Variance Reduction - Stochastic Structural Dynamics - Lecture Slides, Slides of Stochastic Processes

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Monte Carlo simulation approach-

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^ ^

^



^

^ 

^ 

^ 

^ ^

^

^

^ 

^ ^

^ 

 ^

 ^

(^ )^01

(^21)

( )

1

(^

)

1

Var^

(^

)

( )^

0

( )^

;^ ( )

.

0

f^

X^

X

g xn

i i

n

F^

F

F^

F i

X

F^

V^

F^

h

V X

V

F

P^

p^ x dx

I g x

p^

x dx I^

g^ X

I^ g^

X n

P^

P

P^

P^

n

n

I^ g x

p^ x

P^

F x h

x dx F x

P^

F^ X

h^ x

I^ g v

p^ v

h^ v

P



^





  

^



^

^

^ 



^



 ^

^





^ 

^





^

^

^



^

 ^

 ^ 





 

Probability of failure Variance reduction

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Sub

‐set

simulations

using

Markov

Chain

Monte

Carlo

(MCMC)

-^ S K Au and J L Beck, 2001, Estimation of small failureprobabilities in high dimension by subset simulation, ProbabilisticEngineering Mechanics, 16, 263-277•^ J S Liu, 2001, Monte Carlo strategies in scientific computing,Springer, NY.

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Basic idea^ Small failure probability can be expressed as a product  of larger conditional failure probabilities.These larger conditional failure probabilities can be estimated with lesser computation

al effort.

The method is applicable to a wide class of problems

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^ ^

^



^ ^

    ^  ^ ^

^ ^

  ^ ^

^  ^

 ^

 ^ ^

^ 

• 0

0,

0, *

1

max

max ,

,^ ,^0

F

t^ T m m^

t^ T

m N n n^ n F^

X

P^

P^ z t

z

t^

T

P^

z t^

z

P^ Z

X

z

P^ g

X

Z^

X^

z t

g^ X

z

Z

X

X^

a^ b

z

P^

I^ g^

x^

 p^ x dx

 



^  

^

^

^  

^

^

^

^

^

^

^

^

^

^

^

^

 ^

^

^

^



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^ ^

^ 

   1 2 ˆ

ˆ^

ˆ^ is an unbiased and consistent estimator of P

with

minimum variance. The optimal variance is given by

(^

F^ F

X N^

i

F

i F^

F

F^

F

P^ I P

g^ x^

p^ x dx

P^

I^ g^

X

N P P^

P n   

^ 

^

^

^

^

^

   Remark

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^ ^

^



^

^

^ 

(^11)

1 1

1

1

If^ -s are configured such that

|^

and

are much larger than

, then we will be able to estimate

in terms of product of "large" probabilities.Suppose,

m F^

i^

i i i^

i^

i

F

F

F P^

P F^

P F^

F

F^

P F

F

P F

P

P

 P

 





^



Remarks

^

^ ^

^ ^

^ ^

^ ^

^ ^



6

1

1

1

1

1

1

then we could obtain an estimate of

as 10

Estimation of probability of failure of the order of 0.1 can beeasily done using MCS because the failure events here are m

P^ F

^

^

^

^

^

^



^

^

^

^

ore

frequent.

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^ ^

^



^  ^

(^11) 

1 1 1 1

can be estimated using a "brute force" Monte Carlo.|^ ,

,^

1 can be estimated using MCMC.

m F^

i^

i i i^

i

P^

P F

P F

F

P F P F

F

i^

m



 

^



Remarks (continued)

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^  (^1) th

• 2

4. This leads to 200 points in failure region of . Rank order the value

of^ (^

) at these 200 points and identify the 20

ranked member and denote

it by^

. Define a new performance

g^ X

Steps (Continued) g Xg

^ ^

^ 

^ 

^ ^

^ 

^  2

2

2

2

2

2

1

1

1

function

.

Define

0 ˆ Clearly,

Estimate of

0 |^

0 0.1.

5. Repeat this exercise till

is reached.

6. Obtain the final probability of failure by using

F

m

F^

i

g^ X g X^

g

F^

g^ X P

P^ g^

X^

g^ X

F^ F

P^ P F

P F



^



^

 ^

 ^

 ^



^

^

^ 

^





^

^

 (^11)

| m

i i

F

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RemarksThe definition of

-s (as in the present illustrative explanation)

ensures that

-s are all equal to 0.1. Estimates for sampling variance can be deduced.Choice of proposal density functio

i

i F

F P   ^

n:

In standard normal space, typically shifted normal pdf.

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^ 

^

^

^ 

Number of samples: 200 at each subsetProposal pdf

|^

~N^

i^

i

q^

X^

x^

x^

I

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0

1

2

3

4

5

6

010 -1 10 -2 10 -3 10 -4 10 -5 10

level^ 

Failure probability

Level

 





^ 

1 P  F

5

Blue line:Simulation with 10

samples

(^1)  10 2 ^10

(^3)  10

(^4)  10

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 ^

^ 

^ 

^

     25 1 0 10

cos^

sin 1

~ iid N 0,

;^ ~ iid N 0, 2

,^

2 max What is P

n^

n^

n^

n

n n^

n

n^

k n m t

m X^ t^

a^

t^ b

t

a^

b

a^

b^ n k n X^

X^ t X

^

^    

^

^

^

^

^

^

^

^

^

^ 

 Example Question

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^ 

^

^

^ 

5

Number of samples: 200 per subsetProposal pdf

|^

~N^

Brute force Monte Carlo with 10

samples

i^

i

q^

X^

x^

x^

I

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