International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Efficient Monte Carlo algorithm for rare failure event simulation Patelli, Edoardo; Au, Siu Kui 2015-07

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Efficient Monte Carlo Algorithm For Rare Failure Event SimulationEdoardo PatelliLecturer, Institute for Risk and Uncertainty, University of Liverpool, United KingdomSiu Kui AuProfessor, Institute for Risk and Uncertainty, University of Liverpool, United KingdomABSTRACT: Studying failure scenarios allows one to gain insights into their cause and consequence,providing information for effective mitigation, contingency planning and improving system resilience.A new efficient algorithm is here proposed to solve applications where an expensive-to-evaluate computermodel is involved. The algorithms allows to generate the conditional samples for the Subset simulationby representing each random variable by an arbitrary number of hidden variables. The resulting algorithmis very simple yet powerful and it does not required the use of the Markov Chain Monte Carlo method.The proposed algorithm has been implemented in a open source general purpose software, OpenCossanallowing the solution of large scale problems of industrial interest by taking advantages of High Per-formance Computing facilities. The applicability and flexibility of the proposed approach is shown bysolving a number of different problems.1. INTRODUCTIONRare failure events of safety critical systems (suchas in Nuclear and Aviation industry) can have hugeimpacts as shown by recent accidents (e.g. TohokuEarthquake and the consequent Fukushima-Daiichiaccident).Assessing risk quantitatively requires the quan-tification of the probability of occurrence of a spe-cific event by properly propagating the uncertaintythrough the model that predicts the quantities of in-terest. In principle, rare failure events can be in-vestigated through Monte Carlo simulation (see e.g.Liu (2001)). However, this is computationally pro-hibitive for complex systems because it requires alarge number of samples to obtain one failure sam-ple. Checking whether each sample fails requiresevaluating of the model and the calculation of out-put quantities, which is generally computationallyexpensive for complex systems.Advanced Monte Carlo methods aim at estimat-ing rare failure probabilities more efficiently thandirect Monte Carlo (see e.g. Schuëller (2009)). Un-fortunately, high dimension and model complexitymake it extremely difficult to improve the efficiencyof Monte Carlo algorithms purely based on priorknowledge, leaving algorithms that adapt the gener-ation of samples during simulation the only choice.The estimation of small probabilities of failure fromcomputer simulations is a classical problem in engi-neering, and the Subset Simulation algorithm pro-posed by Au and Beck (2001) has become one ofthe most popular method to solve it thanks to itssignificant savings in the number of simulations toachieve a given accuracy of estimation, with respectto many other Monte Carlo approaches.In this paper, a new efficient algorithm for Sub-set simulation is proposed to generate conditionalsamples without using Markov Chain Monte Carlo.This allows to simplify the Subset simulation andto remove one of the main sequential componentsof Subset simulation. The proposed algorithm hasbeen implemented in a open source general pur-pose software, OpenCossan (Patelli et al., 2014).The computational framework allows the applica-tion of the novel algorithm to solve large scale ex-amples of practical interest and taking advantagesof High Performance Computing facilities. A num-ber of academic examples and real applications ofindustrial interest are solved and presented to showthe applicability of the proposed approach.112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20152. SUBSET SIMULATIONIn this section a short description of the originalSubset algorithm and the proposed algorithm is pre-sented.2.1. Original algorithm: Subset-MCMCSubset simulation (Au and Beck, 2001) is an ad-vanced Monte Carlo method aimed at estimatingrare failure probabilities more efficiently than di-rect Monte Carlo. The method has been alreadyapplied efficiently to a wide range of applications(e.g. Alvarez et al. (2014); Chiachio et al. (2014);Ching and Hsieh (2007)) since it is not based onany geometrical assumption about the topology ofthe failure domain.The key idea of Subset simulation is to decom-pose the failure event F into a sequence of nestedfailure events: F = Fm ∈ Fm−1 ∈ ·· · ∈ F1 : F =∩iFi. The probability of failure is expressed as theproduct of P(F1) and the conditional probabilitiesP(Fk+1/Fk),k = {1, . . . ,m−1}:P(F) = P(∩iFi) = P(F1)m−1∏i=1P(Fk+1/Fk). (1)During subset simulation the threshold values,δ1, . . . ,δm, are adaptively generated so that theconditional failure probabilities Fk are sufficientlylarge. Hence, by choosing m and Fk appropriately,the conditional probabilities can be evaluated effi-ciently by direct simulation.The challenging part of the Subset algorithm isthe realization of the conditional samples X that aredistributed according the conditional probabilitiesP(x|Fi). In the original implementation of Subset,the generation of conditional samples are obtainedadopting a independent-component Markov ChainMonte Carlo (MCMC) algorithm. Using a mod-ified Metropolis algorithm Metropolis and Ulam(1949), the chains are generated in two steps. Firsta next state is sampled from a proposal distribu-tion x′ ← pi(x). Then, the candidate solution isaccepted if its associated performance function isgreater than the intermediate threshold level δk oth-erwise the candidate sample is rejected.In general, controlling the MCMC algorithm isnot a trivial task. Furthermore, the generation of theconditional samples based on Markov Chains repre-sents the bottleneck for the parallelization and scal-ability of the algorithms on cluster and grid com-puting.2.2. Proposed algorithm: Subset-∞The main idea of the proposed algorithm (hereafterindicated as Subset-∞) is to define an equivalentproblem where each random variable Xi is repre-sented by an arbitrary number of hidden variables.In fact, Gaussian variables can be represented by aninfinite number of Gaussian variables since a lin-ear combination of Gaussian variables is still Gaus-sian. In addition, any problem can be representedin the so-called Standard Normal Space by meansof a transformation (see e.g. (Nataf, 1962)) whereeach input variable is represented by an indepen-dent Gaussian distribution with 0 mean and unitarystandard deviation.Studying the limiting behaviour of theindependent-component MCMC, Au (2015)has demonstrated that the conditional distributionof the candidate samples is a Gaussian distri-bution with mean and variance that depends onthe proposal distribution. Thanks to this result,conditional samples can be directly sampled froman appropriated Gaussian distribution withoutresorting to the MCMC algorithm and the selectionof the proposal distribution.2.2.1. Numerical implementationThe main advantage of the Subset-∞ algorithmrelies on its simplicity and scalability. In fact,the Subset-∞ does not require the construction ofMarkov Chains but only the capability to generaterealizations in the standard normal space from nor-mal distributions.The following pseudo-algorithm generates sam-ples distributed as the target conditional distributionof failure F(x|Fk) in the Standard Normal space:1. assume a sample X(i) is distributed as F(x|Fk)2. Calculate a =√1− s2 where s = [s1, . . . ,sn]represents the vector of chosen variances foreach component of the candidate Xn3. Generate the candidate X′ ∼ N(a ·X(i),s)4. The new sample X(i+1) = X′ if X′ ∈ Fk other-wise X(i+1) = X(i) if X′ /∈ Fk212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20155. Repeat for all samples6. Repeat for all Subset levelThe proposed algorithm is generally applicable forany finite dimensional problem that can be repre-sented in standard normal space.The proposed algorithm has been implementedin OPENCOSSAN software (Patelli et al., 2014).OPENCOSSAN is a collection of open source algo-rithms, methods and tools released under the LGPLlicense (Free Software Foundation, 2007), and un-der continuous development at the Institute for Riskand Uncertainty at the University of Liverpool, UK.The source code is freely available upon requestat the web address to the modularity and structure of the soft-ware organized in classes, i.e. data structures con-sisting of data fields and methods together withtheir interactions and interfaces, the implementa-tion of the Subset-∞ algorithm has required the im-plementation of only few lines of code.2.2.2. Parallelization and scalabilityGenerally, reliability analysis requires to evaluateof the model a large number of time. Although Sub-set simulation allows a significant reduction of thenumber of model calls, the wall clock time requiredby the analysis can only be further reduced resort-ing to the parallel execution of the code. Multipleindependent instances of the solver are executed si-multaneously for different realizations of the input,allowing for a reduction of the analyses time with-out any loss of accuracy.MCMC is an inherently serial algorithm andhence it requires complex algorithms to parallelizethe samples generation such as e.g. the specu-lative computing approaches (Pellissetti, 2009) orpartitioning and weight estimation scheme (Van-Derwerken and Schmidler, 2013). In contrast, theproposed ∞-algorithm is easily paralellelizable andtwo types of parallelization can be used. The firsttype of parallelization is used to speed-up the anal-ysis in case the solver (i.e. the model to be eval-uated) is a coded in Matlab. In this case, a spe-cial job on a pool of MATLAB workers is cre-ated on each multi-core machine, connecting theMATLAB client to the parallel pool (e.g. usingthe command parpool). Features from the MAT-LAB parallel toolbox e.g., parfor, can be used todistribute the tasks on the MATLAB clients. Thistype of parallelism can be implemented on eachsingle computational node. In case the analysisrequires the call of an external solver (such as aFE/CFD analysis) the multi-thread, shared mem-ory parallelism capabilities of the external softwareneed to be adopted. The second level of paralleliza-tion exploits cluster and grid computing, i.e. theavailability of machines connected in an heteroge-neous network. In this case, the total number ofsimulations is slitted in a multiple number of inde-pendent batch jobs. The jobs are then submittedto the job scheduler/manager and distributed effi-ciently on the available machines of the grid/cluster.Using OPENCOSSAN framework, the paral-lelization of the analysis is straightforward. Inde-pendent multiple stream and sub-stream are gen-erated by combined multiple recursive generator(mrg32k3a). Then, OPENCOSSAN creates inde-pendent jobs by compiling portion of OPENCOS-SAN using mcc and then distribute the compiledcode to the node of the cluster (workers). Hence,it is possible to execute jobs in parallel withoutthe necessity to install MATLAB on each compu-tational node of the cluster, but only accessing theMATLAB runtime libraries.3. NUMERICAL EXAMPLESIn this Section different numerical examples arepresented to show the applicability of the Subset-∞ algorithm.3.1. Simple synthetic modelThe purpose of this first numerical example is toshow the applicability of the new proposed algo-rithm using a very simple problem. Although, it isa very simple example and consider only one input,it represents an extreme case for the proposed ap-proach since it has been designed to be efficient forhigh dimensional problems. Nevertheless, it allowsto show the applicability of the methods and clearlyvisualize and compare the conditional samples ob-tained using the Subset-∞ algorithm with samplesobtained by means of the standard Subset-MCMCapproach.312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015In this example the reliability problem is definedas the probability of the random variable X1 to ex-ceed a constant threshold. The limit state functionis defined as:g1(X) = X1−3 (2)where X1 is a normally distributed variable with0 mean and 1 standard deviation. The estimationof the probability of failure has been computed bymeans of the Subset simulation adopting the stan-dard MCMC implementation and the proposed ∞-algorithm, respectively. The results are summa-rized in Table 2. The Subset-MCMC simulationshave been performed using an target conditionalfailure probability of 0.1, a uniform proposal distri-bution with width 0.4 while Subset-∞ simulationshave been performed using a variance s = 0.5. Themain Subset parameters are summarized in Table 1.Table 1: Parameters of the Subset algorithm used forsolving the numerical example simple synthetic model.Parameter ValueInitial samples size 100Target cond. failure probability 0.1Proposal dist. (Subset-MCMC) U(-0.2,0.2)Variance s (Subset-∞) 0.5Table 2: Estimated failure probability of the simplesynthetic problem.Analytical MCMC Subset-∞pˆ f 1.3 ·10−3 1.32 ·10−3 1.61˙0−3Std - 9.53 ·10−4 1.08 ·10−3Samples - 440 160Figures 1-2 show the values of the computedperformance function g1 as a function of the sub-set level using MCMC and ∞ algorithm, respec-tively. The horizontal lines represent the identifiedthreshold values, δ1, . . . ,δm; the stars represent re-jected samples while circles show the accepted con-ditional samples.Figure 3 shows the effect of the proposed vari-ance s for the Subset-∞ algorithm on the estimationof the failure probability. For each value of s 20Subset levels0 1 2 3 4Vg 1-10123456 Subset-MCMCFigure 1: Performance function estimated by means ofthe Subset-MCMC algorithm.Subset levels0 1 2 3 4Vg 1-10123456 Subset- 1Figure 2: Performance function estimated by means ofthe Subset-∞ algorithm.s0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1p f10 -910 -810 -710 -610 -510 -410 -310 -210 -1 Subset- 1Figure 3: Effect of the proposed variance s of the esti-mation of Pˆf using the Subset-∞ algorithm.412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015independent runs of Subset simulation have beenperformed. The red dashed line represents the an-alytical solution, the yellow dots the Pˆf estimationperformed by the Subset-∞.3.2. Simple synthetic model in high-dimensionIn order to test the performance of the proposed ap-proach, the dimensionality of the simple problempresented in Section 3.1 has been incremented upto 2500 variables. The same limit state function ofthe previous model has been used (i.e. Eq. (2)).Hence, only the input X1 controls the failure proba-bility.Subset simulations have been repeated 10 timesand the minimum, median and maximum of thefailure estimation is summarized in Table 3 usingMCMC algorithm with a uniform proposed distri-bution of width 0.4 (in standard normal space) andin Table 4 using the canonical algorithm with aproposed variance of s = 0.5, respectively. Fig-ure 4 shows the estimation of the failure probabil-ity for different number of input variables using theSubset-MCMC and Subset-∞, respectively. As ex-pected, both algorithms show similar results. Theyseem to be insensitive to the dimensionality of theproblem.Table 3: Estimation of the failure probability computedusing Subset-MCMC for different number of input vari-ables. Subset simulation has been repeated 10 times.Dimension Min Median Max10 2.37 ·10−5 8.13 ·10−4 3.1 ·10−3100 2.88 ·10−4 2.17 ·10−3 6.3 ·10−31000 8.48 ·10−5 1.20 ·10−3 4.5 ·10−3Table 4: Estimation of the failure probability computedusing Subset-∞ algorithm for different number of in-put variables. Subset simulation has been repeated 10times.Dimension Min Median Max10 2.80 ·10−4 1.35 ·10−3 4.5 ·10−3100 1.50 ·10−4 1.15 ·10−3 5.6 ·10−31000 2.4 ·10−4 5.45 ·10−4 4.7 ·10−3Number of variables10 0 10 1 10 2 10 3 10 4p f10 -510 -410 -310 -2 Simple synthetic modelSubset-MCMCSubset- ∞Figure 4: Estimation of the failure probability, Pˆf ,for the simple synthetic model estimated by means ofSubset-MCMC and Subset-∞ simulation for differentnumber of input variables, respectively.Figure 5 shows the computational time (wallclock time) required by the Subset simulations asa function of the number of variables. The compu-tational time has been calculated using OpenCossanrevision 769 on a quad-core Intel Core i5-4590TCPU @2.00GHz. As shown in Figure 5, the Subset-∞ algorithm is one order of magnitude faster thanthe standard Subset-MCMC procedure.Number of variables10 1 10 2 10 3Time [s]10 -110 010 110 210 310 410 5 Computational time (wall clock time) SubSet-MCMCSubSet- ∞Figure 5: Computational time required by the Subset-MCMC and Subset-∞ as a function of the number ofinput variables, respectively.3.3. Simple frameIn this numerical example applies the reliabil-ity analysis of a simple structural frame systemsketched in Figure 6 and originally presented in512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Schuëller et al. (1989) is presented. The structuralparameters are summarized in Table 5.5X155X5X3X2 X4X6 X7Figure 6: Simple structural frame.The structural failure criterion is defined by thedevelopment of any of the 3 possible collapsemechanisms as illustrated in Figure 7. Hence, thelimit state function for the structure is the combina-tion of the 3 limit state functions that describe eachcollapse mechanism. This can be represented as asystems with components in parallel where the fail-ure of a single component produce the failure of theentire system. Hence the performance function ofthe system is:g2(X) = min(g2a,g2b,g2c) (3)whereg2a(X) = X1 +2X3 +2X4 +X5−5X6−5X7 (4)g2b(X) = X1 +2X2 +X4 +X5−5X6 (5)g2c(X) = X2 +2X3 +X4−5X6 (6)Figure 7: Possible failure modes of the simple struc-tural frame.Subset simulation has been used to estimate thefailure probability using the parameters shown inTable 1. The results of the reliability analysis aresummarized in Table 6. The results show that theproposed Subset-∞ approach is able to handle multinon linear limit state functions (in Standard NormalSpace).Table 5: Input variables of the simple structural frame.All the inputs are considered uncorrelated.Variable Distribution Mean StdXi={1:5} Lognormal 60 6.0X6 Type I - largest 20 6.0X7 Type I - largest 20 7.5Table 6: Estimated failure probability of the simplestructural frame.MC SS-MCMC Subset-∞pˆ f 1.27 ·10−4 3.18 ·10−5 1.26 ·10−4CoV 8.87 ·10−2 6.43 ·10−1 4.87 ·10−1Samples 1 ·106 2.07 ·103 1.8 ·1033.4. A multi-storey building modelThe applicability of the Subset simulation for solv-ing problems of industrial interest is showing byperforming the reliability analysis of a multi-storeybuilding modelled with ABAQUS. The FE-modelof the structure as shown in Fig. 8, involves approx-imately 8,200 elements and 66,300 DOFs, wheresolid elements (C3D8I) are used for the foundation,the mesh of the floors consists solely of quadrilat-eral elements (S4) and each of the 16 columns ofall floors are modelled with 2-node beam elements(B31).Figure 8: FE-model of a multi-storey building withindication of the loading and the observed column ofinterest. The FE model has been adapted from Ref. ?.The load case under investigation is the combi-612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015nation of self weight and simplified lateral windload modelled by deterministic concentrated staticforces acting on the nodes of one edge of each floorand on the upper part of the staircase where themagnitudes increase with the height of the build-ing. Failure is defined as the exceedance of theyield stress in a bar element of the column of the5th floor indicated by the arrow in Fig. 8. Hence,the performance function is defined byg(θ) = σmax−σ(θ), (7)where σmax defines the maximum stress level toler-ated by the column (i.e. the resistance) and σ(θ) isthe element stress as extracted from the output fileof the FE-analysis (i.e. the demand). θ representsthe vector of 244 uncertainty structural parameterssummarized in Table 7.Table 7: Random variables used for modelling the un-certainties within the multi-storey building.Parameter ValueColumn Resistance N(7.5e7,1.0e7) PaColumns section U[0.36,0.44] mE-modulus LN(3.5e10,3.5e9) PaDensity LN(2500,250) kg/m3Poisson ratio LN(0.25,0.025)The reliability analysis has been performed us-ing Subset simulation. The estimation by meansof Monte Carlo simulation is infeasible due to thelarge sample size needed to trustworthy identify thefailure region and the computational cost associ-ated to the analysis of this large FE-model. Subsetsimulations have been run with 100 initial samples,intermediated failure probability of 0.1 and a uni-form proposal distribution with a width of 0.4 forstandard Subset-MCMC and a proposal variance ofs = 0.5 for Subset-∞. The results are summarizedin Table 8.Figures 9 and 10 show the computed column re-sistance versus the maximum of the element stressin the column as extracted from the output file of theFE-analysis for different levels during Subset sim-ulation. The Subset-MCMC and Subset-∞ providesimilar results. They are both applicable to solveTable 8: Reliability analysis of a FE multi-story build-ing by means of Subset simulation.Subset-MCMC Subset-∞pˆ f 8.58 ·10−6 2.4 ·10−7CoV 1.24 1.06Samples 550 600large scale problems of practical interest. As al-ready pointed out, the Subset-∞ algorithm is com-putational faster and more parallelizable than theclassic Subset-MCMC and hence it allows to re-duce the wall-clock time required by the analysis.Column resistance σmax ×10 72 3 4 5 6 7 8 9 10σ(θ)×10 722.533.544.55 Subset-MCMC Level 1 (MCS)Level 2 (rejected)Level 2 (accepted)Level 3 (rejected)Level 3 (accepted)Level 4 (rejected)Level 4 (accepted)Level 5 (rejected)Level 5 (accepted)Level 6 (rejected)Level 6 (accepted)Figure 9: Samples generated by the Subset-MCMCsimulation. The lines represent the identified thresholdlevels.Column resistance σmax ×10 72 3 4 5 6 7 8 9 10σ(θ)×10 722.533.544.55 Subset- ∞ Level 1 (MCS)Level 2 (rejected)Level 2 (accepted)Level 3 (rejected)Level 3 (accepted)Level 4 (rejected)Level 4 (accepted)Level 5 (rejected)Level 5 (accepted)Level 6 (rejected)Level 6 (accepted)Figure 10: Samples generated by the Subset-∞ simula-tion. The lines represent the identified threshold levels.712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20154. CONCLUSIONSIn this paper, a new efficient algorithm for rare fail-ure event simulation has been presented. In theproposed algorithm, denoted as Subset-∞, each ran-dom variable is represented by an large number ofhidden (Gaussian) variables. As a consequence, theconditional samples can be obtained directly forman appropriate Gaussian distribution without resort-ing to the Markov Chain Monte Carlo method.The proposed algorithm has been implementedin OPENCOSSAN. A number of different numeri-cal examples have been presented to show the flex-ibility and applicability of the approach. In fact,Subset-∞ shows the same accuracy and efficiencyof the classic Subset-MCMC. However, Subset-∞ ismuch simple and faster. In addition, the approach isfully scalable on parallel machines since it does nothave the drawback of the sequential Markov ChainMonte Carlo.5. REFERENCESAlvarez, D. A., Hurtado, J. E., and Uribe, F. (2014). “Es-timation of the lower and upper probabilities of failureusing random sets and subset simulation.” Vulnerabil-ity, Uncertainty, and Risk, American Society of CivilEngineers, 905–914.Au, I. (2015). “Subset simulation in finite-infinite di-mensional space. part i: Theory.Au, S.-K. and Beck, J. (2001). “Estimation of small fail-ure probabilities in high dimensions by subset simu-lation.” Probabilistic Engineering Mechanics, 16(4),263–277.Chiachio, M., Beck, J. L., Chiachio, J., and Rus, G.(2014). “Approximate bayesian computation by sub-set simulation.” arXiv preprint arXiv:1404.6225.Ching, J. and Hsieh, Y. (2007). “Approximate reliability-based optimization using a three-step approach basedon subset simulation.” Journal of Engineering Me-chanics, 133(4), 481–493.Free Software Foundation (2007). “Free software foun-dation, gnu lesser general public license, version 3,<>.Liu, J. (2001). Monte Carlo Strategies in Scientific Com-puting. Springer Series in Statistics. Springer.Metropolis, N. and Ulam, S. (1949). “The monte carlomethod.” Journal of the American Statistical Associ-ation, 44, 335–341.Nataf, A. (1962). “Détermination des distributiondont les marges sont données.” Comptes rendus del’academie des sciences, 225, 42–43.Patelli, E., Broggi, M., Angelis, M., and Beer, M.(2014). “Opencossan: An efficient open tool for deal-ing with epistemic and aleatory uncertainties.” Vul-nerability, Uncertainty, and Risk, American Societyof Civil Engineers, 2564–2573.Patelli, E., H.M.Panayirci, Broggi, M., Goller, B., Beau-repaire, P., Pradlwarter, H. J., and Schuëller, G. I.(2012). “General purpose software for efficient un-certainty management of large finite element models.”Finite Elements in Analysis and Design, 51, 31–48.Pellissetti, M. (2009). “Parallel processing in struc-tural reliability.” Journal of Structural Engineeringand Mechanics, 32(1), 95–126.Schuëller, G. (2009). “Efficient Monte Carlo simula-tion procedures in structural uncertainty and reliabil-ity analysis - recent advances.” Journal of StructuralEngineering and Mechanics, 32(1), 1–20.Schuëller, G. I., Bucher, C., Bourgund, U., and Ouy-pornprasert, W. (1989). “On efficient computa-tional schemes to calculate structural failure probabil-ities.” Journal of Probabilistic Engineering Mechan-ics, 4(1), 10–18.VanDerwerken, D. N. and Schmidler, S. C. (2013). “Par-allel markov chain monte carlo.” ArXiv e-prints.8


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