FERUM 4.0 User's Guide - IFMA

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Parameters specific to a DS analysis are listed hereafter.% Simulation analysis (MC,IS,DS,SS) and distribution analysis optionsanalysisopt.num_sim = 200; % Number of directions (DS)analysisopt.rand_generator = 1; % 0: default rand matlab function% 1: Mersenne Twister (to be preferred)% Directional Simulation (DS) analysis optionsanalysisopt.dir_flag = ’det’; % ’det’: deterministic points uniformly distributed% on the unit hypersphere using eq_point_set.m% function% ’random’: random points uniformly distributed% on the unit hypersphereanalysisopt.rho = 8; % Max search radius in standard normal space for% Directional Simulation analysisanalysisopt.tolx = 1e-5; % Tolerance for searching zeros of g functionanalysisopt.keep_a = 0; % Flag for storage of a-values which gives axes% along which simulations are carried outanalysisopt.keep_r = 0; % Flag for storage of r-values for which g(r) = 03.6 Subset SimulationStarting from the premise that the failure event F={g(x,θ g )≤0} is a rare event, S.-K. Au and J.L. Beck proposedto estimate P(F) by means of more frequent intermediate conditional failure events{F i }, i= 1,..., m(called subsets) so that F 1 ⊃ F 2 ⊃ ...⊃ F m = F[AB01]. The m-sequence of intermediate conditional failureevents is selected so that F i ={g(x,θ g )≤ y i }, where y i ’s are decreasing values of the limit-state function andy m = 0. As a result, the failure probability p f = P(F) is expressed as a product of the following m conditionalprobabilities:m∏p f = P(F)= P(F m )= P(F m | F m−1 )P(F m−1 )=...= P(F 1 ) P(F i | F i−1 ) (13)i=25Unknownlimit−state5Unknownlimit−stateu 20u 20First threshold y 1−5−5 0 5u 1(a)First threshold y 1−5−5 0 5u 1(b)5Unknownlimit−state5Unknownlimit−stateu 20u 20Second threshold y 2Last threshold y =y =0 m 3−5 0 5−5−5 0 5−5u 1(c)u 1(d)Figure 8: Main steps in Subset Simulation.12
Each subset event F i (and the related threshold value y i ) is determined so that its corresponding conditionalprobability equals a sufficiently large valueα, in order to be efficiently estimated with a rather smallnumber of samples (in practiceα≈0.1-0.2, set byanalysisopt.pf_target parameter). In essence, thereis a trade-off between minimizing the number m of subsets by choosing rather small intermediate conditionalprobabilities and maximizing the same probabilities so that they can be estimated efficiently by simulations.The first threshold y 1 is obtained by a crude MCS, so that P(F 1 )≈α (see Figure 8, subplot (a)). For furtherthresholds, new sampling points corresponding to{ F i | F i−1 } conditional events are obtained from MarkovChains Monte Carlo (MCMC), based on a modified Metropolis-Hastings algorithm, see green star points in figure8, subplot (b) corresponding to i= 2 case. The process is repeated until a negative threshold y i is found.This is therefore the last step (i=m) and y m is set to zero. The corresponding probability P(F m | F m−1 ) is thenevaluated. See Figure 8, subplot (d). The last step is reached for m=i= 3 in the present case. The coefficientof variation of the failure probability estimated from SS can be evaluated by intermediate coefficients ofvariation, weighted by the correlation that exists between the samples used for the estimation of intermediateconditional probabilities, please refer to reference[AB01] for more details.For a Subset Simulation analysis, select option 23 in FERUM 4.0. Parameters specific to this specificmethod are listed hereafter.% Simulation analysis (MC,IS,DS,SS) and distribution analysis optionsanalysisopt.num_sim = 10000; % Number of samples per subset step (SS)analysisopt.rand_generator = 1; % 0: default rand matlab function% 1: Mersenne Twister (to be preferred)% Subset Simulation (SS) analysis optionsanalysisopt.width = 2; % Width of the proposal uniform pdfsanalysisopt.pf_target = 0.1; % Target probability for each subset stepanalysisopt.flag_cov_pf_bounds = 1; % 1: calculate upper and lower bounds of the% coefficient of variation of pfanalysisopt.ss_restart_from_step = -inf;% 0: no calculation% i>=0 : restart from step i% -inf : all steps, no record (default)% -1 : all steps, record allanalysisopt.flag_plot = 0; % 1: plots at each step (2 r.v. examples only)% 0: no plotsanalysisopt.flag_plot_gen = 0; % 1: intermediate plots for each MCMC chain% (2 r.v. examples only)% 0: no plots3.7 Global Sensitivity AnalysisGlobal sensitivity analysis aims at quantifying the impact of the variability in each (or group of) input variateson the variability of the output of a model in apportioning the output model variance to the variance in theinput variates. Sobol’ indices[Sob07] are the most usual global sensitivity measures. They can be evaluatedin FERUM 4.0.We consider here a model given by:Y= g(X)= g X 1 , X 2 ,..., X nwhere X=(X 1 , X 2 ,..., X n ) is a vector of n independent random input variates, g is a deterministic model andY is a scalar random output.In order to determine the importance of each input variate, we consider how the variance of the output Ydecreases when variate X i is fixed to a given x i * value:where V(•) denotes the variance function.(14)V Y| X i = x i * (15)Since x i * value is unknown, we take the expectation of Equation (15) and, by virtue of the law of totalvariance, we can write:V E Y| X i = V(Y)− E V Y| Xi(16)13
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FERUM 4.0 User's Guide - IFMA

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