def analyse_variance(example_name):
dpms = DiscretePMS()
dpms.name = example_name
n_particles = 2**np.arange(2, 7)
mean_vec = []
var_vec = []
n_mc = 50
for n in n_particles:
if dpms.name == "linear":
# masses
m_r = 2.
m_q = 10. / n
# stiffness
kappa_r = 1.
kappa_q = 1 / n
# forces
U_r = lambda r, dr: 0.5 * kappa_r * r**2
T_r = lambda r, dr: 0.5 * m_r * dr**2
U_q = lambda q, dq: 0.5 * kappa_q * q**2
T_q = lambda q, dq: 0.5 * m_q * dq**2
# constraint
g = lambda r, q: q - r
t_end = 23
r0 = 1.
dr0 = 0.
mean = 2.0
scale = 1.0
if dpms.name == "onion":
# masses
m_r = 2.
m_q = 1. / n
# stiffness
kappa_r = 1.
kappa_q = 0.0 / n
# forces
U_r = lambda r, dr: 0.5 * kappa_r * r**2
T_r = lambda r, dr: 0.5 * m_r * dr**2
U_q = lambda q, dq: 0.5 * kappa_q * q**2
T_q = lambda q, dq: 0.5 * m_q * dq**2
# constraint
factor = 1.
# g = lambda r, q : factor*q/(1+r**2) - q**3
g = lambda r, q: -factor * sp.sin(0.1 * q) / (1 + r**2) + 0.1 * q
t_end = 20
r0 = 1.
dr0 = 0.
mean = 0.9
scale = 1.0
r_end = np.zeros(shape=(n_mc, ))
dpms.init_equations(T_r, U_r, T_q, U_q, g)
# colors in plots
col_min = np.array([0., 0.6, 0.9])
col_max = np.array([0.6, 0., 0.9])
for i in range(0, n_mc):
# initial distribution
Q0 = np.random.normal(loc=mean, scale=scale, size=(n, ))
dpms.init_state(r0, dr0, Q0, t_end)
dpms.simulate()
r_end[i] = dpms.r[-1]
p = i / (n_mc - 1)
plt.plot(dpms.sol.t,
dpms.r,
color=p * col_min + (1. - p) * col_max,
lw=0.3)
# calucate mean and variance at t_end
cur_mean = 1 / n_mc * np.sum(r_end)
cur_var = 1 / (n_mc - 1) * np.sum((r_end - cur_mean)**2)
mean_vec.append(cur_mean)
var_vec.append(cur_var)
plt.xlabel(r"$t$")
plt.ylabel(r"$r(t)$")
plt.title("heavy system (%d samples, %d particles)" % (n_mc, n))
plt.savefig(
gen_name(dpms, "disc") + ("_mc%d_n%d" % (n_mc, n)) + ".pdf")
plt.show()
plt.loglog(n_particles,
var_vec,
label=r"$\mathrm{Var}[ r(t_\text{end} ) ]$")
plt.loglog(n_particles, 1. / n_particles, label=r"$n^{-1}$")
plt.xlabel(r"$\log(n)$")
plt.title("variance of the heavy system")
#plt.ylabel(r"$\log$")
plt.legend()
plt.savefig(gen_name(dpms, "disc") + ("_variance_mc%d" % (n_mc)) + ".pdf")
plt.show()
# stiffness kappa_r = 1. kappa_q = 0.0 / n # forces U_r = lambda r, dr: 0.5 * kappa_r * r**2 T_r = lambda r, dr: 0.5 * m_r * dr**2 U_q = lambda q, dq: 0.5 * kappa_q * q**2 T_q = lambda q, dq: 0.5 * m_q * dq**2 # constraint factor = 1. # g = lambda r, q : factor*q/(1+r**2) - q**3 # g = lambda r, q : q**2 - 1/(1+(r-0.5)**2) # start within the circle! # g = lambda r, q : q - r g = lambda r, q: q**2 + (r - 2)**2 # looks nice and smooth! # g = lambda r, q : # initial distribution r0 = 3. dr0 = 0. Q0 = np.random.normal(loc=2, scale=0.2, size=(n, )) t_end = 20 dpms.init_equations(T_r, U_r, T_q, U_q, g) dpms.init_state(r0, dr0, Q0, t_end, n_eval=1000) dpms.simulate(method='BDF') save_plots(plt, dpms, "singular", "disc")
# forces U_r = lambda r, dr: 0.5 * kappa_r * r**2 T_r = lambda r, dr: 0.5 * m_r * dr**2 U_q = lambda q, dq: 0.5 * kappa_q * q**2 T_q = lambda q, dq: 0.5 * m_q * dq**2 # constraint factor = 1. # g = lambda r, q : factor*q/(1+r**2) - q**3 g = lambda r, q: -factor * sp.sin(0.1 * q) / (1 + r**2) + 0.1 * q # g = lambda r, q : q**2 - 1/(1+(r-0.5)**2) # start within the circle! # g = lambda r, q : q - r # g = lambda r, q : q**2 + r**2 # looks nice and smooth! # g = lambda r, q : # initial distribution r0 = 1. dr0 = 0. Q0 = np.random.normal(loc=2.5, scale=0.3, size=(n, )) t_end = 24 #60 dpms.init_equations(T_r, U_r, T_q, U_q, g) dpms.init_state(r0, dr0, Q0, t_end, n_eval=400) dpms.simulate() dpms.name = "onion" save_plots(plt, dpms, "disc")
# forces U_r = lambda r, dr: 0.5 * kappa_r * r**2 T_r = lambda r, dr: 0.5 * m_r * dr**2 U_q = lambda q, dq: 0.5 * kappa_q * q**2 T_q = lambda q, dq: 0.5 * m_q * dq**2 # constraint factor = 1. # g = lambda r, q : factor*q/(1+r**2) - q**3 # g = lambda r, q : q**2 - 1/(1+(r-0.5)**2) # start within the circle! # g = lambda r, q : q - r g = lambda r, q: q**2 + (r - 2)**2 # looks nice and smooth! # g = lambda r, q : # initial distribution r0 = 3. dr0 = 0. Q0 = np.random.normal(loc=2., scale=0.2, size=(n, )) t_end = 8 dpms.init_equations(T_r, U_r, T_q, U_q, g) dpms.init_state(r0, dr0, Q0, t_end, n_eval=8000) G_noise = 0.2 * np.concatenate([np.array([0., 0.]), np.ones((n, ))]) dpms.simulate(G=lambda y, t: np.diag(G_noise)) save_plots(plt, dpms, "singular_stoch_new", "disc")