|
| 1 | +r""" |
| 2 | +Proximal operators |
| 3 | +================== |
| 4 | +
|
| 5 | +This example demonstrates the use of the :py:module:`pylops_mpi.proximal` |
| 6 | +module, and more specifically how to create and apply PyProximal operators |
| 7 | +to distributed array. |
| 8 | +
|
| 9 | +""" |
| 10 | +import numpy as np |
| 11 | +from mpi4py import MPI |
| 12 | +from matplotlib import pyplot as plt |
| 13 | + |
| 14 | +import pylops |
| 15 | +import pyproximal |
| 16 | + |
| 17 | +import pylops_mpi |
| 18 | + |
| 19 | +np.random.seed(42) |
| 20 | +plt.close("all") |
| 21 | +comm = MPI.COMM_WORLD |
| 22 | +rank = comm.Get_rank() |
| 23 | +size = comm.Get_size() |
| 24 | + |
| 25 | +############################################################################### |
| 26 | +# Let's start with so-called separable proximal operators. These are functionals |
| 27 | +# whose proximal operator can be computed in a element-wise fashion. As such, |
| 28 | +# no special implementation is required for the distributed counterpart of |
| 29 | +# those operators. Instead, we can simply wrap the PyProximal operator into |
| 30 | +# a :py:class:`pylops_mpi.proximal.MPIProxOperator`. |
| 31 | +# |
| 32 | +# We take the :py:class:`pyproximal.proximal.L1` norm as an example. |
| 33 | + |
| 34 | +n = 10 |
| 35 | +arr = pylops_mpi.DistributedArray(global_shape=n * size, |
| 36 | + partition=pylops_mpi.Partition.SCATTER) |
| 37 | +arr[:] = rank * np.arange(n) |
| 38 | + |
| 39 | +l1 = pyproximal.proximal.L1(sigma=2.0) |
| 40 | +l1d = pylops_mpi.proximal.MPIProxOperator(l1) |
| 41 | + |
| 42 | +# Call |
| 43 | +f = l1d(arr) |
| 44 | + |
| 45 | +# Proximal |
| 46 | +prox = l1d.prox(arr, .1) |
| 47 | +proxdlocal = prox.asarray() |
| 48 | + |
| 49 | +dprox = l1d.proxdual(arr, .1) |
| 50 | +dproxdlocal = dprox.asarray() |
| 51 | + |
| 52 | +arrlocal = arr.asarray() |
| 53 | +if rank == 0: |
| 54 | + flocal = l1(arrlocal) |
| 55 | + proxlocal = l1.prox(arrlocal, .1) |
| 56 | + dproxlocal = l1.proxdual(arrlocal, .1) |
| 57 | + print("||x||_1: ", f, flocal) |
| 58 | + print("prox_||x||_1: ", all(proxdlocal == proxlocal)) |
| 59 | + print("proxd_||x||_1: ", all(dproxdlocal == dproxlocal)) |
| 60 | + |
| 61 | +############################################################################### |
| 62 | +# We repeat now the same with the :py:class:`pyproximal.proximal.Box` operator. |
| 63 | + |
| 64 | +arr = pylops_mpi.DistributedArray(global_shape=n * size, |
| 65 | + partition=pylops_mpi.Partition.SCATTER) |
| 66 | + |
| 67 | +arr[:] = 3 * np.ones(n) |
| 68 | +if rank == 0: |
| 69 | + arr[n//2] = 20 # outside of the box |
| 70 | + |
| 71 | +box = pyproximal.Box(lower=1., upper=5.) |
| 72 | +boxd = pylops_mpi.proximal.MPIProxOperator(box) |
| 73 | + |
| 74 | +# Call |
| 75 | +f = boxd(arr) |
| 76 | + |
| 77 | +# Proximal |
| 78 | +prox = boxd.prox(arr, .1) |
| 79 | +proxdlocal = prox.asarray() |
| 80 | + |
| 81 | +# Dual-Proximal |
| 82 | +dprox = boxd.proxdual(arr, .1) |
| 83 | +dproxdlocal = dprox.asarray() |
| 84 | + |
| 85 | +arrlocal = arr.asarray() |
| 86 | +if rank == 0: |
| 87 | + flocal = box(arrlocal) |
| 88 | + proxlocal = box.prox(arrlocal, .1) |
| 89 | + dproxlocal = box.proxdual(arrlocal, .1) |
| 90 | + print("Box(x): ", f, flocal) |
| 91 | + print("prox_Box ", all(proxdlocal == proxlocal)) |
| 92 | + print("proxd_Box ", all(dproxdlocal == dproxlocal)) |
| 93 | + |
| 94 | +############################################################################### |
| 95 | +# We move on now to a operator that is not separable and must be fully |
| 96 | +# re-implemented in a distributed fashion, namely the |
| 97 | +# :py:class:`pylops_mpi.proximal.MPIL2` norm. |
| 98 | +# |
| 99 | +# More precisely, when ``Op`` and ``b`` are passed to this operator, |
| 100 | +# its proximal does call for the solution of a distributed inverse problem. |
| 101 | +# |
| 102 | +# However, let's start with the simplest case: :math:`||\mathbf{x}||_2^2` |
| 103 | + |
| 104 | +arr = pylops_mpi.DistributedArray(global_shape=n * size, |
| 105 | + partition=pylops_mpi.Partition.SCATTER) |
| 106 | + |
| 107 | +arr[:] = rank * np.arange(n) |
| 108 | + |
| 109 | +l2 = pyproximal.L2(sigma=2.0) |
| 110 | +l2d = pylops_mpi.proximal.MPIL2(sigma=2.0) |
| 111 | + |
| 112 | +# Call |
| 113 | +f = l2d(arr) |
| 114 | + |
| 115 | +# Proximal |
| 116 | +prox = l2d.prox(arr, .1) |
| 117 | +proxdlocal = prox.asarray() |
| 118 | + |
| 119 | +# Gradient |
| 120 | +grad = l2d.grad(arr) |
| 121 | +graddlocal = grad.asarray() |
| 122 | + |
| 123 | +arrlocal = arr.asarray() |
| 124 | +if rank == 0: |
| 125 | + flocal = l2(arrlocal) |
| 126 | + proxlocal = l2.prox(arrlocal, .1) |
| 127 | + gradlocal = l2.grad(arrlocal) |
| 128 | + print("||x||_2^2: ", f, flocal) |
| 129 | + print("prox_||x||_2^2: ", all(proxdlocal == proxlocal)) |
| 130 | + print("grad_||x||_2^2: ", all(graddlocal == gradlocal)) |
| 131 | + |
| 132 | +############################################################################### |
| 133 | +# Next we move onto the more general case |
| 134 | +# :math:`||\mathbf{Op} \mathbf{x} - \mathbf{b}||_2^2` |
| 135 | + |
| 136 | +solver="cgls" |
| 137 | +Op = pylops.Diagonal(np.ones(n * size)) |
| 138 | +Opd = pylops_mpi.MPIBlockDiag([pylops.Diagonal(np.ones(n)),]) |
| 139 | + |
| 140 | +b = pylops_mpi.DistributedArray(global_shape=n * size, |
| 141 | + partition=pylops_mpi.Partition.SCATTER) |
| 142 | +b[:] = rank * np.ones(n) |
| 143 | +blocal = b.asarray() |
| 144 | + |
| 145 | +x0 = arr.zeros_like() |
| 146 | +x0local = x0.asarray() |
| 147 | + |
| 148 | +l2 = pyproximal.L2( |
| 149 | + Op=Op, b=blocal, sigma=2.0, |
| 150 | + solver=solver, x0=x0local, |
| 151 | + kwargs_solver=dict(show=True)) |
| 152 | +l2d = pylops_mpi.proximal.MPIL2( |
| 153 | + Op=Opd, b=b, sigma=2.0, |
| 154 | + solver=solver, x0=x0, |
| 155 | + kwargs_solver=dict(show=True if rank==0 else False)) |
| 156 | + |
| 157 | +# Call |
| 158 | +f = l2d(arr) |
| 159 | + |
| 160 | +# Proximal |
| 161 | +prox = l2d.prox(arr, .1) |
| 162 | +proxdlocal = prox.asarray() |
| 163 | + |
| 164 | +# Gradient |
| 165 | +grad = l2d.grad(arr) |
| 166 | +graddlocal = grad.asarray() |
| 167 | + |
| 168 | +arrlocal = arr.asarray() |
| 169 | +if rank == 0: |
| 170 | + flocal = l2(arrlocal) |
| 171 | + proxlocal = l2.prox(arrlocal, .1) |
| 172 | + gradlocal = l2.grad(arrlocal) |
| 173 | + print("||Op . x - b||_2^2: ", f, flocal) |
| 174 | + print("prox_||Op . x - b||_2^2 - norm diff=", |
| 175 | + np.linalg.norm(proxdlocal - proxlocal)) |
| 176 | + print("grad_||Op . x - b||_2^2: ", all(graddlocal == gradlocal)) |
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