marker="x",
label="noisy pressure drop")
plt.xlabel("$t_n$", fontsize=24)
plt.ylabel("$\delta p_h^n$", fontsize=24)
plt.yticks(fontsize=20)
plt.xticks(fontsize=20)
plt.legend(loc=4, borderaxespad=0.0, prop={'size': 20}, frameon=False)
plt.tight_layout()
plt.show(block=True)
### instantiate an MCMC sampler
mcmc = MCMC(density, proposal_density, draw_from_proposal,
initial_point_mcmc)
# run the MCMC sample
mcmc.run(max_iter=10000, burning=1000)
# write the samples into a file
#mcmc.write("samples-N3-fric0.075-wNoise.dat")
# plotting the sampled friction coefficients
for idx, coef in enumerate(mcmc.density_samples):
pipe.get_lambda_average(coef)
plt.plot(pipe.mesh, pipe.lambda_avg)
pipe_true.get_lambda_average(true_friction)
plt.plot(pipe_true.mesh,
pipe_true.lambda_avg,
'--',
color="black",
linewidth=5.0,
label="True friction function")
#plt.plot(pipe.timeslices, pipe.pressure_drop, "o-")
if i in [1, 9, 11, 19, 25, 41]:
plt.plot(pipe_tmp.mesh,
pipe_tmp.lambda_avg,
linestyle=linestyles[counter % len(linestyles)],
label="$N=" + str(i / 2) + "$")
counter += 1
plt.legend(loc="best", prop={'size': 16}, frameon=False)
plt.yticks(fontsize=20)
plt.xticks(fontsize=20)
plt.xlabel("$x$", fontsize=24)
plt.ylabel("$\lambda^N(x)$", fontsize=24)
plt.tight_layout()
plt.savefig("results/figure_fric_bump.pgf")
plt.show()
# clean memory
collect()
pipe.info()
### instantiate an MCMC sampler
mcmc = MCMC(density, proposal_density, draw_from_proposal,
initial_point_mcmc)
# run the MCMC sample
mcmc.run(max_iter=200000, burning=5000)
# write the samples into a file
mcmc.write(filename, write_prob=True)
class MCMCBayesianInversion(BayesianInversion):
'''
GasBayesianInversion solves a Bayesian Inverse
problem for Gas pipe friction coefficients.
'''
def __init__(self, c_sound, t_final, x_l, x_r, dx, expan_coef,
boundary_eps, time_ins):
BayesianInversion.__init__(self, c_sound, t_final, x_l, x_r, dx,
expan_coef, boundary_eps, time_ins)
def uni_prior(self, x):
'''
uniform prior
Parameters
----------
x: float * len(x)
'''
uni_prior_up = self.uni_prior_range["up"]
uni_prior_down = self.uni_prior_range["down"]
volume = 1.0
for i in range(len(x)):
volume *= abs(uni_prior_up[i] - uni_prior_down[i])
normalize = 1.0 / volume
for i in range(len(x)):
if (((x[i] < uni_prior_up[i]) &
(x[i] > uni_prior_down[i])) == False):
return 0.0
return normalize
def proposal_density(self, new, old):
'''
proposal density function
Parameters
----------
new: float * len(new)
old: float * len(old)
'''
length = len(new)
out = empty(length, dtype=float)
for i in range(length):
out[i] = 1.0/(2*pi*self.sigma_normal**2) * \
exp( -0.5*(new[i] - old[i])**2 / self.sigma_normal**2 )
return out
def draw_from_proposal(self, old):
'''
draw from the proposal
Parameters
----------
old: float * len(old)
'''
return normal(old, self.sigma_normal, size=len(old))
def likelihood(self, x):
'''
computes the likelihood function
Parameters
----------
pipe: SemiLinSystem
x: float * len(x)
'''
pipe = self.pipe
pipe.run(x, progress_bool=False)
# check if we have negative friction
pipe.get_current_lambda_average()
for i in pipe.lambda_avg:
if i < 0.0: return 0.0
S = pipe.get_presure_drop(time_instance=self.time_ins, inplace=False)
Proj_y_obs = self.EvalY_obs(pipe.timeslices, self.y_obs,
self.y_obs_times)
out = exp(-0.5 * dot(S - Proj_y_obs, S - Proj_y_obs) / self.epsilon)
#print (out, self.epsilon)
return out
def density(self, x):
'''
density function for the MCMC:
multiplication of the likelihood and the prior distribution
Parameters
----------
x: float * len(x)
'''
PRIOR = self.uni_prior(x)
if PRIOR > 10.0**-8:
return self.likelihood(x) * PRIOR
else:
return 0.0
def run(self,
conf,
y_obs,
y_obs_times,
max_iter=2000,
burning=500,
jupyter=False):
'''
run_mcmc takes an epsilon and compute the y_obs and then run mcmc using those observations.
Returns
-------
samples: list
'''
from UQuant.mcmc import MCMC
self.epsilon = conf["epsilon"]
self.sigma_normal = conf["sigma_normal"]
self.initial_point_mcmc = conf["initial_point_mcmc"]
self.uni_prior_range = conf["uni_prior_range"]
self.y_obs = y_obs
self.y_obs_times = y_obs_times
self.sampler = MCMC(self.density, self.proposal_density,
self.draw_from_proposal, self.initial_point_mcmc)
self.sampler.run(max_iter=max_iter, burning=burning, jupyter=jupyter)
# pipe.run([0.075])
# pipe.get_presure_drop(inplace=True)
# plt.plot(pipe.timeslices, pipe.pressure_drop, "-x")
# #plt.plot(pipe_true.timeslices, pipe_true.pressure_drop, "-o")
# #plt.plot(y_obs_times, y_obs, "-o")
# plt.plot(pipe.timeslices, EvalY_obs(pipe.timeslices), "-*")
# plt.show(block=False)
for i in range(0, 51):
if i % 10 == 1:
pipe = SemiLinSystem(c_sound, t_final, x_l, x_r, dx, expan_coef,
boundary_eps)
pipe.info()
mcmc = MCMC(density, proposal_density, draw_from_proposal,
initial_point_mcmc)
mcmc.run(2000, burning=500)
samples[dx] = mcmc.density_samples
dx += 0.0001
gc.collect() # collect un-used objects
# plt.legend(loc=2, borderaxespad=0.0)
# plt.show(block=False)
Samples = {}
for idx in samples:
Samples[idx] = []
for point in samples[idx]:
Samples[idx].append(point[0])
for idx in Samples: