Environmental Engineering Reference
In-Depth Information
Using the product rule of probability (e.g., Ang and Tang, 2007),
Equation 7.1
is rewrit-
ten as
m
m
∏
∏
…
PE
()
=
P
()
E
=
P
()
E
PE EE
(
|
E
)
=
P
()
E
PE E
(
|
)
(7. 2)
m
1
i
i
-
12
i
-
1
1
i
i
-
1
i
=
2
i
=
2
where
P
(
E
1
) =
P
(
Y
>
y
1
) and
P
(
E
i
|
E
i
−1
) = {
P
(
Y
>
y
i
|
Y
>
y
i
−1
),
i
= 2, …,
m
} is equal to
P
(
E
i
|
E
i
−1
E
i
−2
…
E
1
) in
Equation 7.2
because
E
i
−1
,
E
i
−2
, …,
E
1
are a sequence of nested events, that
is,
EE
…
i
. In implementations,
y
1
,
y
2
, …,
y
m
are generated adaptively using
information from simulated samples so that the sample estimate of
P
(
E
1
) and {
P
(
E
i
|
E
i
−1
),
i
= 2, …,
m
} always corresponds to a common specified value of conditional probability
p
0
(Au and Beck, 2001, 2003; Au et al., 2010). The efficient generation of conditional
samples is pivotal to the success of Subset Simulation, and it is made possible through
the machinery of MCMC. In MCMC, Metropolis algorithm (Metropolis et al., 1953) is
used here, and successive samples are generated from a specially designed Markov chain
whose limiting stationary distribution tends to the target PDF as the length of Markov
chain increases.
⊃⊃⊃
E
1
2
−
1
7.2.2 Simulation procedures
Subset Simulation starts with Direct MCS, in which
N
direct MCS samples are generated.
The
Y
values of the
N
samples are calculated and ranked in an ascending order. The (1 −
p
0
)
N
th value in the ascending list of
Y
values is chosen as
y
1
, and hence, the sample estimate
for
P
(
E
1
) =
P
(
Y
>
y
1
) is
p
0
. In other words, there are
p
0
N
samples with
E
1
=
Y
>
y
1
among the
N
samples generated from Direct MCS. Then, the
p
0
N
samples with
E
1
=
Y
>
y
1
are used as
“seeds” for the application of MCMC to simulate
N
additional conditional samples given
E
1
=
Y
>
y
1
. The
p
0
N
seed samples are then discarded so that there are a total of
N
samples
with
E
1
=
Y
>
y
1
. The
Y
values of the
N
samples with
E
1
=
Y
>
y
1
are ranked again in an
ascending order, and the (1 −
p
0
)
N
th value in the ascending list of
Y
values is chosen as
y
2
,
which defines the
E
2
=
Y
>
y
2
. Note that the sample estimate for
P
(
E
2
|
E
1
) =
P
(
Y
>
y
2
|
Y
>
y
1
)
is also equal to
p
0
. Similarly, there are
p
0
N
samples with
E
2
=
Y
>
y
2
. These samples provide
“seeds” in MCMC to simulate additional
N
conditional samples with
E
2
=
Y
>
y
2
. Then, the
p
0
N
seed samples are discarded, so that there are
N
conditional samples with
E
2
=
Y
>
y
2
.
The procedure is repeated
m
times until the probability space of interest (i.e., the sample
domain with
Y
>
y
m
) is achieved. Note that the Subset Simulation procedures contain
m
+ 1 steps, including one Direct MCS to generate unconditional samples and
m
steps of
MCMC to simulate conditional samples. The
m
+ 1 steps of simulations are referred to as
“
m
+ 1 levels” in Subset Simulation (Au and Beck, 2001, 2003; Au et al., 2010). Finally,
N
+
m
(1 −
p
0
)
N
samples are obtained from the
m
+ 1 levels of simulations.
A polynomial function example is used here to demonstrate the Subset Simulation
procedures described above. Consider a reliability analysis with a performance function
GX X
= 3
1
2
, where
X
1
= a Gaussian random variable with a mean of 1 and standard
deviation of 0.05 and
X
2
= a Gaussian random variable with a mean of 1 and standard
deviation of 0.1. Failure occurs when
G
≥ 15. A Subset Simulation run with
N
= 50,
m
= 1,
p
0
= 0.1, and
Y
=
G
is performed to evaluate the failure probability in the example. Detailed
steps of Subset Simulation are described below:
2
) values of the 50 MCS samples;
2. Calculate the
Y
(i.e.,
YG XX
== +
3
9
1
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