An Adaptive Line Sampling Method for Reliability Analysis

Abstract

Rare event probability estimation imposes serious difficulties to the conventional simulation methods like crude Monte Carlo. Several methods have been put forward to address this issue. Line sampling, directional simulation, importance sampling and subset simulation are more known among other methods. This paper aims at improving the line sampling method. The proposed method iteratively adapts the line sampling procedure to the limit state surface and at the end yields an estimation of the failure probability with the required coefficient of variation. The proposed method is compared to the conventional line sampling via well-known academic problems. The results show considerable improvement over the conventional line sampling method.

Appendix

Here we present the proof of Eqs. (9) to (11). It is evident that changing the important direction does not affect the probability estimation, so averaging the reliability results derived from different important directions in different iterations also delivers an estimation of the failure probability [Eq. (9)]. We choose the averaging weights so as to minimize the coefficient of variation for the averaged estimation. Assuming the independence of estimations drawn from different iterations, the standard deviation of the averaged estimation is:

$$\sigma = \sqrt {\mathop \sum \limits_{i = 1}^{k} w_{i}^{2} \sigma_{i}^{2} }$$

(15)

Dividing both sides by P _f produces Eq. (11). By application of the Lagrange multiplier method and applying the constraint of $\mathop \sum \limits_{i = 1}^{k} w_{i} = 1$, the optimal weights will be derived as Eq. (10).

Derivation of Eq. (7).

We know that for identically distributed and independent random variables X _i , the variance of their arithmetic mean $\bar{X}$ is:

$${\text{Var}}\left( {\bar{X} = \frac{1}{N}\mathop \sum \limits_{i = 1}^{N} X_{i} } \right) = \frac{1}{N}{\text{Var}}\left( X \right)$$

(16)

The variance of all the X_i random variables is equal to each other and is denoted by the term Var(X). By definition of Var(X) as ${\text{Var}}\left( X \right) = E\left[ {\left( {X - E\left( X \right)} \right)^{2} } \right]$, it can be estimated as:

$${\text{Var}}\left( X \right) \approx \frac{1}{N}\mathop \sum \limits_{i = 1}^{N} \left( {X_{i} - \bar{X}} \right)^{2}$$

(17)

Thus, we get an estimation for the variance of $\bar{X}$ as:

$${\text{Var}}\left( {\bar{X}} \right) \approx \frac{1}{{N^{2} }}\mathop \sum \limits_{i = 1}^{N} \left( {X_{i} - \bar{X}} \right)^{2}$$

(18)

To derive the Eq. (7), it is sufficient to replace $X_{i}$ by $\varPhi \left( { - \left| {c_{i} } \right|} \right)$.