3. Proposed Methods for Uncertainty Propagation
3.2 Optimization-Based Confidence Intervals for CDF and Reliability Estimates In this chapter, uncertainty analysis is carried out using the probabilistic
techniques for reliability analysis described in the last section. We treat epistemic and aleatory uncertainties separately, performing reliability analysis conditioned on a realization of the distribution parameters. Thus there are two sets of uncertain variables in the problem. The first set of uncertain variables, x, has aleatory or irreducible
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uncertainty and is basic to the limit state function, i.e., these variables correspond to quantities such as capacity and load for a structure. The second set of variables has epistemic uncertainty, and is the distribution parameters θ, selected from a set of admissible values Θ. It should be noted that given the presence of epistemic uncertainty, the failure probability is itself uncertain because of the uncertainty in the distributions of the basic random variables. It is desired to determine bounds on this failure probability, given uncertainty in the distribution parameters. Explicit and separate treatment of the epistemic and aleatory variables allows for the calculation of probability distributions of and confidence intervals for the failure probability.
In general, the aleatory uncertainty is propagated using any appropriate probabilistic technique. However, the failure probability is conditioned on a set of distribution parameter values. This conditioning has necessitated nested methods for uncertainty propagation, where a set of distribution parameters would be selected first, and then given these distribution parameters, a reliability analysis would be performed.
Mehta et al (1993) proposed formulations that allow for the use of FORM in such a nested manner. The most general problem of calculating bounds on the failure probability would thus be stated as
Θ θ
θ
θ
∈ . .
. . .
) ( max min/
t s
t r w
PF
(9)
In the reliability analysis, the distributions of the basic random variables x are conditioned on θ. The cumulative distribution functions for the basic random variables with uncertain probability distributions are calculated by conditioning on a particular
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realization of the uncertain distribution parameters. Their optimum values are chosen to minimize or to maximize the failure probability.
If FORM is to be used in confidence bounds calculation, then the MPP is given below as a mathematical programming problem with the following generalized statement:
Θ
∈
= θ
x
θ x
θ x
0 ) (
. .
)}
, ( min { max min/
g t s
β
(10)
This nested optimization problem can be decoupled and expressed as:
)
*, ( max / min arg
*
) 0 ) (
*) , ( ( min arg
*
θ θ x
θ x x x
Θ θ
x
β β
∈
=
=
= where
g
(11)
Each optimization problem in Eq. (11) is solved iteratively until convergence.
If the uncertainty in the distribution parameters is represented probabilistically, then it is possible to use the approach of Eq. (11) to calculate confidence bounds on the failure probability. In calculating these confidence bounds, it is useful to define a transformation of the distribution parameters to the standard normal parameter space uθ. Once this transformation is defined, the second optimization problem can be defined such that the set Θ becomes a hypersphere in the transformed space of radius βT. With this definition, Eq. (11) then becomes
}
| )
*, ( max{
/ min arg
*
} ) 0 ) (
*) , ( { min arg
*
T
where
g
β β
β
θ =
=
=
=
θ u θ x
θ x x x
θ x
(12)
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We note that the solution of Eq. (12) guarantees that to first order accuracy the probability of the reliability index associated with the system’s limit state exceeding β(x*, θ*) is Φ(±βT). Hence the solution of the problem gives confidence bounds on the failure probability with the 1 - α/2 confidence level equal to Φ(-|βT|).
If the failure probability of an entire series or parallel system is of concern, MCS could be used directly with Eq. (9) where the failure or safety of all components in the system is evaluated for each randomly generated sample point. Alternatively, the MPP for each component could be determined using FORM for each limit state function, and the system reliability would become the objective function for the second optimization problem in Eq. (12).
It should be noted that there are no system response function evaluations required for the inverse FORM analysis with the epistemic variables. In other words, if expensive structural or CFD codes are required to evaluate the limit state function for the purposes of reliability analysis, in this decoupled formulation, no evaluations are required to find the values of the distribution parameters which minimize or maximize the likelihood of the MPP. This is because the second problem (the inverse FORM problem) in Eq. (11) manipulates the transformation to normality only, and does not involve solution of the first problem (the direct FORM problem) in which limit state functions are required to evaluate the gradients of the objectives and constraints. The inverse FORM reliability analysis finds the worst-case values of the distribution parameters so that the failure probability is maximized, or best case parameters such that the failure probability is minimized. When the two optimization problems converge, we have first order estimates
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of the failure probability by solving the reliability analysis, where the expensive function evaluations are encountered, only a few times in this decoupled formulation.
As in direct FORM for the case of certain probability distributions, sensitivity analysis can be performed on both the epistemic and aleatory uncertainties using the sensitivity vector α. The interpretation of the sensitivity vector α (see Eq. 8) for the aleatory random variables is much the same as in the case with probability distributions with no randomness. However, the alpha vector for the distribution parameters also lends important information to the decision maker. This vector gives an indication of the sensitivity of the failure probability to the uncertainty in each distribution parameter.
Sensitivities of distribution parameters near zero indicate that the outcome of the design problem is unlikely to change, regardless of the value of the distribution parameter. High sensitivities, however, indicate the distribution parameter has a large influence on the reliability estimate. This information can be used in determining the variables for which to pursue more intensive data collection.