ASRAnetStructural Reliability & Risk Assessment4-8 July 2016Wuhan, ChinaLecture 17: Monte Carlo SimulationProfessor Purnendu K. DasB.E., M.E., PhD, CEng, CMarEng, FRINA, FIStructE, FIMarEST
Structural Reliability & Risk Assessment 4 – 8 July 2016 Wuhan, China Lecture 17: Monte Carlo Simulation Professor Purnendu K. Das B.E., M.E., PhD, CEng, CMarEng, FRINA, FIStructE, FIMarEST 1
CONTENTSMONTECARLOSIMULATION- GenerationofRandomNumbers-Generationofrandomnumberswithagiventypeofdistribution-Generationofrandomnumberswithastandardnormaldistribution-Generationofrandomnumberswithlog-normaldistribution-ErrorEstimationofMonteCarloSimulationOTHERSIMULATIONBASEDMETHODS-ImportanceSampling Method-SelectionofImportanceSamplingFunctionEXAMPLESCLOSINGREMARKS
CONTENTS • MONTE CARLO SIMULATION – Generation of Random Numbers – Generation of random numbers with a given type of distribution – Generation of random numbers with a standard normal distribution – Generation of random numbers with log-normal distribution – Error Estimation of Monte Carlo Simulation • OTHER SIMULATION BASED METHODS – Importance Sampling Method – Selection of Importance Sampling Function • EXAMPLES • CLOSING REMARKS 2
CONTENTSMONTECARLOSIMULATION-GenerationofRandomNumbers-Generationofrandomnumberswithagiventypeofdistribution- Generationofrandomnumberswithastandardnormal distribution-Generationofrandomnumberswithlog-normaldistributionErrorEstimationofMonteCarloSimulationOTHERSIMULATIONBASEDMETHODSImportanceSamplingMethod-Selectionof ImportanceSamplingFunctionEXAMPLESCLOSINGREMARKS3
CONTENTS • MONTE CARLO SIMULATION – Generation of Random Numbers – Generation of random numbers with a given type of distribution – Generation of random numbers with a standard normal distribution – Generation of random numbers with log-normal distribution – Error Estimation of Monte Carlo Simulation • OTHER SIMULATION BASED METHODS – Importance Sampling Method – Selection of Importance Sampling Function • EXAMPLES • CLOSING REMARKS 3
P = P(g(X)≤ 0)=J... fx(X)dXg(x)<0Where X = (X1,., Xi..., Xn) are random variablesInMonteCarlosimulationPf = P(g(X)≤0) ~ rN
g 0 Pf P g 0 . f x ( ) d X X X X Where X = (X1,., Xi., Xn) are random variables t f f N n P P g X 0 In Monte Carlo simulation 4
g(X)= R - LForexample, ifWerandomlygeneratenumbersforRandL,LetussaythenumbergeneratedareasfollowsRLg(X)2019>020.518.5>019.819.3>019.419.505
For example, if gX R L We randomly generate numbers for R and L, Let us say the number generated are as follows: R L g(X) 20 19 >0 20.5 18.5 >0 19.8 19.3 >0 19.4 19.5 0 5
So thefailure probabilityof theabove exampleis:1/5N1Z1g(x,)≤0]P = ( (..[1g(x)≤of,(X)dXNi=lThe main issues in Monte Carlo simulation are:1. How to generate random numbers for a giventype of distribution2.How to estimate error of Monte Carlo simulation3.How to determine the total number of samples6
So the failure probability of the above example is: 1/5 1. How to generate random numbers for a given type of distribution 2.How to estimate error of Monte Carlo simulation 3.How to determine the total number of samples The main issues in Monte Carlo simulation are: Nt i 1 i t f x I g 0 N 1 P . I g X 0 f (X) dX X 6
CONTENTSMONTECARLOSIMULATION-GenerationofRandomNumbers-Generationofrandomnumberswithagiventypeofdistribution- Generationofrandomnumberswithastandardnormal distribution-Generationofrandomnumberswithlog-normaldistributionErrorEstimationofMonteCarloSimulationOTHERSIMULATIONBASEDMETHODSImportanceSamplingMethod-Selectionof ImportanceSamplingFunctionEXAMPLESCLOSINGREMARKS
CONTENTS • MONTE CARLO SIMULATION – Generation of Random Numbers – Generation of random numbers with a given type of distribution – Generation of random numbers with a standard normal distribution – Generation of random numbers with log-normal distribution – Error Estimation of Monte Carlo Simulation • OTHER SIMULATION BASED METHODS – Importance Sampling Method – Selection of Importance Sampling Function • EXAMPLES • CLOSING REMARKS 7
The so-called pseudo random numbergenerators are used.A linear congruential generatori+1 =aL; +c(Mod N)Where N is called modulus, a is multiplier (integer)cis increment (integer). N is a very large number,say 10e5. L, is the initial number, which is called'seed number', and chosen by the analyst.8
A linear congruential generator L aL c j1 j The so-called pseudo random number generators are used. Where N is called modulus, a is multiplier (integer), c is increment (integer). N is a very large number, say 10e5. L0 is the initial number, which is called ‘seed number’, and chosen by the analyst. (Mod N) 8
Afew of important points:(1) These are not ‘true' random numbers. But they are almostas good as 'true' random numbers.(2) These generators can generate uniformly distributedrandom numbers over [0,1](3).Different random number generators have differentperformance.(4) Don't blindly use the random numbergenerators providedby a computer system.Choose yourself
A few of important points: (1) These are not ‘true’ random numbers. But they are almost as good as ‘true’ random numbers. (2) These generators can generate uniformly distributed random numbers over [0,1] (3). Different random number generators have different performance. (4) Don’t blindly use the random number generators provided by a computer system. Choose yourself. 9
CONTENTSMONTECARLOSIMULATION-GenerationofRandomNumbers-Generationofrandomnumberswithagiventypeofdistribution- Generationofrandomnumberswithastandardnormal distribution-Generationofrandomnumberswithlog-normaldistributionErrorEstimationofMonteCarloSimulationOTHERSIMULATIONBASEDMETHODSImportanceSamplingMethod-Selectionof ImportanceSamplingFunctionEXAMPLESCLOSINGREMARKS10
CONTENTS • MONTE CARLO SIMULATION – Generation of Random Numbers – Generation of random numbers with a given type of distribution – Generation of random numbers with a standard normal distribution – Generation of random numbers with log-normal distribution – Error Estimation of Monte Carlo Simulation • OTHER SIMULATION BASED METHODS – Importance Sampling Method – Selection of Importance Sampling Function • EXAMPLES • CLOSING REMARKS 10