Convergence on the value of Pi using Monte Carlo vs QMC

convergence_pi.py

"""Convergence of Pi using QMC.

---------------------------

MIT License

Copyright (c) 2022 Pamphile Tupui ROY

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"""
import numpy as np
from scipy.stats import qmc
import matplotlib.pyplot as plt

n_conv = 999

MIN_POWER = 4
POWER_MAX = 17
ns_gen = 2 ** np.arange(MIN_POWER, POWER_MAX + 1)

dim = 2


def func(x):
    radius = (x[:, 0]**2 + x[:, 1]**2) ** .5
    in_circle = len(np.where(radius <= 1)[0])
    return 4 * in_circle / x.shape[0]


def conv_method(samples, func, n_conv):
    evals = [func(sample) for sample in samples]
    
    ref = np.pi
    squared_errors = (ref - np.array(evals)) ** 2
    rmse = (np.sum(squared_errors) / n_conv) ** 0.5

    return rmse


# Analysis


def sampler_mc(ns, dim):
    sampler_ = lambda x: np.random.random((x, dim))
    samples_ = [sampler_(ns) for _ in range(n_conv)]
    return np.array(samples_)


def sampler_sobol(ns, dim):
    def _sampler_sobol(ns):
        engine = qmc.Sobol(d=dim, scramble=True)
        return engine.random(ns)

    samples_ = [_sampler_sobol(ns) for _ in range(n_conv)]
    return np.array(samples_)


samplers = {
    "MC": [sampler_mc, []],
    "Sobol'": [sampler_sobol, []],
}


for ns in ns_gen:
    print(f'-> ns={ns}')

    for sampler_ in samplers:
        sample_ = samplers[sampler_][0](ns, dim)

        conv_res = conv_method(sample_, func, n_conv)

        samplers[sampler_][1].append(conv_res)


for sampler_ in samplers:
    samplers[sampler_][1] = np.array(samplers[sampler_][1])

fig, ax = plt.subplots(figsize=(5, 5))

for sampler_ in samplers:
    ax.plot(
        ns_gen, np.log2(samplers[sampler_][1]),
        label=sampler_
    )

ax.set_xlabel(r'$N_s$')

ax.set_xscale('log')

ax.set_xticks(ns_gen)
ax.set_xticklabels([fr'$2^{{{ns}}}$' for ns in np.arange(MIN_POWER, POWER_MAX + 1)])
ax.set_ylabel(r'$\log (\epsilon (\mu))$')
fig.legend(labelspacing=0.7, bbox_to_anchor=(0.9, 0.9))
fig.tight_layout()