Line sampling

Firstly the importance direction must be determined. This can be achieved by finding the design point, or the gradient of the limit state function.

A set of samples is generated using Monte Carlo simulation in the standard normal space. For each sample ${\displaystyle {\boldsymbol {x}}}$, the probability of failure in the line parallel to the important direction is defined as:

${\displaystyle p_{f}({\boldsymbol {x}})=\int _{-\infty }^{+\infty }I({\boldsymbol {x}}+\beta \cdot {\boldsymbol {\alpha }})\varphi (\beta )\,d\beta }$

where ${\displaystyle I(\cdot )}$  is equal to one for samples contributing to failure, and is zero otherwise:

${\displaystyle I_{f}({\boldsymbol {x}})={\begin{cases}1&{\text{if }}{\boldsymbol {x}}\in \Omega _{f}\\0&{\text{else}}\end{cases}}}$

${\displaystyle {\boldsymbol {\alpha }}}$  is the important direction, ${\displaystyle \varphi }$  is the probability density function of a Gaussian distribution (and ${\displaystyle \beta }$  is a real number). In practice the roots of a nonlinear function must be found to estimate the partial probabilities of failure along each line. This is either done by interpolation of a few samples along the line, or by using the Newton–Raphson method.

The global probability of failure is the mean of the probability of failure on the lines:

${\displaystyle {\tilde {p}}_{f}={\frac {1}{N_{L}}}\sum _{i=1}^{N_{L}}p_{f}^{(i)}}$

where ${\displaystyle N_{L}}$  is the total number of lines used in the analysis and the ${\displaystyle p_{f}^{(i)}}$  are the partial probabilities of failure estimated along all the lines.

For problems in which the dependence of the performance function is only moderately non-linear with respect to the parameters modeled as random variables, setting the importance direction as the gradient vector of the performance function in the underlying standard normal space leads to highly efficient Line Sampling. In general it can be shown that the variance obtained by line sampling is always smaller than that obtained by conventional Monte Carlo simulation, and hence the line sampling algorithm converges more quickly.[1] The rate of convergence is made quicker still by recent advancements which allow the importance direction to be repeatedly updated throughout the simulation, and this is known as adaptive line sampling.[2]

An illustration of the line sampling algorithm. Two line samples are shown approaching the limit state surface.

The algorithm is particularly useful for performing reliability analysis on computationally expensive industrial black box models, since the limit state function can be non-linear and the number of samples required is lower than for other reliability analysis techniques such as subset simulation.[3] The algorithm can also be used to efficiently propagate epistemic uncertainty in the form of probability boxes, or random sets.[4][5] A numerical implementation of the method is available in the open source software OpenCOSSAN.[6]

• Rare event sampling
• Curse of dimensionality
• Quantitative risk assessment