def exact_solution_KP4459():
"""Exactly solvable test system from Kloeden & Platen ch 4.4 eqn (4.59)"""
h = 0.0004
tspan = np.arange(0.0, 5.0, h)
N = len(tspan)
y0 = np.array([1.0])
a = -0.02
b1 = 1.0
b2 = 2.0
# Ito version:
def f(y, t):
return np.array([a * y[0]])
def G(y, t):
return np.array([[b1 * y[0], b2 * y[0]]])
# Stratonovich version of the same equation:
def f_strat(y, t):
return np.array([(a - (b1**2 + b2**2) / 2.0) * y[0]])
# Generate Wiener increments and repeated integrals for one sample path:
dW = sdeint.deltaW(N - 1, 2, h)
__, I = sdeint.Iwik(dW, h)
J = I + 0.5 * h * np.eye(2).reshape((1, 2, 2)) # since m==2
# compute exact solution y for that sample path:
y = np.zeros(N)
y[0] = y0
Wn = np.array([0.0, 0.0])
for n in range(1, N):
tn = tspan[n]
Wn += dW[n - 1, :]
y[n] = y0 * np.exp((a - (b1**2 + b2**2) / 2.0) * tn + b1 * Wn[0] +
b2 * Wn[1])
return (dW, I, J, f, f_strat, G, y0, tspan, y)
def exact_solution_KP4446():
"""Exactly solvable test system from Kloeden & Platen ch 4.4 eqn (4.46)
By making this a fixture, the exact solution can be re-used in several
tests without re-calculating it.
"""
h = 0.0004
tspan = np.arange(0.0, 10.0, h)
N = len(tspan)
y0 = 0.1
a = 1.0
b = 0.80
# Ito version:
def f(y, t):
return -(a + y*b**2)*(1 - y**2)
def G(y, t):
return b*(1 - y**2)
# Stratonovich version of the same equation:
def f_strat(y, t):
return -a*(1 - y**2)
# Generate Wiener increments and repeated integrals for one sample path:
dW = sdeint.deltaW(N-1, 1, h)
__, I = sdeint.Iwik(dW, h) # Ito repeated integrals
J = I + 0.5*h # Stratonovich repeated integrals (m==1)
# Compute exact solution y for that sample path:
y = np.zeros(N)
y[0] = y0
Wn = 0.0
for n in range(1, N):
tn = tspan[n]
Wn += dW[n-1]
y[n] = (((1 + y0)*np.exp(-2.0*a*tn + 2.0*b*Wn) + y0 - 1.0)/
((1 + y0)*np.exp(-2.0*a*tn + 2.0*b*Wn) + 1.0 - y0))
return (dW, I, J, f, f_strat, G, y0, tspan, y)
def exact_solution_KP4459():
"""Exactly solvable test system from Kloeden & Platen ch 4.4 eqn (4.59)"""
h = 0.0004
tspan = np.arange(0.0, 5.0, h)
N = len(tspan)
y0 = np.array([1.0])
a = -0.02
b1 = 1.0
b2 = 2.0
# Ito version:
def f(y, t):
return np.array([a*y[0]])
def G(y, t):
return np.array([[b1*y[0], b2*y[0]]])
# Stratonovich version of the same equation:
def f_strat(y, t):
return np.array([(a - (b1**2 + b2**2)/2.0)*y[0]])
# Generate Wiener increments and repeated integrals for one sample path:
dW = sdeint.deltaW(N-1, 2, h)
__, I = sdeint.Iwik(dW, h)
J = I + 0.5*h*np.eye(2).reshape((1, 2, 2)) # since m==2
# compute exact solution y for that sample path:
y = np.zeros(N)
y[0] = y0
Wn = np.array([0.0, 0.0])
for n in range(1, N):
tn = tspan[n]
Wn += dW[n-1,:]
y[n] = y0*np.exp((a - (b1**2 + b2**2)/2.0)*tn + b1*Wn[0] + b2*Wn[1])
return (dW, I, J, f, f_strat, G, y0, tspan, y)
def exact_solution_R74():
"""exactly solvable test system from Roessler2010 eqn (7.4)"""
d = 5
m = 4
h = 0.0004
tspan = np.arange(0.0, 10.0, h)
N = len(tspan)
A = np.ones((d, d)) * 0.05
B0 = np.ones((d, d)) * 0.01
for i in range(0, d):
A[i, i] = -1.5
B0[i, i] = 0.2
B = np.vstack([B0[np.newaxis, ...]] * m)
y0 = np.ones(d)
# Ito version:
def f(y, t):
return np.dot(A, y)
def G(y, t):
return np.dot(B, y).T
G_separate = [lambda y, t: np.dot(B0, y)] * m
# Stratonovich version of the same equation:
S = np.ones((d, d)) * -0.0086
for i in range(0, d):
S[i, i] = -0.0808
def f_strat(y, t):
return np.dot(A + S, y)
# Generate Wiener increments and repeated integrals for one sample path:
dW = sdeint.deltaW(N - 1, m, h) # shape (N-1, m)
__, I = sdeint.Iwik(dW, h)
J = I + 0.5 * h * np.eye(m).reshape((1, m, m))
# compute exact solution y for that sample path:
y = np.zeros((N, d))
y[0] = y0
Wn = np.zeros(m)
term1 = A - 0.5 * np.einsum('kij,kjl', B, B)
for n in range(1, N):
tn = tspan[n]
Wn += dW[n - 1]
term2 = np.einsum('kij,k', B, Wn)
y[n] = linalg.expm(term1 * tn + term2).dot(y0)
return (dW, I, J, f, f_strat, G, G_separate, y0, tspan, y)
def exact_solution_R74():
"""exactly solvable test system from Roessler2010 eqn (7.4)"""
d = 5
m = 4
h = 0.0004
tspan = np.arange(0.0, 10.0, h)
N = len(tspan)
A = np.ones((d, d))*0.05
B0 = np.ones((d, d))*0.01
for i in range(0, d):
A[i, i] = -1.5
B0[i, i] = 0.2
B = np.vstack([B0[np.newaxis,...]]*m)
y0 = np.ones(d)
# Ito version:
def f(y, t):
return np.dot(A, y)
def G(y, t):
return np.dot(B, y).T
G_separate = [lambda y, t: np.dot(B0, y)] * m
# Stratonovich version of the same equation:
S = np.ones((d, d))*-0.0086
for i in range(0, d):
S[i, i] = -0.0808
def f_strat(y, t):
return np.dot(A + S, y)
# Generate Wiener increments and repeated integrals for one sample path:
dW = sdeint.deltaW(N-1, m, h) # shape (N-1, m)
__, I = sdeint.Iwik(dW, h)
J = I + 0.5*h*np.eye(m).reshape((1, m, m))
# compute exact solution y for that sample path:
y = np.zeros((N, d))
y[0] = y0
Wn = np.zeros(m)
term1 = A - 0.5*np.einsum('kij,kjl', B, B)
for n in range(1, N):
tn = tspan[n]
Wn += dW[n-1]
term2 = np.einsum('kij,k', B, Wn)
y[n] = linalg.expm(term1*tn + term2).dot(y0)
return (dW, I, J, f, f_strat, G, G_separate, y0, tspan, y)
def exact_solution_KP4446():
"""Exactly solvable test system from Kloeden & Platen ch 4.4 eqn (4.46)
By making this a fixture, the exact solution can be re-used in several
tests without re-calculating it.
"""
h = 0.0004
tspan = np.arange(0.0, 10.0, h)
N = len(tspan)
y0 = 0.1
a = 1.0
b = 0.80
# Ito version:
def f(y, t):
return -(a + y * b**2) * (1 - y**2)
def G(y, t):
return b * (1 - y**2)
# Stratonovich version of the same equation:
def f_strat(y, t):
return -a * (1 - y**2)
# Generate Wiener increments and repeated integrals for one sample path:
dW = sdeint.deltaW(N - 1, 1, h)
__, I = sdeint.Iwik(dW, h) # Ito repeated integrals
J = I + 0.5 * h # Stratonovich repeated integrals (m==1)
# Compute exact solution y for that sample path:
y = np.zeros(N)
y[0] = y0
Wn = 0.0
for n in range(1, N):
tn = tspan[n]
Wn += dW[n - 1]
y[n] = (((1 + y0) * np.exp(-2.0 * a * tn + 2.0 * b * Wn) + y0 - 1.0) /
((1 + y0) * np.exp(-2.0 * a * tn + 2.0 * b * Wn) + 1.0 - y0))
return (dW, I, J, f, f_strat, G, y0, tspan, y)