Spatio-Temporal simulation¶
In [1]:
import numpy as np
import gstlearn as gl
import gstlearn.document as gdoc
from scipy.special import gamma
from IPython.display import Markdown
In [2]:
Markdown(gdoc.loadDoc("Spatio_Temporal.md"))
Out[2]:
Spatio-Temporal Model
This Spatio_temporal model corresponds to the following family of SPDEs :
$$\left(\frac{\partial}{\partial t} + c\left[(1-\nabla H. \nabla)^{\alpha} + v.\nabla\right]\right)Z(s,t)=\sqrt{c}W_T(t)\otimes X_S(s)$$
where
$c>0$ is a time scale parameter
$H$ is an anisotropic matrix
$v$ is a velocity vector
$W_T$ is a temporal white-noise
$X_S$ is a spatially structured noise, solution of
$$(1-\nabla H_S \nabla)X_S = \mathcal{W}_S$$
In [3]:
from sksparse.cholmod import cholesky
import scipy as sc
from scipy.sparse import diags
In [4]:
scale = 4.
kappa = 1./scale
kappa2 = kappa**2
dt = .1
dx = 1.
c = 40
sqc = np.sqrt(c)
sqdt = np.sqrt(dt)
In [5]:
nx = [100,100]
m = gl.Model.createFromParam(gl.ECov.BESSEL_K,range=1,param=2,flagRange=False)
mesh = gl.MeshETurbo(nx ,[dx,dx])
S = gl.ShiftOpCs(mesh,m)
Smat = S.getS().toTL()
TildeC12 = diags(np.sqrt(S.getTildeC()))
invTildeC12 = diags(1./np.sqrt(S.getTildeC()))
G = TildeC12 @ Smat @ TildeC12
M = TildeC12 @ TildeC12
K = (kappa2 * M + G) @ (kappa2 * M + G) @ (kappa2 * M + G)
param = 3
P = M + c*dt * K
cholP = cholesky(P)
def evalInvA(x) :
return cholP.solve_A(M @ x)
def evalInvB(x) :
return sqc * sqdt * cholP.solve_A(TildeC12 @ x)
# TODO: Reason for the warning ??!
# https://stackoverflow.com/a/48273717
/tmp/ipykernel_4894/2759233675.py:13: CholmodTypeConversionWarning: converting matrix of class csr_matrix to CSC format cholP = cholesky(P)
In [6]:
np.random.seed(123)
x = np.random.normal(size=Smat.shape[0])
for i in range(100):
u = np.random.normal(size=Smat.shape[0])
x = evalInvA(x) + evalInvB(u)
In [7]:
anim = True
if anim :
%matplotlib notebook
import matplotlib.pyplot as plt
from matplotlib.animation import FuncAnimation
#x = np.random.normal(size=Smat.shape[0])
xtot = np.zeros(shape = [Smat.shape[0],50])
xtot[:,0] = x
fig, ax = plt.subplots(figsize=(2,2))
ln = plt.imshow(x.reshape(nx), 'BrBG')
def init():
ln.set_data(x.reshape(nx))
return ln
def update(frame):
u = np.random.normal(size=Smat.shape[0])
xtot[:,frame] = evalInvA(xtot[:,frame-1]) + evalInvB(u)
ln.set_data(xtot[:,frame].reshape(nx))
return ln
if anim :
ani = FuncAnimation(fig, update, frames = 50,
init_func=init, blit=False,interval=10)
p = plt.show()
In [8]:
%matplotlib inline
In [9]:
model = gl.Model.createFromParam(gl.ECov.BESSEL_K,range = scale,param = param,flagRange=False)
rangev = 3.5 * model.getCova(0).getRange()
h = np.linspace(0,rangev,100)
points = [gl.SpacePoint([0.,i]) for i in h]
cov = [model.eval(points[0],i) for i in points]
In [10]:
def correc(kappa,alpha,d=2):
return gamma(alpha-d/2)/ (gamma(alpha)*(4*np.pi)**(d/2)*kappa**(2*alpha-d))
In [11]:
d = 2
param = 3
alpha = param+d/2
hmax = 3.5 * model.getCova(0).getRange()
N = 2**8
ind = np.concatenate([np.arange(int(N/2),N-1,2),np.arange(0,int(N/2),2)])
a= np.pi * (N-1) / hmax
v = np.linspace(-1.,1.,N)
u = a/2 * v
deltau=a/(N-1)
xi = np.meshgrid(u,u)
normxi = np.array([i**2 + j**2 for i in u for j in u]).reshape((len(u),len(u)))
grxi = c * (kappa2 + normxi)**alpha * correc(kappa,param+d/2,d)
nres = 200
time = np.arange(0,nres,1)
result = np.zeros(shape=[len(ind),nres])
for k in time :
fourier = np.exp(-k *dt * np.abs(grxi))/( 1./c * (2*np.pi)**d*2*grxi)
B = np.real(np.fft.fftn(fourier,norm="backward"))
result[:,k] = B[0,:][ind]
In [12]:
X= np.pi * v[np.arange(0,N,2)] /deltau
points = [gl.SpacePoint([0.,i]) for i in X]
cov = [model.eval(gl.SpacePoint([0.,0]),i)/2. for i in points]
plt.plot(X,cov)
for i in range(10):
covZ=result[:,i * 10]*deltau**d
plt.plot(X, covZ)
lim = 1.5 * model.getCova(0).getRange()
plt.xlim([-lim,lim])
a = plt.axvline(x=0.)
In [13]:
i = plt.imshow(result)
In [14]:
v
Out[14]:
array([-1. , -0.99215686, -0.98431373, -0.97647059, -0.96862745,
-0.96078431, -0.95294118, -0.94509804, -0.9372549 , -0.92941176,
-0.92156863, -0.91372549, -0.90588235, -0.89803922, -0.89019608,
-0.88235294, -0.8745098 , -0.86666667, -0.85882353, -0.85098039,
-0.84313725, -0.83529412, -0.82745098, -0.81960784, -0.81176471,
-0.80392157, -0.79607843, -0.78823529, -0.78039216, -0.77254902,
-0.76470588, -0.75686275, -0.74901961, -0.74117647, -0.73333333,
-0.7254902 , -0.71764706, -0.70980392, -0.70196078, -0.69411765,
-0.68627451, -0.67843137, -0.67058824, -0.6627451 , -0.65490196,
-0.64705882, -0.63921569, -0.63137255, -0.62352941, -0.61568627,
-0.60784314, -0.6 , -0.59215686, -0.58431373, -0.57647059,
-0.56862745, -0.56078431, -0.55294118, -0.54509804, -0.5372549 ,
-0.52941176, -0.52156863, -0.51372549, -0.50588235, -0.49803922,
-0.49019608, -0.48235294, -0.4745098 , -0.46666667, -0.45882353,
-0.45098039, -0.44313725, -0.43529412, -0.42745098, -0.41960784,
-0.41176471, -0.40392157, -0.39607843, -0.38823529, -0.38039216,
-0.37254902, -0.36470588, -0.35686275, -0.34901961, -0.34117647,
-0.33333333, -0.3254902 , -0.31764706, -0.30980392, -0.30196078,
-0.29411765, -0.28627451, -0.27843137, -0.27058824, -0.2627451 ,
-0.25490196, -0.24705882, -0.23921569, -0.23137255, -0.22352941,
-0.21568627, -0.20784314, -0.2 , -0.19215686, -0.18431373,
-0.17647059, -0.16862745, -0.16078431, -0.15294118, -0.14509804,
-0.1372549 , -0.12941176, -0.12156863, -0.11372549, -0.10588235,
-0.09803922, -0.09019608, -0.08235294, -0.0745098 , -0.06666667,
-0.05882353, -0.05098039, -0.04313725, -0.03529412, -0.02745098,
-0.01960784, -0.01176471, -0.00392157, 0.00392157, 0.01176471,
0.01960784, 0.02745098, 0.03529412, 0.04313725, 0.05098039,
0.05882353, 0.06666667, 0.0745098 , 0.08235294, 0.09019608,
0.09803922, 0.10588235, 0.11372549, 0.12156863, 0.12941176,
0.1372549 , 0.14509804, 0.15294118, 0.16078431, 0.16862745,
0.17647059, 0.18431373, 0.19215686, 0.2 , 0.20784314,
0.21568627, 0.22352941, 0.23137255, 0.23921569, 0.24705882,
0.25490196, 0.2627451 , 0.27058824, 0.27843137, 0.28627451,
0.29411765, 0.30196078, 0.30980392, 0.31764706, 0.3254902 ,
0.33333333, 0.34117647, 0.34901961, 0.35686275, 0.36470588,
0.37254902, 0.38039216, 0.38823529, 0.39607843, 0.40392157,
0.41176471, 0.41960784, 0.42745098, 0.43529412, 0.44313725,
0.45098039, 0.45882353, 0.46666667, 0.4745098 , 0.48235294,
0.49019608, 0.49803922, 0.50588235, 0.51372549, 0.52156863,
0.52941176, 0.5372549 , 0.54509804, 0.55294118, 0.56078431,
0.56862745, 0.57647059, 0.58431373, 0.59215686, 0.6 ,
0.60784314, 0.61568627, 0.62352941, 0.63137255, 0.63921569,
0.64705882, 0.65490196, 0.6627451 , 0.67058824, 0.67843137,
0.68627451, 0.69411765, 0.70196078, 0.70980392, 0.71764706,
0.7254902 , 0.73333333, 0.74117647, 0.74901961, 0.75686275,
0.76470588, 0.77254902, 0.78039216, 0.78823529, 0.79607843,
0.80392157, 0.81176471, 0.81960784, 0.82745098, 0.83529412,
0.84313725, 0.85098039, 0.85882353, 0.86666667, 0.8745098 ,
0.88235294, 0.89019608, 0.89803922, 0.90588235, 0.91372549,
0.92156863, 0.92941176, 0.9372549 , 0.94509804, 0.95294118,
0.96078431, 0.96862745, 0.97647059, 0.98431373, 0.99215686,
1. ])
In [15]:
result[:,0].max()
Out[15]:
357.4611315408981