Importance sampling for MC simulation (“Importance-weighted random walk”) M i i i b a x b a x x f M x x f dx x x x f dx x f A 1 ) ( ) ( ) ( 1 ) ( ) ( ) ( ) ( ) ( ) ( Sampling points from a uniform distribution may not be the best way for MC. When most of the weight of the integral comes from a small range of x where f(x) is large, sampling more often in this region would increase the accuracy of the MC.
19
Embed
Importance sampling for MC simulation (“Importance-weighted random walk”)
Importance sampling for MC simulation (“Importance-weighted random walk”). Sampling points from a uniform distribution may not be the best way for MC. - PowerPoint PPT Presentation
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Importance sampling for MC simulation(“Importance-weighted random walk”)
M
i i
ib
ax
b
a x
xf
Mx
xfdxx
x
xfdxxfA
1)()(
)(1
)(
)()(
)(
)()(
Sampling points from a uniform distribution may not be the best way for MC.When most of the weight of the integral comes from a small range of x where f(x) is large, sampling more often in this region would increase the accuracy of the MC.
Example: Importance of small region: Measuring the depth of the NileSystematic
quadrature or uniform sampling
Importance sampling(importance weighted random
walk)
Frenkel and Smit, Understanding Molecular Simulations
Cyrus Levinthal formulated the “Levinthal paradox” (late 1960’s): -Consider a protein molecule composed of (only) 100 residues, -each of which can assume (only) 3 different conformations. -The number of possible structures of this protein yields 3100 = 5×1047. -Assume that it takes (only) 100 fs to convert from one structure to another. -It would require 5×1034 s = 1.6×1027 years to ”systematically” explore all possibilities.-This long time disagrees with the actual folding time (μs~ms). Levinthal’s paradox
Example: Importance of small region:Energy funnel in protein folding
To decrease the error of MC simulation
(f) = standard deviation in the observable O, i.e., y = f(x), itself
f measures how much f(x) deviates from its average over the integration region. ~ independent of the number of trials M (or N) ~ estimated from one simulation
where,OOtrue M
)deviation standard , trialsofnumber ( M~ cost
O=f(x)
a b x
A
<f>
x2 x1xi xM
O=f(x)
a b x
A
<f>
O
0 1 p(O)
O
0 1 p(O)
vs.
(f = 0)(f > 0) non-ideal, real case
the ideal case
M
i i
ib
ax
b
a x
xf
Mx
xfdxx
x
xfdxxfA
1)()(
)(1
)(
)()(
)(
)()(
Importance sampling
flat function
sharp(probabilit
y) distributio
n
broad distributio
n
fluctuating, varying function f
f
From N-step uniform sampling
From normalized importance sampling
weighted by w(x)
3 in accurac
y
Importance sampling for MC simulation: Example
Lab 3: Importance sampling for MC simulation
* Calculate the normalization constant N for each probability distribution function (x).
What’s new in Lab 3: Importance sampling* Include “cpu.h”, compile “cpu.c”, and call “cpu()” to measure the cpu time of the run.tstart = cpu();