Solving ODE and PDE by Monte Carlo Method

1. Solving ODE and PDEby Monte Carlo Method University of Washington InsukJoh

2. Table of Contents • Introduction to Monte Carlo Method • Solving ODE: General Case • Solving ODE: Gambler’s Ruin • Solving PDE: Laplace Equation • Solving PDE: General Elliptic Equation • Conclusion

3. Table of Contents • Introduction to Monte Carlo Method • Solving ODE: General Case • Solving ODE: Gambler’s Ruin • Solving PDE: Laplace Equation • Solving PDE: General Elliptic Equation • Conclusion

4. Monte Carlo Method • Numerical method • Uses random sampling to obtain distribution of probabilistic entity • Useful to obtain solution of implicit equation • Useful if deterministic model is NOT available

5. Monte Carlo Method • Example Problem: Calculate the value of π.

6. Monte Carlo Method • Example Problem: Calculate the value of π. It is known that Asquare = 1 , Acircle = π/4 Use random sampling. Count the number of dots in the circle.

7. Table of Contents • Introduction to Monte Carlo Method • Solving ODE: General Case • Solving ODE: Gambler’s Ruin • Solving PDE: Laplace Equation • Solving PDE: General Elliptic Equation • Conclusion

8. ODE: General Case • ODE form where

9. ODE: General Case • Step 1: Change the form of equation • Step 2: Generate random samples. N = (# of all samples) • Step 3: Count # of samples greater than k, S = (# of samples greater than k) • Step 4: Calculate y(j) • Step 5: Repeat Step 2 ~ Step 4

10. ODE: General Case • Example • Error less than 1 %

11. Table of Contents • Introduction to Monte Carlo Method • Solving ODE: General Solution • Solving ODE: Gambler’s Ruin • Solving PDE: Laplace Equation • Solving PDE: General Elliptic Equation • Conclusion

12. ODE: Gambler’s Ruin • 2nd Order ODE • Problem of probability for winning a game where v(x,t) = probability of a guy A wins the game p = probability of person A wins one round x = stakes for game t = tth round

13. ODE: Gambler’s Ruin • Two equations, probability p and coefficient β are related by • The solution has the form of

14. ODE: Gambler’s Ruin • Algorithm

15. Table of Contents • Introduction to Monte Carlo Method • Solving ODE: General Case • Solving ODE: Gambler’s Ruin • Solving PDE: Laplace Equation • Solving PDE: General Elliptic Equation • Conclusion

16. PDE: Laplace Equation • Laplace Equation in k-dimension

17. PDE: Laplace Equation • Finite difference method

18. PDE: Laplace Equation

19. PDE: Laplace Equation • Random walk : A drunk person walks from node P. The probability to choose any one of the four neighboring node is equal, which is ¼. If he reaches a boundary node, the random walk ends, and a new random walk starts.

20. PDE: Laplace Equation • The solution where M = number of boundary nodes N = number of random walk trials Ni= number of visit at the boundary node i f(Qi) = boundary condition at boundary node i

21. Table of Contents • Introduction to Monte Carlo Method • Solving ODE: General Case • Solving ODE: Gambler’s Ruin • Solving PDE: Laplace Equation • Solving PDE: General Elliptic Equation • Conclusion

22. PDE: General Elliptic Equation • Elliptic Equation where B11 > 0 , B22 > 0 , and B11B22 – B122 > 0

23. PDE: General Elliptic Equation • Finite difference method Where , , , , 5 neighboring nodes with different probability!

24. Table of Contents • Introduction to Monte Carlo Method • Solving ODE: General Case • Solving ODE: Gambler’s Ruin • Solving PDE: Laplace Equation • Solving PDE: General Elliptic Equation • Conclusion

25. Conclusion • Advantage • No suffering from complicated geometry • Useful for high number of dimensions (curse of dimensionality) • Useful for functions without explicit form • No suffering from complicated geometry • Disadvantage • Requires large number of step size and sample size • Converges slowly

26. Conclusion • To improve accuracy • Larger number of random sampling • Smaller steps (∆x) • For special case, change parameter (