
axs[0].axis("tight")
axs[0].set_title(r"$x$")
plt.colorbar(im, ax=axs[0])
im = axs[1].imshow(Y, interpolation="nearest")
axs[1].axis("tight")
axs[1].set_title(r"$y$")
plt.colorbar(im, ax=axs[1])
plt.tight_layout()
plt.subplots_adjust(top=0.8)

###############################################################################
# And here with some Zero blocks
#
# .. math::
#    \mathbf{D_{Block}} =
#     \begin{bmatrix}
#        \mathbf{D_v} & \mathbf{0} & -1*\mathbf{D_h} \\
#        \mathbf{D_h} & 2*\mathbf{D_h}   & \mathbf{0} \\
#     \end{bmatrix}, \qquad
#    \mathbf{y} =
#     \begin{bmatrix}
#        \mathbf{D_v} \mathbf{x_1} - \mathbf{D_h} \mathbf{x_3} \\
#        \mathbf{D_h} \mathbf{x_1} + 2*\mathbf{D_h} \mathbf{x_2}
#     \end{bmatrix}

# With MatrixMult operator filled with zeros
Zop = pylops.MatrixMult(np.zeros(D2vop.shape))
Bop = pylops.Block([[D2vop, Zop, -1 * D2hop], [D2hop, 2 * D2hop, Zop]])
Y = np.reshape(Bop * X.ravel(), (2 * Nv, Nh))

# With Zero operator
Zop = pylops.Zero(D2vop.shape[0])
B1op = pylops.Block([[D2vop, Zop, -1 * D2hop], [D2hop, 2 * D2hop, Zop]])
Y1 = np.reshape(B1op * X.ravel(), (2 * Nv, Nh))

print("Y == Y1:", np.allclose(Y, Y1))

###############################################################################
# Finally we can use the *block-diagonal operator* to apply three operators
# to three different subset of the model and data
#
# .. math::
#    \mathbf{D_{BDiag}} =
#     \begin{bmatrix}
#        \mathbf{D_v}  & \mathbf{0}       &  \mathbf{0}  \\
#        \mathbf{0}    & 0.5*\mathbf{D_v} &  \mathbf{0}  \\
#        \mathbf{0}    & \mathbf{0}       &  -\mathbf{D_h}
#     \end{bmatrix}, \qquad
#    \mathbf{y} =
#     \begin{bmatrix}
#        \mathbf{D_v}     \mathbf{x_1}  \\
#        0.5*\mathbf{D_v} \mathbf{x_2}  \\
#        -\mathbf{D_h}  \mathbf{x_3}
#     \end{bmatrix}

BD = pylops.BlockDiag([D2vop, 0.5 * D2vop, -1 * D2hop])
Y = np.reshape(BD * X.ravel(), (3 * Nv, Nh))

fig, axs = plt.subplots(1, 2, figsize=(10, 3))
fig.suptitle("Block-diagonal", fontsize=14, fontweight="bold", y=0.95)
im = axs[0].imshow(X, interpolation="nearest")
axs[0].axis("tight")
axs[0].set_title(r"$x$")
plt.colorbar(im, ax=axs[0])
im = axs[1].imshow(Y, interpolation="nearest")
axs[1].axis("tight")
axs[1].set_title(r"$y$")
plt.colorbar(im, ax=axs[1])
plt.tight_layout()
plt.subplots_adjust(top=0.8)

###############################################################################
# If we consider now the case of having a large number of operators inside a
# blockdiagonal structure, it may be convenient to span multiple processes
# handling subset of operators at the same time. This can be easily achieved
# by simply defining the number of processes we want to use via ``nproc``.

X = np.zeros((Nv * 10, Nh))
for iv in range(10):
    X[int(Nv / 2) + iv * Nv, int(Nh / 2)] = 1

BD = pylops.BlockDiag([D2vop] * 10, nproc=2)
print("BD Operator multiprocessing pool", BD.pool)
Y = np.reshape(BD * X.ravel(), (10 * Nv, Nh))
BD.pool.close()

fig, axs = plt.subplots(1, 2, figsize=(10, 3))
fig.suptitle("Block-diagonal", fontsize=14, fontweight="bold", y=0.95)
im = axs[0].imshow(X, interpolation="nearest")
axs[0].axis("tight")
axs[0].set_title(r"$x$")
plt.colorbar(im, ax=axs[0])
im = axs[1].imshow(Y, interpolation="nearest")
axs[1].axis("tight")
axs[1].set_title(r"$y$")
plt.colorbar(im, ax=axs[1])
plt.tight_layout()
plt.subplots_adjust(top=0.8)

###############################################################################
# Finally we use the *Kronecker operator* and replicate this example on
# `wiki <https://en.wikipedia.org/wiki/Kronecker_product>`_.
#
# .. math::
#    \begin{bmatrix}
#        1  & 2  \\
#        3  & 4 \\
#    \end{bmatrix} \otimes
#    \begin{bmatrix}
#        0  & 5  \\
#        6  & 7 \\
#    \end{bmatrix} =
#    \begin{bmatrix}
#         0 &  5 &  0 & 10 \\
#         6 &  7 & 12 & 14 \\
#         0 & 15 &  0 & 20 \\
#        18 & 21 & 24 & 28 \\
#    \end{bmatrix}
A = np.array([[1, 2], [3, 4]])
B = np.array([[0, 5], [6, 7]])
AB = np.kron(A, B)

n1, m1 = A.shape
n2, m2 = B.shape

Aop = pylops.MatrixMult(A)
Bop = pylops.MatrixMult(B)

ABop = pylops.Kronecker(Aop, Bop)
x = np.ones(m1 * m2)

y = AB.dot(x)
yop = ABop * x
xinv = ABop / yop

print(f"AB = \n {AB}")

print(f"x = {x}")
print(f"y = {y}")
print(f"yop = {yop}")
print(f"xinv = {xinv}")

###############################################################################
# We can also use :py:class:`pylops.Kronecker` to do something more
# interesting. Any operator can in fact be applied on a single direction of a
# multi-dimensional input array if combined with an :py:class:`pylops.Identity`
# operator via Kronecker product. We apply here the
# :py:class:`pylops.FirstDerivative` to the second dimension of the model.
#
# Note that for those operators whose implementation allows their application
# to a single axis via the ``axis`` parameter, using the Kronecker product
# would lead to slower performance. Nevertheless, the Kronecker product allows
# any other operator to be applied to a single dimension.
Nv, Nh = 11, 21

Iop = pylops.Identity(Nv, dtype="float32")
D2hop = pylops.FirstDerivative(Nh, dtype="float32")

X = np.zeros((Nv, Nh))
X[Nv // 2, Nh // 2] = 1
D2hop = pylops.Kronecker(Iop, D2hop)

Y = D2hop * X.ravel()
Y = Y.reshape(Nv, Nh)

fig, axs = plt.subplots(1, 2, figsize=(10, 3))
fig.suptitle("Kronecker", fontsize=14, fontweight="bold", y=0.95)
im = axs[0].imshow(X, interpolation="nearest")
axs[0].axis("tight")
axs[0].set_title(r"$x$")
plt.colorbar(im, ax=axs[0])
im = axs[1].imshow(Y, interpolation="nearest")
axs[1].axis("tight")
axs[1].set_title(r"$y$")
plt.colorbar(im, ax=axs[1])
plt.tight_layout()
plt.subplots_adjust(top=0.8)