Diagonal

This example shows how to use the pylops_gpu.Diagonal operator to perform Element-wise multiplication between the input vector and a vector \(\mathbf{d}\).

In other words, the operator acts as a diagonal operator \(\mathbf{D}\) whose elements along the diagonal are the elements of the vector \(\mathbf{d}\).

import torch
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.gridspec as pltgs

import pylops_gpu

plt.close('all')

Let’s define a diagonal operator \(\mathbf{d}\) with increasing numbers from 0 to N and a unitary model \(\mathbf{x}\).

N = 10
d = torch.arange(N, dtype=torch.float32)
x = torch.ones(N, dtype=torch.float32)

Dop = pylops_gpu.Diagonal(d)

y = Dop * x
y1 = Dop.H * x

gs = pltgs.GridSpec(1, 6)
fig = plt.figure(figsize=(7, 3))
ax = plt.subplot(gs[0, 0:3])
im = ax.imshow(Dop.matrix(), cmap='rainbow', vmin=0, vmax=N)
ax.set_title('A', size=20, fontweight='bold')
ax.set_xticks(np.arange(N-1)+0.5)
ax.set_yticks(np.arange(N-1)+0.5)
ax.grid(linewidth=3, color='white')
ax.xaxis.set_ticklabels([])
ax.yaxis.set_ticklabels([])
ax.axis('tight')
ax = plt.subplot(gs[0, 3])
ax.imshow(x[:, np.newaxis], cmap='rainbow', vmin=0, vmax=N)
ax.set_title('x', size=20, fontweight='bold')
ax.set_xticks([])
ax.set_yticks(np.arange(N-1)+0.5)
ax.grid(linewidth=3, color='white')
ax.xaxis.set_ticklabels([])
ax.yaxis.set_ticklabels([])
ax = plt.subplot(gs[0, 4])
ax.text(0.35, 0.5, '=', horizontalalignment='center',
        verticalalignment='center', size=40, fontweight='bold')
ax.axis('off')
ax = plt.subplot(gs[0, 5])
ax.imshow(y[:, np.newaxis], cmap='rainbow', vmin=0, vmax=N)
ax.set_title('y', size=20, fontweight='bold')
ax.set_xticks([])
ax.set_yticks(np.arange(N - 1) + 0.5)
ax.grid(linewidth=3, color='white')
ax.xaxis.set_ticklabels([])
ax.yaxis.set_ticklabels([])
fig.colorbar(im, ax=ax, ticks=[0, N], pad=0.3, shrink=0.7)

Out:

<matplotlib.colorbar.Colorbar object at 0x7fbc6bef0ba8>

Similarly we can consider the input model as composed of two or more dimensions. In this case the diagonal operator can be still applied to each element or broadcasted along a specific direction. Let’s start with the simplest case where each element is multipled by a different value

nx, ny = 3, 5
x = torch.ones((nx, ny), dtype=torch.float32)
print('x =\n%s' % x)

d = torch.arange(nx*ny, dtype=torch.float32).reshape(nx, ny)
Dop = pylops_gpu.Diagonal(d)

y = Dop * x.flatten()
y1 = Dop.H * x.flatten()

print('y = D*x =\n%s' % y.reshape(nx, ny))
print('xadj = D\'*x =\n%s ' % y1.reshape(nx, ny))

Out:

x =
tensor([[1., 1., 1., 1., 1.],
        [1., 1., 1., 1., 1.],
        [1., 1., 1., 1., 1.]])
y = D*x =
tensor([[ 0.,  1.,  2.,  3.,  4.],
        [ 5.,  6.,  7.,  8.,  9.],
        [10., 11., 12., 13., 14.]])
xadj = D'*x =
tensor([[ 0.,  1.,  2.,  3.,  4.],
        [ 5.,  6.,  7.,  8.,  9.],
        [10., 11., 12., 13., 14.]])

And we now broadcast

nx, ny = 3, 5
x = torch.ones((nx, ny), dtype=torch.float32)
print('x =\n%s' % x)

# 1st dim
d = torch.arange(nx, dtype=torch.float32)
Dop = pylops_gpu.Diagonal(d, dims=(nx, ny), dir=0)

y = Dop * x.flatten()
y1 = Dop.H * x.flatten()

print('1st dim: y = D*x =\n%s' % y.reshape(nx, ny))
print('1st dim: xadj = D\'*x =\n%s ' % y1.reshape(nx, ny))

# 2nd dim
d = torch.arange(ny, dtype=torch.float32)
Dop = pylops_gpu.Diagonal(d, dims=(nx, ny), dir=1)

y = Dop * x.flatten()
y1 = Dop.H * x.flatten()

print('2nd dim: y = D*x =\n%s' % y.reshape(nx, ny))
print('2nd dim: xadj = D\'*x =\n%s ' % y1.reshape(nx, ny))

Out:

x =
tensor([[1., 1., 1., 1., 1.],
        [1., 1., 1., 1., 1.],
        [1., 1., 1., 1., 1.]])
1st dim: y = D*x =
tensor([[0., 0., 0., 0., 0.],
        [1., 1., 1., 1., 1.],
        [2., 2., 2., 2., 2.]])
1st dim: xadj = D'*x =
tensor([[0., 0., 0., 0., 0.],
        [1., 1., 1., 1., 1.],
        [2., 2., 2., 2., 2.]])
2nd dim: y = D*x =
tensor([[0., 1., 2., 3., 4.],
        [0., 1., 2., 3., 4.],
        [0., 1., 2., 3., 4.]])
2nd dim: xadj = D'*x =
tensor([[0., 1., 2., 3., 4.],
        [0., 1., 2., 3., 4.],
        [0., 1., 2., 3., 4.]])