1. Importance sampling for MC simulation
(“Importance-weighted random walk”)
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.

2. Example: Importance of small region: Measuring the depth of the Nile
Systematic quadrature
or uniform sampling
Importance sampling
(importance weighted random walk)
Frenkel and Smit, Understanding Molecular Simulations

3. Example: Importance of small region: Energy funnel in protein folding
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

4. To decrease the error of MC simulation
~ cost
 (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
fluctuating, varying function
sharp
(probability) distribution
O=f(x) a b x A
<f> x2 x1 xi xM O 1
p(O)
O=f(x) O 1
p(O)
broad distribution
flat function f f
<f> A
vs. a b x
(f > 0)
non-ideal, real case
the ideal case
(f = 0)
Importance sampling

5. Importance sampling for MC simulation: Example
From N-step
uniform sampling
3 in accuracy
From normalized importance sampling
weighted by w(x)

6. Lab 3: Importance sampling for MC simulation

7. 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(); ~
tend = cpu();
printf("CPU time: %5.5f s, CPU time/measure: %g s", tend - tstart, (tend - tstart) / M);
* Calculate the normalization constant N for each probability distribution function (x).
* Include “ran3.h” & “fran3.h”; compile fran3.c; call “fexp” or “flin” defined in fran3.
r = ran3(&seed);
f_x = exp(-x*x);
r = ran3(&seed);
rho = fexp(r, &x);
f_x = exp(-x*x) / rho;
r = ran3(&seed);
rho = flin(r, &x);
f_x = exp(-x*x) / rho;
* Display histograms for the distribution of x values generated by fran3 (for the step 4 only).
r = ran3(&seed);
f_x = exp(-x*x);
i_hist = (int) (x * inv_dx);
if (i_hist < n_hist)
hist[i_hist] += 1.0;
r = ran3(&seed);
rho = fexp(r, &x);
f_x = exp(-x*x) / rho;
i_hist = (int) (x * inv_dx);
if (i_hist < n_hist)
hist[i_hist] += 1.0;
r = ran3(&seed);
rho = flin(r, &x);
f_x = exp(-x*x) / rho;
i_hist = (int) (x * inv_dx);
if (i_hist < n_hist)
hist[i_hist] += 1.0;

8. Plot with gnuplot, excel, origin, … Fit to a function.
* Display histograms for the distribution of x values generated by fran3. (full version)
/* number of bins in histogram*/
n_hist = 50;
/* Allocate memory for histogram */
hist = (double *) allocate_1d_array(n_hist, sizeof(double));
/* Initializae histogram */
for (i_hist = 0; i_hist < n_hist; ++i_hist) hist[i_hist] = 0.0;
/* Size of the ihistogram bins */
dx = 1.0 / n_hist; /* 1.0 is the size of the interval */
inv_dx = 1.0 / dx;
/* histogram accumulated */
i_hist = (int) (x * inv_dx);
if (i_hist < n_hist)
hist[i_hist] += 1.0;
/* Write histogram. */
fp = fopen("hist_2.dat", "w");
for (i_hist = 0; i_hist < n_hist; ++i_hist) {
x = (i_hist + 0.5) * dx;
fprintf(fp, "%g %g", x, hist[i_hist] / M); }
fclose(fp);
Plot with gnuplot, excel, origin, …
Fit to a function.
What is the resulted function?

9. Sampling a non-uniform & discrete
Further reading:
Sampling a non-uniform & discrete
probability distribution {pi} (tower sampling)
(Ref) Gould, Toboshnik, Christian, Ch. 11.5

10. Further reading:
Sampling a non-uniform & continuous
probability distribution (x)

11. Example

15. Lab 3: Importance sampling for MC simulation
bad
distribution function
w(x)
(x)
p(x)
Normalized!
constant
distribution function
(uniform)
Normalized!
good
distribution function
Normalized!
Integrand f(x)

16. Results: Importance sampling for MC simulation
One experiment with M measures
n = 100 experiments, each with M measures

17. Results: Importance sampling for MC simulation
One experiment with M measures
n = 100 experiments, each with M measures

18. = Importance-weighted average
Importance sampling
= Importance-weighted average
Analogy:
Throw a dice with the results of {1, 2, 2, 2, 2, 4, 5, 6}.
(Discrete) probability distribution {pi, i=1-6}
= {1, 4, 0, 1, 1, 1} / 8 where 8 =
= {1/8, 1/2, 0, 1/8, 1/8, 1/8}
Mean value <A>
= 3 = 24/8
= ( ) / 8
= (1 + 2x ) / 8
= 1x1/8 + 2x4/8 + 3x0/8 + 4x1/8 + 5x1/8 + 6x1/8

19. Beyond 1D integrals: A system of N particles in a container
of a volume V in contact with a thermostat T (constant NVT)
external constraint
Particles interact with each other through a potential energy U(rN) (~ pair potential).
U(rN) is the potential energy of a microstate {rN} = {x1, y1, z1, …, xN, yN, zN}.
(rN) is the probability to find a microstate {rN} under the constant-NVT constraint.
(=1/kT) or for discrete microstates
Partition function Z (required for normalization) = the weighted sum of all the microstates compatible with the constant-NVT condition
or for discrete microstates
Average of an observable O, <O>, over all the microstates compatible with constant NVT
ensemble average
“canonical ensemble”
for discrete microstates