CH11_SEC05_1_Invariance.ipynb

Kernel: Python 3

import numpy as np
import matplotlib.pyplot as plt
from matplotlib import rcParams, cm
from scipy import integrate
from mpl_toolkits.mplot3d import Axes3D 


plt.rcParams['figure.figsize'] = [12, 12]
plt.rcParams.update({'font.size': 18})

L = 20
n = 100
x = np.linspace(-L,L,n)
y = np.copy(x)

X,Y = np.meshgrid(x,y)

Xd = np.zeros((n**2,n))
for jj in range(n):
    u = np.tanh(np.sqrt(np.power(X,2)+np.power(Y,2))) * \
            np.cos(np.angle(X+(1j)*Y)- \
            np.sqrt(np.power(X,2)+np.power(Y,2)) + \
           (jj+1)/10)
    f = np.exp(-0.01*(np.power(X,2) + np.power(Y,2)))
    uf = u * f
    Xd[:,jj] = uf.reshape(-1)

plt.pcolor(x,y,uf,cmap='hot')
plt.show()

U,S,VT = np.linalg.svd(Xd,full_matrices=0)
V = VT.T

plt.rcParams['figure.figsize'] = [12, 6]

[plt.plot(V[:,k],linewidth=2,label='mode {}'.format(k+1)) for k in range(4)]

plt.legend(loc='lower right')
plt.show()

fig, axs = plt.subplots(2,1)
axs[0].plot(100*S/np.sum(S),'ko',ms=10)
axs[0].set_ylabel('Singular Values')
axs[1].semilogy(100*S/np.sum(S),'ko',ms=10)
axs[1].set_ylabel('Singular Values (log)')

for ax in axs:
    ax.set_xlim(-1,40)

plt.show()

plt.rcParams['figure.figsize'] = [12, 12]
fig,axs = plt.subplots(2,2)
axs = axs.reshape(-1)

for jj in range(4):
    mode = np.reshape(U[:,jj],(n,n))
    axs[jj].pcolor(X,Y,mode,cmap='gray')
    axs[jj].axis(False)

u = np.tanh(np.sqrt(np.power(X,2)+np.power(Y,2))) * \
            np.cos(np.angle(X+(1j)*Y)- \
            np.sqrt(np.power(X,2)+np.power(Y,2)))
f = np.exp(-0.01*(np.power(X,2)+np.power(Y,2)))
uf = u*f

plt.rcParams['figure.figsize'] = [12,4]
fig,axs = plt.subplots(1,3)
axs[0].pcolor(x,y,uf,cmap='gray')
axs[1].pcolor(x,y,np.abs(uf),cmap='gray')
axs[2].pcolor(x,y,np.power(uf,5),cmap='gray')

for ax in axs:
    ax.axis(False)

## Translation
n = 200
L = 20
x = np.linspace(-L,L,n) # space
y = np.copy(x)
m = 41
T = 10
t = np.linspace(0,T,m) # time
c = 3 # wave speed

X = np.zeros((n,m))
for jj in range(m):
    X[:,jj] = np.exp(-np.power(x+15-c*t[jj],2))
    
U,S,VT = np.linalg.svd(X)
V = VT.T

plt.rcParams['figure.figsize'] = [12,12]
fig = plt.figure()
ax = fig.add_subplot(1,1,1,projection='3d')
ax.view_init(elev=70, azim=-70)

for jj in range(m):
    ax.plot(x,t[jj]*np.ones_like(x),X[:,jj],'k',linewidth=0.75)

U2,S2,V2T = np.linalg.svd(X)
V2 = V2T.T

fig, axs = plt.subplots(2,1)
axs[0].plot(100*S2/np.sum(S2),'ko',ms=10)
axs[0].set_ylabel('Singular Values')
axs[1].semilogy(100*S2/np.sum(S2),'ko',ms=10)
axs[1].set_ylabel('Singular Values (log)')

for ax in axs:
    ax.set_xlim(-1,40)

plt.show()