A Comparison of Monte Carlo Methods with Systematic Point Selection

1. A Comparison of Monte Carlo Methods with Systematic Point Selection Katherine Stammer Intro to Systems Engineering

2. Overview Using Monte Carlo Methods to calculate π Using systematically selected points to calculate π Mathematical analysis of the methods Experimental Results Comparison of the two methods Conclusion

3. Using Monte Carlo to Calculate Pi A circle with diameter one is placed inside a one by one square Area of the circle is π /4 ( ) Area of the square is one Points are randomly selected within the square and are evaluated to see whether or not they land inside the circle. The ratio of points inside the circle to the total number of points is equal to the area of the circle. π can be found by multiplying this ratio by 4. • 1 1

4. Hypothesis My hypothesis was that π could be more accurately calculated by selecting a grid of evenly spaced points to test rather than by randomly selecting points. http://people.sc.fsu.edu/~burkardt/m_src/voronoi_new/hundred_points.png http://www.ysbl.york.ac.uk/~cowtan/clipper/doc/map_p1.png

5. Systematically Selecting Points • 1 1 An evenly spaced grid of points are selected to be tested. Similarly, the ratio of points inside the circle to the total number of points is equal to the area of the circle π can be found by multiplying this ratio by 4

6. Mathematical Analysis Monte Carlo Systematic Point Selection Accuracy is of the order where n is the number of points in the circle For a given trial: Accuracy is of the order n where n is the number of points in the circle Example: If 10,000 points are tested, the number of points within the circle is 7,827. When this four digit number is divided by 10,000 the result is 0.7827.

7. Testing • Developed a program in Matlab that randomly selected points • Selected 100 points • Tested these points to see if they were inside the circle • Computed π • Repeated with a step size of 100 points until reaching 100,000 points • Developed a program in Matlab that iterated across two dimensions • Selected a grid of 100 evenly spaced points • Tested these points to see if they were inside the circle • Computed π • Repeated with a step size of 100 points until reaching 100,000 points

8. Experimental Results: Convergence of Methods Red is Systematic Point Selection and Blue is the Monte Carlo Method This graph shows the values of π as calculated using from 100 to 10,000 points in increments of 100 Number of Points Tested

9. Experimental Results: Convergence of Methods Red is Systematic Point Selection and Blue is the Monte Carlo Method This graph shows the values of π as calculated using from 100 to 100,000 points in increments of 100 Number of Points Tested

10. Advantages and Disadvantages of Each Method Monte Carlo Systematic Point Selection Takes less computing time per calculation of π Can be used in situations where the shape is unknown Converges more quickly If the form of the shape inside the box is unknown, systematic point selection may give inaccurate results

11. Conclusions for the Calculation of π Systematic Point Selection can provide an accurate method for calculating π. In the case of calculating π, Systematic Point Selection converges more quickly than the Monte Carlo Method. In this case it is known that the shape we are finding the area of is a circle.

12. General Conclusions Systematic Point Selection is a good method to use if the shape of the area you are calculating is known and additional computing time is not a problem. However, the Monte Carlo Method is a better choice if you do not know the shape you are finding the area of or if it is a shape that does not work well with a grid of points (such as many tiny evenly spaced shapes).

13. Thank You Aaron Mosher for his help with Matlab Anatoly Zlotnik for his assistance with the mathematical calculations of error

14. Questions