Uncertainties Analysis

Transcription

1 Uncertainties Analysis Surrogate Models Jean-Marc Martinez Centre d Etudes de Saclay Département Modélisation de Systèmes et Structures Gif sur Yvette Cédex - France Nuclear Science Commitee NEA- OCDE Paris,9 June 2005 p.1/7

2 Uncertainty Analysis We propose uncertainty analysis based on the use of Surrogate Models instead Numerical /Computational Models to run efficient but time consuming methods as like Nuclear Science Commitee NEA- OCDE Paris,9 June 2005 p.2/7

3 Uncertainty Analysis We propose uncertainty analysis based on the use of Surrogate Models instead Numerical /Computational Models to run efficient but time consuming methods as like Monte Carlo simulations PDF estimation of outputs Nuclear Science Commitee NEA- OCDE Paris,9 June 2005 p.2/7

4 Uncertainty Analysis We propose uncertainty analysis based on the use of Surrogate Models instead Numerical /Computational Models to run efficient but time consuming methods as like Monte Carlo simulations PDF estimation of outputs Fast or Sobol techniques Sensitivity Analysis by ANOVA Nuclear Science Commitee NEA- OCDE Paris,9 June 2005 p.2/7

5 Uncertainty Analysis We propose uncertainty analysis based on the use of Surrogate Models instead Numerical /Computational Models to run efficient but time consuming methods as like Monte Carlo simulations PDF estimation of outputs Fast or Sobol techniques Sensitivity Analysis by ANOVA link with Form/Sorm methods analysis for tail distribution and based on surrogate models (linear or quadratic approximation) Nuclear Science Commitee NEA- OCDE Paris,9 June 2005 p.2/7

6 Basis functions of SM 1) We propose to use Neural Networks Nuclear Science Commitee NEA- OCDE Paris,9 June 2005 p.3/7

7 Basis functions of SM 1) We propose to use Neural Networks Approximation and generalization capability thrifty models (Baron s theorem) Nuclear Science Commitee NEA- OCDE Paris,9 June 2005 p.3/7

8 Basis functions of SM 1) We propose to use Neural Networks Approximation and generalization capability thrifty models (Baron s theorem) Efficiency learning process Levenberg-Marquard optimization techniques Nuclear Science Commitee NEA- OCDE Paris,9 June 2005 p.3/7

9 Basis functions of SM 1) We propose to use Neural Networks Approximation and generalization capability thrifty models (Baron s theorem) Efficiency learning process Levenberg-Marquard optimization techniques Robust methods to estimate statistics of errors Cross-validation, Bootstrap techniques Nuclear Science Commitee NEA- OCDE Paris,9 June 2005 p.3/7

10 Basis functions of SM 1) We propose to use Neural Networks Approximation and generalization capability thrifty models (Baron s theorem) Efficiency learning process Levenberg-Marquard optimization techniques Robust methods to estimate statistics of errors Cross-validation, Bootstrap techniques 2) and the basis functions issued from Polynomial Chaos (Ghanem, Antoniadis, Le Maître) Nuclear Science Commitee NEA- OCDE Paris,9 June 2005 p.3/7

11 Learning of SM Learning Set of Surrogate Models can be built from Data Sampling methods Simple Random Sampling Latin Hypercube Sampling Nuclear Science Commitee NEA- OCDE Paris,9 June 2005 p.4/7

12 Learning of SM Learning Set of Surrogate Models can be built from Data Sampling methods Simple Random Sampling Latin Hypercube Sampling Actual Research in our Laboratory (PhD) if the Surrogate Model is not reliable enough, DoCE (Design of Computer Experiment) using Bagging techniques (Bootstrap Aggregating, Breiman 1996) Nuclear Science Commitee NEA- OCDE Paris,9 June 2005 p.4/7

13 Example Sensitivity Analysis (Sobol) on Homma and Saltelli model f(x) = sin(x 1 ) + asin 2 (x 2 ) + bx 4 3sin(x 1 ) with a = 7, b = 0.1 1/(2π) if π x π pdf(x i ) = 0 if not Nuclear Science Commitee NEA- OCDE Paris,9 June 2005 p.5/7

14 Example Sensitivity Analysis (Sobol) on Homma and Saltelli model f(x) = sin(x 1 ) + asin 2 (x 2 ) + bx 4 3sin(x 1 ) with a = 7, b = 0.1 1/(2π) if π x π pdf(x i ) = 0 if not Neural Network Surrogate Model Multi Layer Perceptron trained from 500 simulations LHS Nuclear Science Commitee NEA- OCDE Paris,9 June 2005 p.5/7

15 Results Estimation of Sobol global coefficients Tarantola method (1997) Variables Exact Value Neural Network Real Function f(x) 1000 LHS of 500 simulations x mean = 0.55, σ = 0.15 x mean = 0.45, σ = 0.11 x mean = 0.24, σ = 0.09 Nuclear Science Commitee NEA- OCDE Paris,9 June 2005 p.6/7

16 Results Estimation of Sobol global coefficients Tarantola method (1997) Variables Exact Value Neural Network Real Function f(x) 1000 LHS of 500 simulations x mean = 0.55, σ = 0.15 x mean = 0.45, σ = 0.11 x mean = 0.24, σ = 0.09 exemplec htemp Entries 5000 Mean RMS NeuralNet htemp Entries 5000 Mean RMS exemplec NeuralNet Nuclear Science Commitee NEA- OCDE Paris,9 June 2005 p.6/7

17 BEMUSE, OCDE Project Best Estimate Method - Uncertainty and Sensitivity Evaluation (A. de Crecy, CEA Grenoble) PWR, Transients issued from LOCA Nuclear Science Commitee NEA- OCDE Paris,9 June 2005 p.7/7

18 BEMUSE, OCDE Project Best Estimate Method - Uncertainty and Sensitivity Evaluation (A. de Crecy, CEA Grenoble) PWR, Transients issued from LOCA Selection and quantification of uncertainties Nuclear Science Commitee NEA- OCDE Paris,9 June 2005 p.7/7

19 BEMUSE, OCDE Project Best Estimate Method - Uncertainty and Sensitivity Evaluation (A. de Crecy, CEA Grenoble) PWR, Transients issued from LOCA Selection and quantification of uncertainties Data sampling from LHS CATHARE supervised by KALIF (CCRT) Nuclear Science Commitee NEA- OCDE Paris,9 June 2005 p.7/7

20 BEMUSE, OCDE Project Best Estimate Method - Uncertainty and Sensitivity Evaluation (A. de Crecy, CEA Grenoble) PWR, Transients issued from LOCA Selection and quantification of uncertainties Data sampling from LHS CATHARE supervised by KALIF (CCRT) Neural Network Surrogate Model NeMo Neural network Modelisation software Nuclear Science Commitee NEA- OCDE Paris,9 June 2005 p.7/7

21 BEMUSE, OCDE Project Best Estimate Method - Uncertainty and Sensitivity Evaluation (A. de Crecy, CEA Grenoble) PWR, Transients issued from LOCA Selection and quantification of uncertainties Data sampling from LHS CATHARE supervised by KALIF (CCRT) Neural Network Surrogate Model NeMo Neural network Modelisation software Estimation of Sensibility Coefficients (Sobol) after the realization of all LHS Experiments Nuclear Science Commitee NEA- OCDE Paris,9 June 2005 p.7/7