def setup_check_variance_reduction_model_ensemble_short_column(
nmodels=5,npilot_samples=None):
example = ShortColumnModelEnsemble()
model_ensemble = pya.ModelEnsemble(
[example.models[ii] for ii in range(nmodels)])
univariate_variables = [
uniform(5,10),uniform(15,10),norm(500,100),norm(2000,400),
lognorm(s=0.5,scale=np.exp(5))]
variable=pya.IndependentMultivariateRandomVariable(univariate_variables)
generate_samples=partial(
pya.generate_independent_random_samples,variable)
if npilot_samples is not None:
# The number of pilot samples effects ability of numerical estimate
# of variance reduction to match theoretical value
cov, samples, weights = pya.estimate_model_ensemble_covariance(
npilot_samples,generate_samples,model_ensemble)
else:
# it is difficult to create a quadrature rule for the lognormal
# distribution so instead define the variable as normal and then
# apply log transform
univariate_variables = [
uniform(5,10),uniform(15,10),norm(500,100),norm(2000,400),
norm(loc=5,scale=0.5)]
variable=pya.IndependentMultivariateRandomVariable(
univariate_variables)
example.apply_lognormal=True
cov = example.get_covariance_matrix(variable)[:nmodels,:nmodels]
example.apply_lognormal=False
return model_ensemble, cov, generate_samples
def plot_mlmc_error():
"""
Define function to create plot so we can create it later and add
other estimator error curves.
Note this is only necessary for sphinx-gallery docs. If just using the
jupyter notebooks they create then we do not need this function definition
and simply need to call fig at end of next cell to regenerate plot.
"""
variances, nsamples_history = [], []
npilot_samples = 5
estimator = pya.MLMC
for target_cost in target_costs:
# compute variance using exact covariance for sample allocation
est = estimator(cov, costs)
nhf_samples, nsample_ratios = est.allocate_samples(target_cost)[:2]
variances.append(est.get_variance(nhf_samples, nsample_ratios))
nsamples_history.append(est.get_nsamples(nhf_samples, nsample_ratios))
# compute single fidelity Monte Carlo variance
total_cost = nsamples_history[-1].dot(costs)
variances.append(cov[0, 0] / int(total_cost / costs[0]))
nsamples_history.append(int(total_cost / costs[0]))
# compute variance using approx covariance for sample allocation
# use nhf_samples from previous target_cost as npilot_samples.
# This way the pilot samples are only an additional cost at the first
# step. This code does not do this though for simplicity
cov_approx = pya.estimate_model_ensemble_covariance(
npilot_samples, poly_model.generate_samples, model_ensemble)[0]
est = estimator(cov_approx, costs)
nhf_samples, nsample_ratios = est.allocate_samples(target_cost)[:2]
variances.append(est.get_variance(nhf_samples, nsample_ratios))
nsamples_history.append(est.get_nsamples(nhf_samples, nsample_ratios))
npilot_samples = nhf_samples
fig, axs = plt.subplots(1, 2, figsize=(2 * 8, 6))
pya.plot_acv_sample_allocation(nsamples_history[::3], costs, model_labels,
axs[1])
total_costs = np.array(nsamples_history[::3]).dot(costs)
axs[0].loglog(total_costs, variances[::3], label=r'$\mathrm{MLMC}$')
mc_line = axs[0].loglog(total_costs,
variances[1::3],
label=r'$\mathrm{MC}$')
total_costs = np.array(nsamples_history[2::3]).dot(costs)
axs[0].loglog(total_costs,
variances[2::3],
'--',
label=r'$\mathrm{MLMC^\dagger}$')
axs[0].set_xlabel(r'$\mathrm{Total}\;\mathrm{Cost}$')
axs[0].set_ylabel(r'$\mathrm{Variance}$')
_ = axs[0].legend()
return fig, axs