TVDTemperingRev.py

From Werner KRAUTH

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#
# Simulated tempering for the V-shaped stationary distribution --- Metropolis
# algorithm
#
import random
import pylab
import numpy as np
for n in [10, 20, 40, 80, 160, 320]:
    ReplicaChange = 0.1
    const = 4.0 / n ** 2
    PiStat = {}
    Table = []
    for x in range(1, n + 1):
        Table.append((x, 0))
        Table.append((x, 1))
#
#   factor of 1/2 because the total must be normalized
#
        PiStat[(x, 0)] = 1.0 / float(n) / 2.0
        PiStat[(x, 1)] = const * abs( (n + 1) / 2 - x) / 2.0
    PiStat[(0, 0)] = 0.0
    PiStat[(0, 1)] = 0.0
    PiStat[(n + 1, 0)] = 0.0
    PiStat[(n + 1, 1)] = 0.0
    PTrans   = np.eye(2 * n)
    Pi = np.zeros([2 * n])
    for x in range(1, n + 1):
        for Rep in [0, 1]:
            i = Table.index((x, Rep))
            Pi[i] = PiStat[(x, Rep)]
            for Dir in [-1, 1]:
                if PiStat[(x + Dir, Rep)] > 0.0:
                    j = Table.index((x + Dir, Rep))
                    PTrans[i, j] = min(1.0, PiStat[(x + Dir, Rep)] / PiStat[(x, Rep)]) / 2.0
                    PTrans[i, i] -= PTrans[i, j]
    PReplica = np.zeros((2 * n,2 * n))
    for x in range(1, n + 1):
        i = Table.index((x,0))
        j = Table.index((x,1))
        PReplica[i, j] = ReplicaChange * min(1.0, PiStat[(x, 1)] / PiStat[(x, 0)])
        PReplica[i, i] = 1.0 - PReplica[i, j]
        PReplica[j, i] = ReplicaChange * min(1.0, PiStat[(x, 0)] / PiStat[(x, 1)])
        PReplica[j, j] = 1.0 - PReplica[j, i]
    P = PTrans @ PReplica
    Pit = np.zeros([2 * n])
    Pit[0] = 1.0
    xvalues = []
    yvalues = []
    iter = 0
    while True:
        iter += 1
        Pit = Pit @ P
        TVD = sum(np.absolute(Pi - Pit) / 2.0)
        xvalues.append(iter / float(n ** 2))
        yvalues.append(TVD)
        if TVD < 0.1: break
    pylab.plot(xvalues,yvalues, label='$n =$ '+str(n))
pylab.legend(loc='upper right')
pylab.xlabel("$t/ n^2$ (rescaled time) ")
pylab.ylabel("TVD")
pylab.title("TVD rev tempering on the path graph of $n$ sites")
pylab.show()
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