Titel_SS06

Importance Sampling Simulation Method

As already mentioned the problem in using Equation (6.43) is that the sampling function
fX()x typically is located in a region far away from the region where the indicator function
Ig( )x  0  attains contributions. The success rate in the performed simulations is thus low.

In practical reliability assessment problems where typical failure probabilities are in the order
of 10-3 – 10-6 this in turn leads to the effect that the variance of the estimate of failure
probability will be rather large unless a substantial amount of simulations are performed.

To overcome this problem different variance reduction techniques have been proposed aiming
at, with the same number of simulations to reduce the variance of the probability estimate. In
the following one of the most commonly applied techniques for variance reduction in
structural reliability applications will be briefly considered, namely the importance sampling
method.

The importance sampling method takes basis in the utilisation of prior information about the
domain contribution to the probability integral, i.e. the region that contributes to the indicator
function. Let us first assume that it is known which point in the sample space contributes
the most to the failure probability. Then by centring the simulations on this point, the
important point, a higher success rate in the simulations would be obtained and the variance of
the estimated failure probability would be reduced. Sampling centred on an important point
may be accomplished by rewriting Equation

x^4

(6.40) in the following way:

 

()

() 0 () () 0 ()

F ()

f

PIg fd Ig fd

f

 X X

S

x

xxxx x

x S

x (6.46)

in which is denoted the importance sampling density function. It is seen that the

integral in Equation

fS()x

(6.46) represents the expected value of the term () 0 ()

()

Ig f

f

 X

S

x x

x

where

the components of S are distributed according to fS( )s

()

. The question in regard to the choice

of an appropriate importance sampling functionfSs , however, remains open.

One approach to the selection of an importance sampling density function fS( )s

  *

S x

X

is to select an
-dimensional joint Normal probability density function with uncorrelated components, mean
values equal to the design point as obtained from FORM analysis, i.e. and standard

deviations e.g. corresponding to the standard deviations of the components of , i.e.  

n

SX.

In this case Equation (6.46) may be written as:

 

() 0 () () () 0 ()()

F () ()

PIg f fd Ig d

f

-

-

X S 

S

xxxsx

xs

fXs s s (6.47)

in equivalence to Equation (6.42) leading to:



1

(^1) () 0 ()
()
N
f
j
PIgf
N  -
 s Xs
s

(6.48)

which may be assessed by sampling over realisations of s as described in the above.