Concept of Line Sampling

Line Sampling, introduced in [36], and recently applied in [53], is an advanced simulation method developed to efficiently compute small failure probabilities for high dimensional problems. The method exploits the metric of the Standard Normal Space (SNS) and requires the knowledge of the so-called “important direction”, which is defined as the vector a P Rd pointing towards the failure region. An initial approximation for the

important direction is commonly obtained by computing the gradient of the performance function,∇G, at the origin of the SNS (see e.g. [51]). Simulation methods estimate the failure probability by computing the integral in the transformed SNS

pF “

ż

Rd´1

IFpuq dΦpuq; (3.1)

where, I_F :Rd Ñ t0,1u is the indicator function, which is1 if Gď0 and 0 otherwise,

original state space to the SNS, and Φpuq is the standard normal CDF. Note that all

the variables have to be transformed into the SNS, via T : Rd ÞÑ Rd, in order for the

method to function, as the method makes use of the geometric feature of the SNS. For example, in this space the norm of a point represents the so-called reliability index, which quantifies how many standard deviations that point is away from the median state in the original space. Eq. (3.1) can be written in the form

pF “ ż Rd´1 ˆż8 ´8 IFpuqφpu1qdu1 ˙ d ź i“2 φpuiqdui; (3.2)

for convenient evaluations. Withu1 pointing orthogonally towards the failure domain,

the expansion wpu2:dq “

ş8

´8IFpuq φpu1q du1 of Eq. (3.2) is a function of the d´1

remaining standard normal variables u2:d P Rd´1 and provides a measure of likeli-

hood for the variable u2:d to be in the failure domain. All the points with coor-

dinates uK _{“ t0,u}

2:du lie on the hyperplane orthogonal to the first coordinate u1.

Variable w can be calculated as wpu2:dq “ ΨpF1q, where F1 “ tu1 PR|Gď0u, and

ΨpAq “ş8_´8IApuqφpuqduis the Gaussian measure of any subsetAĂR. Let the scalar

c˚ be the smallest (in magnitude) value of the coordinate _u

1 such that

c˚

“mintu1PR|IFpuq “1u; (3.3)

then, wcan be approximately calculated as wpu2:dq “Φp´|c˚|q. Therefore, the failure

probability can be obtained as the expected value pF “Erwpu2:dqs “ ż Rd´1 wpu2:dq d ź i“2 φpuiqdui. (3.4)

Note that considering the standard normal CDF,Φp´|c˚_|q, in place of the Gaussian

measure,ΨpF1q, the probability,w, can only be overestimated, because it assumes that

no furthersurvivalregions can be found on the line beyondc˚_{. LS provides an estimation}

of Erwsby repeatedly generating points u2:d from the standard normal PDF in Rd´1,

and computing the respective partial probabilities wpu2:dq. For example, generating

NLpoints (lines)u_2:tj_du, j“1,2, .., NL, an estimate of the failure probability is obtained

computing the average

ˆ pF “ 1 NL NL ÿ j wput_2:j_duq. (3.5) The above approach, can be applied by orienting the important direction a as the coordinate u1. The integral of Eq. (3.4) is calculated, in practice, by exploiting the

geometric features of the SNS. Standard normal points on the hyperplane orthogonal to a can be obtained from any generated standard normal point u as follows uK

a “

is oriented as the important direction. In this way, the search for the limit state, for each random point (line) tju, can be set as utajupcq “ uKta ju`c a, where the scalar c

controls the location of the points along the line tju.

Standard implementation of LS operates with a fixed, initially determined impor- tant direction a. For each random point utju_{, the distance from the hyperplane to}

the performance function in the direction of a is identified searching along the lines utajupcq. Moreover, in standard LS the line search is conducted evaluating the perfor-

mance function G on the support sequence c “ tc1, ..., cNcu, to find the value c ˚ _by

means of interpolation, usually requiring from 6 to8 model evaluations per line.