Efficiency and accuracy

Adaptive Line Sampling shows an improvement in efficiency and accuracy superior to the standard version. This is elucidated in a comparative study with a reference solution obtained by direct Monte Carlo simulation. An explicit performance function is used to test the methods, which is expressed as gpxq “ ´?xT_x`a, where _x are _N

RV

independent normal random variables and a is a constant. First, the test is run with just two random variables, but with decreasing probability targets, by selecting different values ofain the performance function g, as shown in Table 3.1.

Note that, in this case, the values of probability pF “ Φp´βq obtained by First

Order Reliability Method [15] are not trustworthy because of the concave shape of the limit state surface. An illustration of the performance of the methods is shown in Figure 3.3a, where the number of required samples from LS and ALS is compared

Table 3.1: Advanced Line Sampling efficiency tests run on the performance function gpxq “ ´ax2

1`x22`a, reference solution is obtained via MC (106 samples).

gpxq “ ´ b

x21`x22`a; x1„Np5,22q, x2„Np2,22q; a“ t10,10.2,10.5,12,14,16u

Direct Monte Carlo Adaptive Line Sampling Line Sampling

β pc˝_{q pˆ}

F CoV p10´2q pˆF CoVp10´2q Ns pˆF CoV Ns

2.307 1.49 10´2 _{0.8 10}´2 _{1.35 10}´2 _{6.0 63 1.32 10}´2 _{5.5 210} 2.407 1.16 10´2 _{0.9 10}´2 _{1.08 10}´2 _{10.8 66 9.75 10}´3 _{5.4 210} 2.557 7.40 10´3 _{1.1 10}´2 _{6.60 10}´3 _{4.5 67 7.00 10}´3 _{5.8 210} 3.304 7.06 10´4 _{3.8 10}´2 _{6.58 10}´4 _{9.2 65 6.69 10}´4 _{13.6 210} 4.307 1.42 10´5 _{26.5 10}´2 _{1.23 10}´5 _{13.9 59 1.18 10}´5 _{8.3 210} 5.307 ´ ´ 9.18 10´8 _{12.4 64 10.14 10}´8 _{14.0 210}

Table 3.2: Adaptive Line Sampling efficiency tests run on the performance function gpxq “

´ ?

xT_{x`a. and reference solution obtained via MC (10</sub>6<sub>samples).</sub> gpxq “ ´?xT<sub>x`a; x}

i„Np2,1q; a“ t7.0,9.3,14.7,18.1,24.8,34.0,52.5u

Direct Monte Carlo Adaptive Line Sampling Line Sampling

NRV β

` c˝˘

ˆ

pF CoV p10´2q pˆF CoVp10´2q Ns pˆF CoV p10´2q Ns

4 3.000 3.27 10´3 1.75 3.52 10´3 19.7 94 2.75 10´3 7.2 215 10 2.975 8.37 10´3 1.10 6.97 10´3 11.1 102 8.57 10´3 16.9 221 30 3.745 4.56 10´3 _{1.48 4.13 10}´3 _{12.9 120 4.27 10}´3 _{15.6 241} 50 3.958 7.44 10´3 1.15 7.87 10´3 17.8 144 7.59 10´3 16.5 261 100 4.800 4.87 10´3 1.43 5.27 10´3 17.5 222 5.92 10´3 19.7 311 200 5.716 5.85 10´3 _{1.30 5.20 10}´3 _{18.3 323 6.16 10}´3 _{17.0 411} 500 7.778 4.15 10´3 1.55 3.44 10´3 21.4 619 3.56 10´3 17.8 711

to direct MC obtained analytically using a fixed value of CoV “ 10´2_{. The ana-}

lytic number of samples required by direct MC, denoted by NM C, can be obtained as

NM C “ p1´pFq{pCoV2¨pFq, wherepF is the target failure probability. A satisfactory

level of accuracy (CoV“5¨10´2_{) is achieved with just65samples using ALS, compared}

to a necessary sample size of 210 samples using LS. The second test is run fixing the

probability targets (approximately to 10´3_{), while progressively increasing the number}

of random variables, as shown in Table 3.2. The results of the second test, as illustrated in Figure 3.3b, show that Monte Carlo is, as expected, insensitive to the number of variables. Line Sampling methods require significantly fewer samples to achieve the same level of accuracy of MC. The slight increase in samples number with the number of variables in Figure 3.3b is due to the calculation of gradient used to initialise the im- portant direction. As expected, in this second test, the probability of failure computed with the First Order Reliability Method is inaccurate, as also shown in Table 3.2. In both cases, Adaptive Line Sampling has shown not only to be accurate but also to be

10−6 10−4 10−2 100 105 1010 1015 p F number of samples

Direct Monte Carlo Line Sampling Adaptive Line Sampling

(a) 4 10 100 500 102 104 106 number of variables number of samples

Direct Monte Carlo Line Sampling Adaptive Line Sampling

(b)

Figure 3.3: Number of samples required from ALS and LS compared to direct MC (a) for

decreasing probability target, and (b) for increasing dimension. Here results from direct MC are obtained fixing theCoV “0.01.

3.2

Chapter summary

In this chapter, both standard and adaptive implementation of Line Sampling have been presented. The advantage of using the adaptive implementation, in terms of efficiency and accuracy, has been shown by means of a numerical example. The need for an adaptive version of the algorithm is motivated by the fact that the accuracy of the estimation is quite sensitive to the chosen direction. With the added feature of direction adaptation the algorithm selects better directions as the simulation proceeds, converging to a better accuracy, which is quantified by means of the estimator variance. The developed adaptive algorithm has also the additional feature of adapting to the shape of the limit state surface, so that if the surface is approximately a plane, within a sufficiently small neighbour of a point, the algorithm requires very few samples (from 2 to 4) per line to complete the analysis. With the latter feature if the limit state surface is a hyperplane only one sample per line is required and the estimation is exact. However, it has to be noted that in order for the estimation to be exact the limit state surface has to be a hyperplane in the Standard Normal Space, which is the case only for Gaussian distributions. The adaptive feature of Line Sampling makes it particularly suited to cope with situations involving Random Sets and imprecise probabilities. Directional changes are required to switch from a distribution function to another, as required by imprecise probability, thus, the adaptive feature adds continuity to the process of assessing lower and upper probability bounds. The advantage of using Adaptive Line Sampling within the confines of imprecise probability will be shown in the next two chapters.

Chapter 4

Efficient Double Loop Approach for

Reliability Estimation: Application

to Finite Element Model

A numerical strategy for the efficient estimation of set-valued failure probabilities, cou- pling Advanced Line Sampling with optimisation methods, is presented in this chapter. The Double Loop approach is implemented in opposition to the Random Set approach, where the inner loop estimates the failure probability by means of Line Sampling, and the outer loop selects two candidates in the epistemic space corresponding to the lower and upper probability bounds, as depicted in Figure 4.1. The proposed strategy knocks down the computational barrier of computing interval failure probabilities, and reduces the cost of a robust reliability analysis by orders of magnitude. The efficiency and applicability of the developed method is demonstrated via numerical examples. The so- lution strategy is integrated into the open-source software for uncertainty quantification and risk analysis OpenCossan [1], allowing its application on large-scale engineering problems (see also [21]).

4.1

Introduction

The proposed strategy implements a double loop approach, thus, the generalised model includes parametric p-boxes, intervals and CDFs. This comes with the only limitation that a parental distribution model has to be defined for the probability distributions. Dependency among variables are accounted for by means of a copula function C :

1 2 i n G(x) g Model evaluation deterministic 0 simulated eCDF Global Optimization {i} a1{i} a2 Epistemic Domain {i} failure domain g 2 3 4 j m-1 m 0 g

upper eCDF bound

lower eCDF bound g{i}

Monte Carlo Simulation Aleatory Domain x x x x 1 ai an {i} {i} {i}

Figure 4.1: Forward propagation with the Double Loop approach