1 / 13

Simulation Time-stepping and Monte Carlo Methods Random Number Generation

Simulation Time-stepping and Monte Carlo Methods Random Number Generation. Shirley Moore CS 1401 Spring 2013 March 26, 2013. Agenda. Announcements Time-stepping simulation Lab 5 Wrapup Using random number methods Monte Carlo simulation Assignments for Tuesday, April 2

caden
Download Presentation

Simulation Time-stepping and Monte Carlo Methods Random Number Generation

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. SimulationTime-stepping and Monte Carlo MethodsRandom Number Generation Shirley Moore CS 1401 Spring 2013 March 26, 2013

  2. Agenda • Announcements • Time-stepping simulation • Lab 5 Wrapup • Using random number methods • Monte Carlo simulation • Assignments for Tuesday, April 2 • Review and practice for Exam 2 • Fun and review! Google Blockly Maze

  3. Announcements • Two talks this Thursday at 12:00 and 1:00 (see course website for details) • Exam 2 on Thursday, practice and review in class and lab today • Unit 3 plan on course website

  4. What is Computer Simulation? • Modeling real-world phenomena with computational algorithms • Normally requires some assumptions and approximations/simplifications • Aside: Are You Living In a Computer Simulation? ORIGINAL Nick Bostrom. Philosophical Quarterly, 2003, Vol. 53, No. 211, pp. 243-255, www.simulation-argument.com • Types of computer simulations • Time-stepping simulations • Monte Carlo simulations

  5. Time-Stepping Simulations • Simulate continuous time with a loop • Each iteration of the loop is called a time-step and models an interval of time (e.g., 1 second, 0.01 second, 1 hour, 1 day, etc.) • Relevant physical quantities are updated each time stamp according to physical laws • Examples • Lab 5 Project Motion Simulation • Climate Change Simulation

  6. 2D Projectile Motion Simulation • In each time step (i.e., loop iteration), the statements in the loop body calculate approximately what happens to the projectile: • The projectile MOVES (changes position where position is given by x,y coordinates) because its velocity (with components in the horizontal x and vertical y directions) is not zero. • The projectile’s vertical velocity CHANGES because of the effect of gravity.

  7. How does position change during one time step? • Change in a given direction is velocity in that direction multiplied by the time step. • e.g., If you are driving a car north at 40 miles an hour (velocity), then in 15 minutes = 0.25 hour (time step), your have gone 40 * 0.25 = 10 miles, so position position + 40 * 0.25 • For the projectile motion simulation xx + vx * dt yy + vy * dt

  8. How does the velocity change during one time step? • Horizontal velocity doesn’t change • Newton’s First Law of Motion • Our model simplifies away air friction • Vertical velocity is reduced by effect of gravity • Newton’s Second Law of Motion • Again simplifying away air friction • Near Earth’s surface, gravity makes free falling objects accelerate at a rate of -9.8 (meters per second) per second • i.e., -9.8 is the change in velocity that occurs each second • So, vyvy + (-.9.8) * dt

  9. Simulation Errors • What are sources of error in our projectile motion simulation? • Model simplifications • Computational approximations

  10. Climate Change Simulation • Climate change as simulated by the NCAR CCSM: http://www.youtube.com/watch?v=d8sHvhLvfBo • Community Earth System Model (CESM) (formerly CCSM):http://www.cesm.ucar.edu • Very, Very Simple Climate Model: http://www.windows2universe.org/earth/climate/cli_model.html

  11. Projectile Motion Calculations • Calculations inside the loop body t = t + dt

  12. Random Number Generation • Math.random() is a Java method that returns a (pseudo)-random double value d such that 0.0 <= d < 1.0 • How can you use this method to generate a random integer in a given range – e.g., 1 to 6? • Write a loop that randomly tosses a 6-sided die.

  13. Monte Carlo Simulation Example // Estimate π using Monte Carlo simulation public class MonteCarloSimulation { public static void main(String[] args) { final int NTRIALS = 1000000; intnHits = 0; double x, y; // Randomly generate NTRIALS points in 2x2 square // centered at origin and count how many // land inside unit circle. Ratio of circle area to // square area is π/4. for (inti = 0; i < NTRIALS; i++) { x = Math.random() * 2.0 – 1; y = Math.random() * 2.0 – 1; if (x*x + y*y <= 1.0) nHits++; } double pi = 4.0 * (nHits/NTRIALS); System.out.println(“PI is “ + pi); } } Actually, the above program has a bug and will always output 0.0. Can you find the bug and fix it?

More Related