May 1982 LIDS-P-1201 CONVERGENCE OF A GRADIENT PROJECTION METHOD
9. Convergence and Monte Carlo Errors. Measuring Convergence to Equilibrium Variation distance where...
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Transcript of 9. Convergence and Monte Carlo Errors. Measuring Convergence to Equilibrium Variation distance where...
9. Convergence and 9. Convergence and Monte Carlo ErrorsMonte Carlo Errors
9. Convergence and 9. Convergence and Monte Carlo ErrorsMonte Carlo Errors
Measuring Convergence to
Equilibrium• Variation distance
1 2 1 2
1 2
|| || max| ( ) ( )|
1 | ( ) ( ) |
2
A
i
P P P A P A
P i P i
where P1 and P2 are two probability distributions, A is a set of states, i is a single state.
Eigenvalue Problem• Consider the matrix S defined by
[S]ij = pi½ W(i->j) pj
-½
then S is real and symmetric and eigenvalues of S satisfy|n| ≤ 1
• One of the eigenvalue must be 0=1 with eigenvector pj
½.
Spectrum Decomposition
• Then we haveUTSU = Λ, or S = U Λ UT
where Λ is a diagonal matrix with diagonal elements k and U is orthonormal matrix, U UT = I.
• W can be expressed in U, P, and Λ asW = P-½UΛUTP½
Evolution in terms of eigen-states
• Pn= P0Wn
= P0 P-½UΛUTP½ P-½UΛUTP½…
= P0 P-½UΛnUTP½
• In component form, this means
Pn(j) = ∑iP0(i) pi
-½pj½∑
kk
n uikujk
Discussion• In the limit n goes to ∞,
Pn(j) ≈ ∑iP0(i) pi-½pj
½ ui0uj0 = pj
• The leading correction to the limit isPn(j) ≈ pj + a 1
n = pj + a e-n/
Exponential Correlation Time
• We define by the next largest eigenvalue
= - 1/log 1
This number characterizes the theoretical rate of convergence in a Markov chain.
Measuring Error
• Let Qt be some quantity of interest at time step t, then sample average is
QN = (1/N) ∑t Qt
• We treat QN as a random variable. By central limit theorem, QN is normal distributed with a mean <QN>=<Q> and variance σN
2 = <QN2>-<QN>2.
<…> standards for average over the exact distribution.
Confidence Interval
• The chance that the actual mean <Q> is in the interval[ QN – σN, QN + σN ]is about 68 percents.
• σN cannot be computed (exactly) in a single MC run of length N.
Estimating Variance
22
1, 1
int
1
var( ) ( ) 1
var( )
N
N t s t st s
N
t N
Q Q Q QN
tQf t
N N
QN
The calculation of var(Q) = <Q2>-<Q>2 and int can be done in a single run of length N.
Error Formula• The above derivation gives the famous
error estimate in Monte Carlo as:
where var(Q) = <Q2>-<Q>2 can be estimated by sample variance of Qt.
intvar( ) 1Error N
QN N
Time-Dependent Correlation function and integrated correlation
time
• We define
and
22( ) s s t s s t
s s
Q Q Q Qf t
Q Q
int0, 1, 2,... 1
( ) 1 2 ( )t t
f t f t
Circular Buffer for Calculating f(t)
Qt, Current time t
Qt-1 Previous time t-1
Earliest time t-(M-1)
We store the values of Qs of the previous M-1 times and the current value Qt
Qs
An Example of f(t)
Time-dependent correlation function for 3D Ising at Tc on a 163 lattice; Swendsen-Wang dynamics.
From J S Wang, Physica A 164 (1990) 240.
Efficient Method for Computing int
We compute int by the formula
int = N σN2/var(Q)
For small value N and then extrapolating N to ∞.
From J S Wang, O Kozan and R H Swendsen, Phys Rev E 66 (2002) 057101.
Exponential and integrated correlation
times
where 1 < 1 is the second largest eigenvalue of W matrix. This result says that exponential correlation time (=-1/log1) is related to the largest integrated correlation time.
1int
1
1sup ( )
1 Q
Q
Critical Slowing Down
Tc T
The correlation time becomes large near Tc. For a finite system (Tc) Lz, with dynamical critical exponent z ≈ 2 for local moves
Relaxation towards Equilibrium
Time t
Magnetization m
T < Tc
T = Tc
T > Tc
Schematic curves of relaxation of the total magnetization as a function of time. At Tc relaxation is slow, described by power law:
m t -β/(zν)
Jackknife Method• Let n be the number of independent
samples• Let c be some estimate using all n
samples• Let ci be the same estimate but using n-
1 samples, with i-th sample removed• Then Jackknife error estimate is
2
1
( )n
J ii
c c