Metropolis harmonic.py
From Werner KRAUTH
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| ==Context== | ==Context== | ||
| - | This page is part of my [[BegRohu_Lectures_2024|2025 Oxford Lectures]] | + | This page is part of my public lectures on [https://www.physics.ox.ac.uk/events/algorithms-and-computations-theoretical-physics-set-lectures Algorithms and Computations in theoretical physics] at the University of Oxford (see the announcement for a syllabus). For more information, see [[Oxford_Lectures_2025|the main page of the public lectures]]. It presents the Metropolis Monte Carlo algorithm invented by Metropolis et al. (1953). For an in-depth discussion, see my below reference. Other programs in the same series are the [[HMC_harmonic.py|Hamiltonian Monte Carlo]] algorithm and the [[Levy_harmonic.py|Lévy construction]], both for the harmonic model. |
| ==Python program== | ==Python program== | ||
| import math, random | import math, random | ||
| - | + | ||
| def U(x): | def U(x): | ||
| U = 0.0 | U = 0.0 | ||
| Line 13: | Line 13: | ||
| U += (x[k] - x_minus) ** 2 / 2.0 | U += (x[k] - x_minus) ** 2 / 2.0 | ||
| return U | return U | ||
| - | + | ||
| N = 8 | N = 8 | ||
| L = 16 | L = 16 | ||
| delta = 1.0 | delta = 1.0 | ||
| - | |||
| x = [L / N * k for k in range(N)] | x = [L / N * k for k in range(N)] | ||
| Iter = 1000000 | Iter = 1000000 | ||
| Line 33: | Line 32: | ||
| U_kp = ((x_plus - x_new) ** 2 + (x_new - x_minus) ** 2) / 2.0 | U_kp = ((x_plus - x_new) ** 2 + (x_new - x_minus) ** 2) / 2.0 | ||
| if random.uniform(0.0, 1.0) < math.exp(- (U_kp - U_k)): x[k] = x_new | if random.uniform(0.0, 1.0) < math.exp(- (U_kp - U_k)): x[k] = x_new | ||
| - | x.append(xend) | ||
| ==Further information== | ==Further information== | ||
| ==References== | ==References== | ||
| - | Krauth, W. XXX | + | Krauth, W. [http://arxiv.org/pdf/2411.11690 Hamiltonian Monte Carlo vs. event-chain Monte Carlo: an appraisal of sampling strategies beyond the diffusive regime]. ArXiv: 2411.11690 |
| + | [[Category:Python]] [[Category:Oxford2025]] | ||
Current revision
Contents |
[edit]
Context
This page is part of my public lectures on Algorithms and Computations in theoretical physics at the University of Oxford (see the announcement for a syllabus). For more information, see the main page of the public lectures. It presents the Metropolis Monte Carlo algorithm invented by Metropolis et al. (1953). For an in-depth discussion, see my below reference. Other programs in the same series are the Hamiltonian Monte Carlo algorithm and the Lévy construction, both for the harmonic model.
[edit]
Python program
import math, random
def U(x):
U = 0.0
for k in range(N):
k_minus = (k - 1) % N
x_minus = x[k_minus]
if k == 0: x_minus -= L
U += (x[k] - x_minus) ** 2 / 2.0
return U
N = 8
L = 16
delta = 1.0
x = [L / N * k for k in range(N)]
Iter = 1000000
for iter in range(Iter):
k = random.randint(0, N - 1) # random slice
k_plus = (k + 1) % N
k_minus = (k - 1) % N
x_plus = x[k_plus]
if k == N - 1: x_plus += L
if k_plus == N: x_plus = x[0] + L
x_minus = x[k_minus]
if k == 0: x_minus -= L
x_new = x[k] + random.uniform(-delta, delta) # new position at slice k
U_k = ((x_plus - x[k]) ** 2 + (x[k] - x_minus) ** 2) / 2.0
U_kp = ((x_plus - x_new) ** 2 + (x_new - x_minus) ** 2) / 2.0
if random.uniform(0.0, 1.0) < math.exp(- (U_kp - U_k)): x[k] = x_new
[edit]
Further information
[edit]
References
Krauth, W. Hamiltonian Monte Carlo vs. event-chain Monte Carlo: an appraisal of sampling strategies beyond the diffusive regime. ArXiv: 2411.11690
