Download presentation
1
Hit-and-Miss (or Rejection) Monte Carlo Method:
a “brute-force” method based on completely random sampling x y A O 1 (xi,yi) Then, how do we throw the stones and count them on computer? 1. Generation of M 2D microstates 2. Measurement (Averaging an observable over M microstates)
2
Lab 1. Calculation of the value of
Simple, Brute-Force, Hit-and-Miss Monte Carlo Method Obtain an approximate value of A (= /4 M/M) by throwing M stones in the square of the total area 1; counting M stones in a quarter of the circle of the radius 1. Question 1 (Formulation of the problem = Restatement of the last page). On computer, this problem can be formulated as obtaining the average (or expectation value) of an observable O over M instantaneous Oi values at each microstate (xi, yi) (i = 1 to M). What values can Oi take? (2) How does the value of Oi depend on (xi, yi)?
3
Lab 1. Calculation of the value of
Simple, Brute-Force, Hit-and-Miss Monte Carlo Method Obtain an approximate value of A (= /4 M/M) by throwing M stones in the square of the total area 1; counting M stones in a quarter of the circle of the radius 1. Question 1 (Formulation of the problem = Restatement of the last page). On computer, this problem can be formulated as obtaining the average (or expectation value) of an observable O over M instantaneous Oi values at each microstate (xi, yi) (i = 1 to M). What values can Oi take? Answer: 0 or 1 (2) How does the value of Oi depend on (xi, yi)? Answer:
4
Lab 1. Calculation of the value of
Question 2 (Algorithm & Flow chart). Draw a flow chart to estimate 4A by averaging over M instantaneous Oi values.
5
Lab 1. Calculation of the value of
Otot = Otot + O O = 0.0 O = 1.0 yes no x=ran3(&seed) y=ran3(&seed) i = 1 i = i + 1 Otot = 0.0 Obar = Otot/M x 4 Question 2 (Algorithm & Flow chart). Draw a flow chart to estimate 4A by averaging over M instantaneous Oi values.
6
Lab 1. Calculation of the value of
Question 3. Write a program to estimate 4A by averaging over M instantaneous O values. Try M = 10,000.
7
Lab 1. Calculation of the value of
Question 3. Write a program to estimate 4A by averaging over M instantaneous O values. Try M = 10,000.
8
Lab 1. Calculation of the value of
Question 6. But, what if we don’t know the true value of what we estimate? Since we can’t estimate the error w.r.t. the true value, how could we estimate the accuracy (error) of the simulation?
9
Lab 1. Calculation of the value of
Variance, Standard Deviation, Error, and Accuracy
10
Lab 1. Calculation of the value of
Variance, Standard Deviation, Error, and Accuracy
11
Lab 1. Calculation of the value of
Answer: We expected that would be similar to ( ), but it’s not the case ( > ). (~constant with M) is much larger than (which decreases as M increases)! Thus, itself is not a good indicator of the accuracy of a simulation. Question 8. Then, what else would be an indicator of the accuracy (error) of the estimation, when we don’t know the true value of what we estimate?
12
Lab 1. Calculation of the value of
13
Lab 1. Calculation of the value of
14
Lab 1. Calculation of the value of
Question 10. But, what if we move onto a very complex problem? Each estimation (simulation) with M trials becomes so time-consuming that we couldn’t afford to running the simulation many times. What should we do in this case? Answer: Let’s try to find a way to estimate the accuracy from one M-trial simulation.
15
Lab 1. Calculation of the value of
16
Lab 1. Calculation of the value of
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.