5.7b Variance-reduction labs

Source: Lab ideas from Computational Physics (Landau, Páez, Bordeianu), §5.7. Rewritten as original drills. Continues 5.7 Monte Carlo variance reduction.

Control variate on $[0,1]$ for $f(x)=e^{-x}$, $g(x)=1-x$, $J=1/2$. Exact value $I=1-e^{-1}$. Use the chapter LCG only.

Fixed LCG

r ← (1103515245 * r + 12345) % 2**31, then uniform r / 2**31. Default seed 1. Advance the state before each draw.

Formula refresh

Warm up: pick the matching equation among lookalikes (same walk as the theory lesson).

Labs

Activity: Plain mean-value MC

Estimate $I=\int_0^1 e^{-x}\,dx$ by $I\simeq\frac{1}{N}\sum_{i=1}^{N} e^{-x_i}$ with $x_i\sim U[0,1)$, with the chapter LCG, seed $1$, $N=1000$. Print with 10 decimals.

No input.

Write-ups

0/4 solved

5.7b.W1

Using “peak”, “zero”, and “waste”, explain:

Why does a peaked integrand waste uniform Monte Carlo samples?

Minimum 3 words. Your answer must include: “peak”, “zero”, “waste”.

0 words/ 3 min

Meet the writing requirements, then you can open the model answer.

Explorers

# control_variate.py — original sketch (not a textbook listing)
M = 2**31

def plain_and_cv(N, seed=1):
    r, s_f, s_res = seed, 0.0, 0.0
    for _ in range(N):
        r = (1103515245 * r + 12345) % M
        x = r / M
        fx = __import__('math').exp(-x)
        s_f += fx
        s_res += fx - (1.0 - x)
    return s_f / N, s_res / N + 0.5

Next: 5.8 Importance sampling.