1. DEPARTMENT OF CIVIL ENGINEERING THE NATIONAL UNIVERSITY OF SINGAPORE Euro-SiBRAM2002 International Colloquium Prague - Czech Republic June 24 to 26, 2002 PKK-1 DIGITAL SIMULATION ALGORITHMS FOR SECOND- ORDER STOCHASTIC PROCESSES PHOON KK, QUEK ST & HUANG SP

2. DEPARTMENT OF CIVIL ENGINEERING THE NATIONAL UNIVERSITY OF SINGAPORE Euro-SiBRAM2002 International Colloquium Prague - Czech Republic June 24 to 26, 2002 PKK-2 WHY BET ON SIMULATION? MOORES LAW - density of transistors doubles every 18 months Computing power will increase 1000-fold after 15 years Common PC already comes with GHz processor, GB memory & hundreds of GB disk

3. DEPARTMENT OF CIVIL ENGINEERING THE NATIONAL UNIVERSITY OF SINGAPORE Euro-SiBRAM2002 International Colloquium Prague - Czech Republic June 24 to 26, 2002 PKK-3 CHALLENGE Develop efficient computer algorithms that can generate realistic sample functions on a modest computing platform Should be capable of handling: 1.stationary or non-stationary covariance fns 2.Gaussian or non-Gaussian CDFs 3.short or long processes

4. DEPARTMENT OF CIVIL ENGINEERING THE NATIONAL UNIVERSITY OF SINGAPORE Euro-SiBRAM2002 International Colloquium Prague - Czech Republic June 24 to 26, 2002 PKK-4 PROPOSAL Use a truncated Karhunen-Loeve (K-L) series for Gaussian process:

5. DEPARTMENT OF CIVIL ENGINEERING THE NATIONAL UNIVERSITY OF SINGAPORE Euro-SiBRAM2002 International Colloquium Prague - Czech Republic June 24 to 26, 2002 PKK-5 K-L PROCESS uncorrelated zero-mean unit variance Gaussian random variables eigenvalues & eigenfunctions of target covariance function C(x 1, x 2 )

6. DEPARTMENT OF CIVIL ENGINEERING THE NATIONAL UNIVERSITY OF SINGAPORE Euro-SiBRAM2002 International Colloquium Prague - Czech Republic June 24 to 26, 2002 PKK-6 KEY PROBLEM are solutions of the homogenous Fredholm integral equation of the second kind Difficult to solve accurately & efficiently

7. DEPARTMENT OF CIVIL ENGINEERING THE NATIONAL UNIVERSITY OF SINGAPORE Euro-SiBRAM2002 International Colloquium Prague - Czech Republic June 24 to 26, 2002 PKK-7 WAVELET-GALERKIN Family of orthogonal Harr wavelets generated by shifting & scaling Basis function over [0,1]

8. DEPARTMENT OF CIVIL ENGINEERING THE NATIONAL UNIVERSITY OF SINGAPORE Euro-SiBRAM2002 International Colloquium Prague - Czech Republic June 24 to 26, 2002 PKK-8

9. DEPARTMENT OF CIVIL ENGINEERING THE NATIONAL UNIVERSITY OF SINGAPORE Euro-SiBRAM2002 International Colloquium Prague - Czech Republic June 24 to 26, 2002 PKK-9 WAVELET-GALERKIN Express eigenfunction as a truncated series of Harr wavelets Apply Galerkin weighting

10. DEPARTMENT OF CIVIL ENGINEERING THE NATIONAL UNIVERSITY OF SINGAPORE Euro-SiBRAM2002 International Colloquium Prague - Czech Republic June 24 to 26, 2002 PKK-10 NUMERICAL EXAMPLE (1) Stationary Gaussian process over [-5, 5] with target covariance:

11. DEPARTMENT OF CIVIL ENGINEERING THE NATIONAL UNIVERSITY OF SINGAPORE Euro-SiBRAM2002 International Colloquium Prague - Czech Republic June 24 to 26, 2002 PKK-11 EIGENSOLUTIONS f(x )

12. DEPARTMENT OF CIVIL ENGINEERING THE NATIONAL UNIVERSITY OF SINGAPORE Euro-SiBRAM2002 International Colloquium Prague - Czech Republic June 24 to 26, 2002 PKK-12 COVARIANCE M = 10 M = 30

13. DEPARTMENT OF CIVIL ENGINEERING THE NATIONAL UNIVERSITY OF SINGAPORE Euro-SiBRAM2002 International Colloquium Prague - Czech Republic June 24 to 26, 2002 PKK-13 NON-GAUSSIAN K-L For= zero-mean process with non- Gaussian marginal distribution = vector of zero-mean unit variance uncorrelated ?? random variables

14. DEPARTMENT OF CIVIL ENGINEERING THE NATIONAL UNIVERSITY OF SINGAPORE Euro-SiBRAM2002 International Colloquium Prague - Czech Republic June 24 to 26, 2002 PKK-14 NON-GAUSSIAN K-L Can estimate using But integrand unknown – evaluate iteratively

15. DEPARTMENT OF CIVIL ENGINEERING THE NATIONAL UNIVERSITY OF SINGAPORE Euro-SiBRAM2002 International Colloquium Prague - Czech Republic June 24 to 26, 2002 PKK-15 NUMERICAL EXAMPLE (2) Stationary non-Gaussian process over [-5, 5] with target covariance & marginal CDF: = 0.5816, = 0.4723, = -2

16. DEPARTMENT OF CIVIL ENGINEERING THE NATIONAL UNIVERSITY OF SINGAPORE Euro-SiBRAM2002 International Colloquium Prague - Czech Republic June 24 to 26, 2002 PKK-16 MARGINAL CDF k = 1 k = 12

17. DEPARTMENT OF CIVIL ENGINEERING THE NATIONAL UNIVERSITY OF SINGAPORE Euro-SiBRAM2002 International Colloquium Prague - Czech Republic June 24 to 26, 2002 PKK-17 CONCLUSIONS K-L has potential for simulation Eigensolutions can be obtained cheaply & accurately from DWT Non-gaussian K-L can be determined by iterative mapping of CDF Theoretically consistent way to generate stationary/non-stationary, Gaussian/non- Gaussian process over finite interval