Line sampling

The basic idea behind line sampling is to refine estimates obtained from the first-order reliability method (FORM), which may be incorrect due to the non-linearity of the limit state function.

in the input parameter space, which points towards the region which most strongly contributes to the overall failure probability.

The importance direction can be closely related to the center of mass of the failure region, or to the failure point with the highest probability density, which often falls at the closest point to the origin of the limit state function, when the random variables of the problem have been transformed into the standard normal space.

Once the importance direction has been set to point towards the failure region, samples are randomly generated from the standard normal space and lines are drawn parallel to the importance direction in order to compute the distance to the limit state function, which enables the probability of failure to be estimated for each sample.

In practice the roots of a nonlinear function must be found to estimate the partial probabilities of failure along each line.

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.

[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] 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.