DIGITAL SIMULATION ALGORITHMS FOR SECOND-ORDER STOCHASTIC PROCESSES
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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
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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
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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
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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:
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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
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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
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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]
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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
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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
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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:
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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)
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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
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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
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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
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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
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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
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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