DEPARTMENT OF CIVIL ENGINEERING • THE NATIONAL UNIVERSITY OF SINGAPORE

Euro-SiBRAM’2002 International Colloquium Prague - Czech Republic June 24 to 26, 2002 PKK-1

DIGITAL SIMULATION DIGITAL SIMULATION ALGORITHMS FOR SECOND-ALGORITHMS FOR SECOND-

ORDER STOCHASTIC ORDER STOCHASTIC PROCESSESPROCESSES

PHOON KK, QUEK ST & HUANG SP

DEPARTMENT OF CIVIL ENGINEERING • THE NATIONAL UNIVERSITY OF SINGAPORE

Euro-SiBRAM’2002 International Colloquium Prague - Czech Republic June 24 to 26, 2002 PKK-2

WHY BET ON SIMULATION?WHY BET ON SIMULATION?

• MOORE’S 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

DEPARTMENT OF CIVIL ENGINEERING • THE NATIONAL UNIVERSITY OF SINGAPORE

Euro-SiBRAM’2002 International Colloquium Prague - Czech Republic June 24 to 26, 2002 PKK-3

CHALLENGECHALLENGE

• 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

DEPARTMENT OF CIVIL ENGINEERING • THE NATIONAL UNIVERSITY OF SINGAPORE

Euro-SiBRAM’2002 International Colloquium Prague - Czech Republic June 24 to 26, 2002 PKK-4

PROPOSAL PROPOSAL

Use a truncated Karhunen-Loeve (K-L) series for Gaussian process:

DEPARTMENT OF CIVIL ENGINEERING • THE NATIONAL UNIVERSITY OF SINGAPORE

Euro-SiBRAM’2002 International Colloquium Prague - Czech Republic June 24 to 26, 2002 PKK-5

K-L PROCESSK-L PROCESS

uncorrelated zero-mean unit variance Gaussian random variables

eigenvalues & eigenfunctions of target covariance function C(x1, x2)

)(k

)x(f , kk

DEPARTMENT OF CIVIL ENGINEERING • THE NATIONAL UNIVERSITY OF SINGAPORE

Euro-SiBRAM’2002 International Colloquium Prague - Czech Republic June 24 to 26, 2002 PKK-6

KEY PROBLEMKEY PROBLEM

are solutions of the homogenous Fredholm integral equation of the second kind

Difficult to solve accurately & efficiently

)x(f , kk

DEPARTMENT OF CIVIL ENGINEERING • THE NATIONAL UNIVERSITY OF SINGAPORE

Euro-SiBRAM’2002 International Colloquium Prague - Czech Republic June 24 to 26, 2002 PKK-7

WAVELET-GALERKINWAVELET-GALERKIN

• Family of orthogonal Harr wavelets generated by shifting & scaling

• Basis function over [0,1]

DEPARTMENT OF CIVIL ENGINEERING • THE NATIONAL UNIVERSITY OF SINGAPORE

Euro-SiBRAM’2002 International Colloquium Prague - Czech Republic June 24 to 26, 2002 PKK-8

0 0.5 1-1

0

1

0 0.5 1-1

0

1

0 0.5 1-1

0

1

0 0.5 1-1

0

1

0 0.5 1-1

0

1

0 0.5 1-1

0

1

0 0.5 1-1

0

1

1= 0,1

2= 1,0 3= 1,1

4= 2,0 5= 2,1 6= 2,2 7= 2,3

j=0

j=1

j=2

DEPARTMENT OF CIVIL ENGINEERING • THE NATIONAL UNIVERSITY OF SINGAPORE

Euro-SiBRAM’2002 International Colloquium Prague - Czech Republic June 24 to 26, 2002 PKK-9

WAVELET-GALERKINWAVELET-GALERKIN

• Express eigenfunction as a truncated series of Harr wavelets

• Apply Galerkin weighting

DEPARTMENT OF CIVIL ENGINEERING • THE NATIONAL UNIVERSITY OF SINGAPORE

Euro-SiBRAM’2002 International Colloquium Prague - Czech Republic June 24 to 26, 2002 PKK-10

NUMERICAL EXAMPLE (1)NUMERICAL EXAMPLE (1)

Stationary Gaussian process over [-5, 5] with target covariance:

DEPARTMENT OF CIVIL ENGINEERING • THE NATIONAL UNIVERSITY OF SINGAPORE

Euro-SiBRAM’2002 International Colloquium Prague - Czech Republic June 24 to 26, 2002 PKK-11

EIGENSOLUTIONSEIGENSOLUTIONS

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

mode

eige

nval

ue

-5 -4 -3 -2 -1 0 1 2 3 4 5-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

xei

genf

uctio

n

firstfifthtenth f(x)

DEPARTMENT OF CIVIL ENGINEERING • THE NATIONAL UNIVERSITY OF SINGAPORE

Euro-SiBRAM’2002 International Colloquium Prague - Czech Republic June 24 to 26, 2002 PKK-12

COVARIANCECOVARIANCE

0 1 2 3 4 5-0.2

0

0.2

0.4

0.6

0.8

1

1.2

lag

cova

rian

ce

10 termstarget

0 1 2 3 4 5-0.2

0

0.2

0.4

0.6

0.8

1

1.2

lag

cova

rian

ce

30 termstarget

M = 10 M = 30

DEPARTMENT OF CIVIL ENGINEERING • THE NATIONAL UNIVERSITY OF SINGAPORE

Euro-SiBRAM’2002 International Colloquium Prague - Czech Republic June 24 to 26, 2002 PKK-13

NON-GAUSSIAN K-LNON-GAUSSIAN K-L

For = zero-mean process with non-Gaussian marginal distribution

= vector of zero-mean unit variance uncorrelated ?? random variables

),x(

)(i

DEPARTMENT OF CIVIL ENGINEERING • THE NATIONAL UNIVERSITY OF SINGAPORE

Euro-SiBRAM’2002 International Colloquium Prague - Czech Republic June 24 to 26, 2002 PKK-14

NON-GAUSSIAN K-LNON-GAUSSIAN K-L

Can estimate using

But integrand unknown – evaluate iteratively

DEPARTMENT OF CIVIL ENGINEERING • THE NATIONAL UNIVERSITY OF SINGAPORE

Euro-SiBRAM’2002 International Colloquium Prague - Czech Republic June 24 to 26, 2002 PKK-15

NUMERICAL EXAMPLE (2)NUMERICAL EXAMPLE (2)

Stationary non-Gaussian process over [-5, 5] with target covariance & marginal CDF:

= 0.5816, = 0.4723, = -2

DEPARTMENT OF CIVIL ENGINEERING • THE NATIONAL UNIVERSITY OF SINGAPORE

Euro-SiBRAM’2002 International Colloquium Prague - Czech Republic June 24 to 26, 2002 PKK-16

MARGINAL CDFMARGINAL CDF

-4 -3 -2 -1 0 1 2 3 40

0.2

0.4

0.6

0.8

1

values

prob

abili

ty

target simulated

-4 -3 -2 -1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

values

prob

abili

ty

simulatedtarget

k = 1 k = 12

DEPARTMENT OF CIVIL ENGINEERING • THE NATIONAL UNIVERSITY OF SINGAPORE

Euro-SiBRAM’2002 International Colloquium Prague - Czech Republic June 24 to 26, 2002 PKK-17

CONCLUSIONSCONCLUSIONS

• 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