Class 22 U 2 1− ~ (0,1) Y U ( ) ( ) ( ) F y F x F x

457.646 Topics in Structural Reliability In-Class Material: Class 22

VI. Simulation Methods

(Ref. Melcher’s textbook Ch3)

 Simulating uniform random variable

U (0,1)

→ Basic in generation of random numbers

→ ( ) sequence from a seed number

→ Desirable to have a ( ) period and ( ) sampling

※ Matlab : rand( )

→ could choose a random number generation algorithm

→ default: Mersenne Twister (Matsumoto & Nishimura 1997)

→ Period:

2

¹⁹⁹³⁶

− 1

→ “Very fast”

Demo

X =[100 1000 10000]V

for i=1:3

X=rand(X (i),1);

subplot(3,1,i) hist(X,sqrt(Xv(i)));

end

V

 Generate random numbers according to CDF Consider

Y ~ U (0,1)

( ) ( ) ( )

Y X

X

F y F x

F x

=

=

∴ = x

⑴ Generate

y i

_i

, 1, =



, N

per ↖

U (0,1)

⑵ Find corresponding

x x

_i

,

_i

= , i = 1,



, N

 Generate general dependent variables

{ X

1

, , X

_n

}

^T

=

X  defined by

joint PDF ( ) joint CDF F ( )

 f

 

X X

x x

1

2 1

1 1

1 1

2 2 1

1 1

( ) ( )

( )

n n

X X X

n X X X n n

y F x

y F x x

y F x x x

− −

 =

 =

 

  =



^





1

2 1

1 1

1

1 1

1

2 2 1

1

1 1

( ) ( )

( )

n n

X X X

n X X X n n

x F y

x F y x

x F y x x

−

−

−

−

−

 = 

 =

 

 = 

^





 Simulation of normally distributed RV’s *(Box & Muller 1958) → homework

※ Matlab mvrnd(M

, ,

Σ

N

⁾

Generate N samples of X

~ N (

M

, )

Σ

 Generate random numbers from Nataf distribution

⑴ Simulate

{ , y

₁ 

, y

_n

}

^T

⑵ Find

{ , x

₁ 

, x

_n

}

^T Using (←)

cf. normrnd

~N( , )

= +

u 0 I X DLu M

i. Find R₀ (Liu & ADK, 1986)

ii. Generate u from

N 0 I ( , )

(or y from

U (0,1)

& transform) iii. Compute Z

~

L u₀ (or Z

~ N ( ,

0 R₀

)

)

iv. Compute ¹( ), i=1, ,n

i Xi

x =F⁻ 

 Monte Carlo Simulation

↖ City in Monaco (“MC project” in 1994)

( ) ( ) 0 f

( )

g

P f d

∪∩ ≤

= ∫

^X

x

x x

f ( ) d

= ∫

^{X x x}

=

average of index function value (w.r.t X

~ F

_X

( )

x )

Simulate x_i

, i=1,



,N

according to

f

_X

( )

x Let

q

_i

= I ( )

x_i ,

i=1,



,N

f

lim

N

P =

→∞

ˆf

P = Estimation of

P

_f using N sample Compare

mean (rand(3,1)) mean (rand(100000,1))

“MCS is an extremely bad method. It should be used only when all alternative methods are worse” –Alan Sokal (1996)

π

?

2

2

1 4 1

4

# 4 # r

r

π π

π

=

∴ = ×

Note: ˆ

Pf is random

↓

How much variability? _ˆ

Pf

δ

q

i : Bernoulli random variable 1 with p=

0 1-p=

[ ] [ ]

i i

E q Var q

=

=

=

=

• E P[ˆ_f]=

“unbiased” estimator of true

P

_f

• Var P[ˆ_f]=

=

=

⇒ _ˆ

1 1

f

f P

f

P N P

δ = = −

Quantifies variation of ˆ Pf

Used as a measure of convergence

※ Minimum No. of Simulation to achieve

δ

Target c.o.v 1 1 f

f

P N_δ P

δ

= ⁻

∴

1

_{2 f}

f

N P

δ

δ P

= −

⋅

e.g

P

_f

= 0.01

※ How to improve accuracy of simulation

1 [ ]

[ ˆ ] 1

ˆ [ ]

[ ]

f i

f i

P q

f i

Var q Var P N

E P E q N

δ = = = ⋅ δ

① Increase N

② Decrease

qi

δ