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multi-dimensional integral in Equation 1 defin-
ing seismic risk and involved in optimization
in Equation 3 cannot be calculated, or even ac-
curately approximated, analytically. An efficient
alternative approach is to estimate the integral by
stochastic simulation
(Taflanidis & Beck, 2008).
Using a finite number,
N
, of samples of
θ
drawn
from some importance sampling density
p
is
(
θ
), an
estimate for (1) is given by the
stochastic analysis
:
selection of importance sampling densities may
preliminary focus on them.
Optimization for the Design
Variables under Uncertainty
The optimal design choice is finally given by the
optimization under uncertainty problem (Spall,
2003):
ˆ
( )
*
1
p
p
(
θ
θ
i
)
ϕ
=
arg min
C
ϕ
(24)
ˆ
( )
ˆ
( ,
N
∑
i
C
ϕ
=
C
ϕ
Ω
)
=
h
( ,
ϕ θ
)
ϕ
∈
Φ
N
i
i
=
1
(
)
is
(23)
The estimate of the objective function for this
optimization involves an unavoidable estimation
error and significant computational cost since
N
evaluations of the model response are needed for
each stochastic analysis. These features that make
this optimization problem challenging. Many nu-
merical techniques and optimization algorithms
have been developed to address such challenges
in design optimization under uncertainty (Royset
& Polak, 2004; Ruszczynski & Shapiro, 2003;
Spall, 2003). Such approaches may involve one
or more of the following strategies: (i) use of
common random numbers to reduce the relative
importance of the estimation error when com-
paring two design choices that are “close” in the
design space, (ii) application of exterior sampling
techniques which adopt the same stream of random
numbers throughout all iterations in the optimi-
zation process, thus transforming the stochastic
problem into a deterministic one, (iii) simultaneous
perturbation stochastic search techniques, which
approximate at each iteration the gradient vector
by performing only two evaluations of the objec-
tive function in a random search direction, and
(iv) gradient-free algorithms (such as evolution-
ary algorithms) which do not require derivative
information. References (Spall, 2003; Taflanidis
& Beck, 2008) provide reviews of appropriate
techniques and algorithms for such optimiza-
tion problems. The very efficient Simultaneous
Perturbation Stochastic Approximation (SPSA)
where vector
θ
i
denotes the sample of the uncer-
tain parameters used in the
i
th
simulation and
Ω
denotes the entire samples set {
θ
i
} used in the
evaluation. As
N
→ ∞
, then Ĉ→C but even for
finite, large enough
N
, Equation 23 gives a good
approximation for the integral in Equation 1. The
importance sampling density
p
is
(
θ
) may be used
to improve the efficiency of this estimation. This
is established by focusing on regions of the
Θ
space that contribute more to the integrand of the
probabilistic integral in Equation 1 (Taflanidis &
Beck, 2008), i.e. by selecting a proposal density
that resembles that integrand. If
p
is
(
θ
)
=p
(
θ
) then
the evaluation in Equation 23 corresponds to
direct Monte Carlo integration. For problems with
large number of model parameters, as the applica-
tion discussed here, choosing efficient importance
sampling densities for all components of
θ
is
challenging and can lead to convergence problems
for the estimator of Equation 23; thus it is prefer-
able to formulate importance sampling densities
only for the important components of
θ
, i.e. the
ones that have biggest influence on the seismic
risk, and use
q
(.)=
p
(.) for the rest (Taflanidis &
Beck, 2008). For seismic risk applications the
characteristics of the hazard, especially the mo-
ment magnitude or epicentral distance are gener-
ally expected to have the strongest impact on the
calculated risk (Taflanidis & Beck, 2009a), so
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