def exa3_44(needFigure=True):
T = 10
N = int(1e4)
a = 1
ne = 512
h = a / ne
epsilon = 1e-3
x = np.linspace(0, a, ne + 1)
u0 = np.exp(-(x - 0.5)**2 / epsilon)
start = timer()
[t, ut] = pde_fem(u0, T, a, N, ne, epsilon, fNagumo)
end = timer()
print("Run time", end - start, "secs")
#
if needFigure:
plt.figure(1)
ch0.PlotSetup()
ax = plt.gca(projection='3d')
[T, X] = np.meshgrid(t, x)
ax.plot_surface(T,
X,
ut,
cmap=cm.coolwarm,
linewidth=0,
antialiased=False)
plt.axis('equal')
ax.set_zlabel(r'$u$')
ax.set_ylabel(r'$x$')
ax.set_xlabel(r'$t$')
plt.savefig('fig3_4a.pdf')
return t, ut
def exa3_33(needFigure=True):
T = 10
N = 1e4
epsilon = 1
a = 20
J = 128
h = a / J
x = np.linspace(0, a, J + 1)
u0 = 1 / (1 + np.exp(-(2 - x) / sqrt(2)))
start = timer()
[t, ut] = pde_fd(u0, T, a, N, J, epsilon, fNagumo, 'N')
end = timer()
print("Run time", end - start, "secs")
[T, X] = np.meshgrid(t, x)
#
if (needFigure == True):
plt.figure(1)
ch0.PlotSetup()
ax = plt.gca()
ax = plt.Axes3D(fig)
#ax.plot_wireframe(T,X,ut,rstride=16,cstride=1000,colors='k')
CS = ax.contourf(X, T, ut, 10, cmap=plt.cm.bone)
#ax.set_zlabel(r'$u$')
ax.set_xlabel(r'$x$')
ax.set_ylabel(r'$t$')
plt.colorbar(CS)
plt.savefig('fig3_4.pdf')
def uniform_mesh_info(ns):
"""
A2.4 Page 69.
"""
h = 1 / ns
x = np.linspace(0, 1, ns + 1)
y = np.copy(x)
# co-ordinates of vertices
xv, yv = np.meshgrid(x, y)
xv = xv.ravel()
yv = yv.ravel()
n2 = ns * ns
nvtx = (ns + 1) * (ns + 1)
ne = 2 * n2
# global vertex labels of individual elements
elt2vert = np.zeros((ne, 3), dtype='int')
vv = np.reshape(np.arange(0, nvtx), (ns + 1, ns + 1), order='F')
v1 = vv[0:ns, 0:ns]
v2 = vv[1:, 0:ns]
v3 = vv[0:ns, 1:]
elt2vert[0:n2, :] = np.vstack((v1.ravel(), v2.ravel(), v3.ravel())).T
v1 = vv[1:, 1:]
elt2vert[n2:, :] = np.vstack((v1.ravel(), v3.ravel(), v2.ravel())).T
plt.figure(2) #
ch0.PlotSetup()
plt.axis('equal')
plt.triplot(xv.ravel(), yv.ravel(), elt2vert, 'k-')
plt.xlabel(r'$x_1$')
plt.ylabel(r'$x_2$')
return xv, yv, elt2vert, nvtx, ne, h
def exa10_42(needFigure=True):
T = 10
N = int(1e3)
a = 1
ne = 512
h = a / ne
epsilon = 1e-3
r = 1
M = 1
sigma = 0.1
x = np.linspace(0, a, ne + 1)
u0 = np.exp(-(x - 0.5)**2 / epsilon)
s0 = np.random.RandomState()
t, u, ut = spde_fem_MhDt(u0, T, a, N, 1, ne, 1, epsilon, fNagumo,
lambda u: sigma * u, r, M, s0)
if needFigure:
plt.figure(1)
ch0.PlotSetup()
ax = plt.gca(projection='3d')
[T, X] = np.meshgrid(t, x)
ax.plot_surface(T,
X,
ut,
cmap=cm.coolwarm,
linewidth=0,
antialiased=False)
ax.set_zlabel(r'$u$')
ax.set_ylabel(r'$x$')
ax.set_xlabel(r'$t$')
plt.savefig('fig10_.pdf')
def exa10_40(needFigure=True):
T=10; N=int(1e3); a=np.array([2*pi,16]); J=np.array([128,64])
alpha=0.1; epsilon=1e-3; sigma=0.1; M=2; kappa=1;
x=np.linspace(0,a[0],J[0]+1); y=np.linspace(0,a[1],J[1]+1)
xx,yy=np.meshgrid(x,y,indexing='ij')
u0=np.sin(xx)*np.cos(pi*yy/8)
u0=np.sin(yy)*np.cos(pi*xx/8)
#
t,u,ut=spde_twod_Gal(u0,T,a,N,kappa,J,epsilon,
fAC,
lambda u: sigma,
alpha,M)
if (needFigure==True):
plt.figure(1)
ch0.PlotSetup()
ax = plt.gca()
xx,yy=np.meshgrid(x,y)
print(xx.shape,ut.shape,x.shape,y.shape)
# ax = plt.gca(projection='3d')
# ax.plot_surface(xx,yy,ut[:,:,-1],
# cmap=cm.bone,linewidth=0,
# antialiased=False)
CS=ax.contourf(x,y,ut[:,:,-1].T,4,cmap=plt.cm.bone)
plt.colorbar(CS)
ax.set_xlabel(r'$x_1$')
ax.set_ylabel(r'$x_2$')
plt.savefig('fig10_11a.pdf')
def exa3_40(needFigure=True):
T = 10
N = 1e3
a = [2 * pi, 16]
J = [128, 256]
epsilon = 1e-3
x = np.linspace(0, a[0], J[0] + 1)
y = np.linspace(0, a[1], J[1] + 1)
[xx, yy] = np.meshgrid(x, y, indexing='ij')
# initial data printed in book
#u0=np.sin(xx)*np.cos(pi*yy/8)
# we think he intended this initial data, which gives Fig 3.8b
u0 = np.sin(yy) * np.cos(pi * xx / 8)
start = timer()
[t, ut] = pde_twod_Gal(u0, T, a, N, J, epsilon, fAC)
end = timer()
print("Run time", end - start, "secs")
if (needFigure == True):
plt.figure(1)
ch0.PlotSetup()
ax = plt.gca(projection='3d')
ax.plot_surface(xx,
yy,
ut[:, :, -1],
cmap=cm.bone,
linewidth=0,
antialiased=False)
plt.axis('equal')
#
ax.set_xlabel(r'$x$')
ax.set_ylabel(r'$y$')
ax.set_zlabel(r'$u$')
#
plt.savefig('fig3_5b.pdf')
def exa10_39(needFigure=True):
T = 2
N = 50
a = 2 * pi
J = 64
r = 1
epsilon = 0.001
sigma = 1
M = 2
kappa = 1
x = np.linspace(0, a, J + 1)
u0 = np.sin(x)
s0 = np.random.RandomState()
tref, uref, ureft = spde_oned_Gal_MJDt(u0, T, a, N, kappa, J, J, epsilon,
fAC, lambda u: sigma, r, M, s0)
if (needFigure == True):
plt.figure(1)
ch0.PlotSetup()
ax = plt.gca()
ax = plt.gca(projection='3d')
[T, X] = np.meshgrid(tref, x)
CS = ax.plot_wireframe(T, X, ureft, rstride=3, cstride=3, colors='k')
#CS=ax.contourf(tref,x,ureft,10,cmap=plt.cm.bone)
#ax.set_zlabel(r'$u$')
ax.set_xlabel(r'$t$')
ax.set_ylabel(r'$x$')
# plt.colorbar(CS)
plt.savefig('fig10_.pdf')
def exa_fbm(H=0.1):
T=10; N=100;
t=np.linspace(0,T,N);
X=fbm(t,H)
plt.figure(1)
ch0.PlotSetup()
plt.plot(t,X,'k-')
plt.xlabel(r'$t$')
plt.ylabel(r'$B^H$')
plt.savefig('fig5_fbm.pdf',bbox_inches='tight')
def exa_bm():
T=10; N=100;
t=np.linspace(0,T,N)
X=bmotion(t)
plt.figure(1)
ch0.PlotSetup()
plt.plot(t,X,'k-')
plt.xlabel(r'$t$')
plt.ylabel(r'X')
plt.savefig('fig5_bm.pdf',bbox_inches='tight')
def fig6_7():
t = np.linspace(0, 20, 200)
Z = quad_sinc(t, 100, 2)
plt.figure(1)
ch0.PlotSetup()
plt.plot(t, np.real(Z))
plt.plot(t, np.imag(Z))
plt.xlabel(r'$t$')
plt.ylabel(r'$Z$')
plt.savefig('fig6_7b.pdf', bbox_inches='tight')
def fig6_9():
N = int(1e3)
dt = 1 / (N - 1)
t, X, Y = circulant_exp(N, dt, 1)
ch0.PlotSetup()
plt.plot(t, X)
plt.plot(t, Y)
plt.xlabel(r'$t$')
plt.ylabel(r'$X$')
plt.savefig('fig6_9.pdf', bbox_inches='tight')
def exa10_10():
dtref=0.01; kappa=100; r=1/2; J=128; a=1;
bj=get_onedD_bj(dtref,J,a,r)
s0=np.random.RandomState()
dW=get_onedD_dW(bj,kappa,0,1,s0)
x=np.linspace(0,a,J-1)
plt.figure(1)
ch0.PlotSetup()
plt.plot(x,np.squeeze(dW))
plt.xlabel(r'$t$')
plt.ylabel(r'$dW$')
plt.savefig('fig10_.pdf',bbox_inches='tight')
def fig7_10c():
grid = np.linspace(0, 10, 200)
u = turn_band_wm(grid, grid, 10, 1, 1)
plt.figure(1)
ch0.PlotSetup()
ax = plt.gca()
#ax.plot_wireframe(T,X,ut,rstride=16,cstride=1000,colors='k')
CS = ax.contourf(grid, grid, u, 10, cmap=plt.cm.bone)
#ax.set_zlabel(r'$u$')
ax.set_xlabel(r'$y$')
ax.set_ylabel(r'$x$')
plt.colorbar(CS)
plt.savefig('fig7_10c.pdf', bbox_inches='tight')
def exa3_8():
u0 = np.array([0.5, 0.1])
T = 1
Nref = 1e6
d = 2
kappa = np.array([10, 1e2, 1e3, 1e4, 1e5])
[dt, errA, errB] = exp_euler_conv(u0, T, Nref, d, f, kappa)
plt.figure(1)
ch0.PlotSetup()
plt.xlabel(r'$\Delta t$')
plt.ylabel(r'error')
plt.loglog(dt, errA, '-k')
plt.savefig('fig3_2.pdf', bbox_inches='tight')
def exa10_43b():
T=1; N=int(1e3); a=1; r=1; sigma=0.5; epsilon=1e-3; M=1;
Nref=1e5; ne=32; h=a/ne; x=np.linspace(0,a,ne+1)
u0=np.exp(-(x-0.5)**2/epsilon)
kappa=np.array([5,10,20,50,100,200,500])
dt,errT=spde_fem_convDt(u0,T,a,Nref,kappa,ne,epsilon,
fNagumo,
lambda u:gquadratic(u,sigma),
r,M)
plt.figure(1)
ch0.PlotSetup()
plt.loglog(dt,errT,'-k*')
plt.xlabel(r'$\Delta t$')
plt.ylabel(r'error')
plt.savefig('fig10_.pdf',bbox_inches='tight')
def exa10_31():
T=0.1; epsilon=0.1; sigma=1; a=1;
M=int(1e4); J=4*2**np.array([2,3,4]);
N=0.25*J**2;
v=np.zeros(N.shape)
for i in range(N.size):
u0=np.zeros(J[i]+1)
v[i]=l2_sq_mct(T,a,int(N[i]),J[i],M,epsilon,sigma)
plt.figure(1)
ch0.PlotSetup()
plt.plot(N,v)
plt.xlabel(r'$J$')
plt.ylabel(r'L2 SQ$')
plt.title(r'Converges to 0.3489')
plt.savefig('exa10_31.pdf',bbox_inches='tight')
def exa8_23():
lam = 1
alpha = 1
sigma = 1
u0 = np.array([0.5, 0])
T = 10
N = 1000
[t, u] = vdp(u0, T, N, alpha, lam, sigma)
plt.figure(1)
ch0.PlotSetup()
plt.plot(t, u[0, :])
plt.plot(t, u[1, :])
plt.xlabel(r'$t$')
plt.ylabel(r'$u$')
plt.savefig('fig8_5.pdf', bbox_inches='tight')
def exa10_43():
T=1; N=int(1e4); a=1; r=1; sigma=0.5; epsilon=1e-3;
neref=512; href=a/neref; x=np.linspace(0,a,neref+1)
u0=np.exp(-(x-0.5)**2/epsilon)
L=np.array([2,4,8,16,32],dtype='int32')
M=1
h,errh=spde_fem_convh(u0,T,a,N,neref,L,epsilon,
fNagumo,
lambda u: gquadratic(u,sigma),
r,M)
plt.figure(1)
ch0.PlotSetup()
plt.loglog(h,errh,'k-*')
plt.xlabel(r'$h$')
plt.ylabel(r'error')
plt.savefig('fig10_.pdf',bbox_inches='tight')
def exa7_41():
fhandle1 = lambda x1, x2: sep_exp(x1, x2, 1 / 5, 1 / 10)
C_red = reduced_cov(201, 401, 1 / 200, 1 / 200, fhandle1)
u1, u2 = circ_embed_sample_2d(C_red, 201, 401)
plt.figure(1)
ch0.PlotSetup()
ax = plt.gca()
x = np.linspace(0, 1, 201)
y = np.linspace(0, 2, 401)
#ax.plot_wireframe(T,X,ut,rstride=16,cstride=1000,colors='k')
CS = ax.contourf(y, x, u1, 10, cmap=plt.cm.bone)
#ax.set_zlabel(r'$u$')
ax.set_xlabel(r'$y$')
ax.set_ylabel(r'$x$')
plt.colorbar(CS)
plt.savefig('fig7_6.pdf', bbox_inches='tight')
def exa10_12():
J=[32,16]; dtref=0.01; kappa=100; a=[2*pi,3*pi]
alpha=0.05; bj=get_twod_bj(dtref,J,a,alpha)
W1,W2=get_twod_dW(bj,kappa,1)
plt.figure(1)
ch0.PlotSetup()
gridx=np.linspace(0,a[0],J[0])
gridy=np.linspace(0,a[1],J[1])
ax = plt.gca()
#ax.plot_wireframe(T,X,ut,rstride=16,cstride=1000,colors='k')
CS=ax.contourf(gridy,gridx,W1[0,:,:],10,cmap=plt.cm.bone)
#ax.set_zlabel(r'$u$')
ax.set_xlabel(r'$x$')
ax.set_ylabel(r'$y$')
plt.colorbar(CS)
plt.savefig('fig10_.pdf')
def fig6_6():
T = 30
N = 2**9
t = np.linspace(0, T, N)
M = N / 4
vector_q = [0.5, 1, 1.5, 2]
ch0.PlotSetup()
for i in range(1, 5):
q = vector_q[i - 1]
X, Y = quad_wm(t, N, M, q)
plt.subplot(2, 2, i)
plt.plot(t, X)
plt.plot(t, Y)
plt.xlabel(r'$t$')
plt.ylabel(r'$X$')
plt.savefig('fig6_6.pdf', bbox_inches='tight')
def exa10_30(needFigure=True):
a=20; J=1024; x=np.linspace(0,a,J+1); u0=1/(1+np.exp(-(2-x)/sqrt(2)));
ell=1; N=int(1e3); T=1; epsilon=1; sigma=1;
t,ut=spde_fd_n_exp(u0,T,a,N,J,epsilon,sigma,ell, fNagumo)
[T,X]=np.meshgrid(t,x)
#
if (needFigure==True):
plt.figure(1)
ch0.PlotSetup()
ax = plt.gca()
#ax.plot_wireframe(T,X,ut,rstride=16,cstride=1000,colors='k')
CS=ax.contourf(X,T,ut,10,cmap=plt.cm.bone)
#ax.set_zlabel(r'$u$')
ax.set_xlabel(r'$x$')
ax.set_ylabel(r'$t$')
plt.colorbar(CS)
plt.savefig('fig10_.pdf')
def exa3_41b():
epsilon = 1
a = 2 * pi
Jref = 1024
N = 1e4
x = np.linspace(0, a, Jref + 1)
u0 = np.sin(4 * x) * np.sin(x)
J = Jref * np.array([1 / 2, 1 / 4, 1 / 8, 1 / 16, 1 / 32, 1 / 64])
T = 10
errJ = pde_oned_Gal_convJ(u0, T, a, N, Jref, J, epsilon, fAC)
plt.figure(1)
ch0.PlotSetup()
plt.loglog(J, errJ, '-k*')
plt.xlabel(r'$J$')
plt.ylabel(r'error')
plt.savefig('fig3_6b.pdf', bbox_inches='tight')
def exa3_41():
epsilon = 1
a = 2 * pi
J = 1024
x = np.linspace(0, a, J + 1)
u0 = np.sin(4 * x) * np.sin(x)
kappa = np.array([5, 10, 20, 50, 100, 200], dtype='int')
T = 10
Nref = int(1e4)
[dt, errT] = pde_oned_Gal_convDt(u0, T, a, Nref, kappa, J, epsilon, fAC)
plt.figure(1)
ch0.PlotSetup()
plt.loglog(dt, errT, '-k*')
plt.xlabel(r'$\Delta t$')
plt.ylabel(r'error')
plt.savefig('fig3_6a.pdf', bbox_inches='tight')
def exa8_25():
d = 2
m = 2
A = np.array([[1, 0], [0, -1]])
sigma = np.array([1, 2])
T = 1
N = 5000
u0 = np.array([1, 1])
t, u = MilsteinDiag(u0, T, N, d, m, lambda u: A.dot(u),
lambda u: sigma * u, lambda u: sigma)
plt.figure(1)
ch0.PlotSetup()
plt.plot(t, u[0, :])
plt.plot(t, u[1, :])
plt.xlabel(r'$t$')
plt.ylabel(r'$u$')
plt.savefig('fig8_6.pdf', bbox_inches='tight')
def fig1_6():
N = 10
J = np.zeros(N)
l2_norm_1 = np.copy(J)
l2_norm_2 = np.copy(J)
h1_norm_1 = np.copy(J)
h1_norm_2 = np.copy(J)
a = 0
b = 1
# domain
for j in range(10):
N = int(pow(2, (j + 3)))
J[j] = N
l2_norm_1[j], h1_norm_1[j] = get_norm(f_u1, a, b, N)
for j in range(10):
N = int(pow(2, (j + 3)))
l2_norm_2[j], h1_norm_2[j] = get_norm(f_u2, a, b, N)
# get symbolic values
[l2_norm_sym_1, h1_norm_sym_1] = sp_u1(a, b)
[l2_norm_sym_2, h1_norm_sym_2] = sp_u2(a, b)
# computer errors
el2_1 = abs(l2_norm_1 - l2_norm_sym_1)
ehs_1 = abs(h1_norm_1 - h1_norm_sym_1)
el2_2 = abs(l2_norm_2 - l2_norm_sym_2)
ehs_2 = abs(h1_norm_2 - h1_norm_sym_2)
# plotting
plt.figure(1)
ch0.PlotSetup()
plt.axis('equal')
plt.subplot(1, 2, 1)
#
plt.loglog(J, el2_1, 'k-')
plt.loglog(J, ehs_1, 'k-.')
plt.xlabel(r'$J$')
plt.ylabel(r'error')
plt.title(r'(a)')
#
plt.subplot(1, 2, 2)
plt.loglog(J, el2_2, 'k-')
plt.loglog(J, ehs_2, 'k-.')
plt.xlabel(r'$J$')
plt.title(r'(b)')
#
plt.tight_layout()
def example_p58():
plt.figure(1)
ch0.PlotSetup()
for i in range(1, 5):
plt.subplot(2, 2, i)
ne = pow(2, i)
uh, A, b, K, M = oned_linear_FEM(ne, np.ones(ne), 10 * np.ones(ne),
np.ones(ne))
plt.plot(np.linspace(0, 1, ne + 1), uh, '-ok')
plt.xlabel(r'$x$')
plt.ylabel(r'$u$')
xx = np.linspace(0, 1, 100)
uex = exact_exa2_3(xx)
plt.plot(xx, uex, '-k')
plt.tight_layout()
plt.savefig('fig2_6.pdf', bbox_inches='tight')
error = test_FEM_error(ne, uh)
print('error', error)
def exa8_28():
u0 = 1
T = 10
N = 40
d = 1
m = 1
theta = 1
r = -8
sigma = 3
t, u = EulerMaruyamaTheta(u0, T, N, d, m, lambda u: r * u,
lambda u: sigma * u, theta)
plt.figure(1)
ch0.PlotSetup()
plt.plot(t, u[0, :])
plt.xlabel(r'$t$')
plt.ylabel(r'$u$')
plt.savefig('fig8_8.pdf', bbox_inches='tight')
def exa3_5(flag=1):
"""
flag=1 Explicit Euler (default)
flag=0 Implicit Euler
"""
u0 = [0.5, 0.1]
T = 10
N = 1000
d = 2
if (flag == 1):
[t, u] = exp_euler(u0, T, N, d, f)
else:
[t, u] = imp_euler(u0, T, N, d, f)
plt.figure(1)
ch0.PlotSetup()
plt.plot(t, u[1, :], 'k-')
plt.xlabel(r'$t$')
plt.ylabel(r'$u_1$')
plt.savefig('fig3_1.pdf', bbox_inches='tight')
def exa3_45():
T = 1
N = int(1e4)
kappa = 1
epsilon = 1e-3
neref = 512
a = 1
L = np.array([2, 4, 8, 16, 32], dtype='int')
href = a / neref
x = np.linspace(0, a, neref + 1)
u0 = np.exp(-(x - 0.5)**2 / epsilon)
h, errh = pde_fem_convh(u0, T, a, N, neref, L, epsilon, fNagumo)
plt.figure(1)
ch0.PlotSetup()
plt.loglog(h, errh, 'k-*')
plt.xlabel(r'$h$')
plt.ylabel(r'error')
plt.savefig('fig3_7b.pdf', bbox_inches='tight')
