图书馆订阅: Guest
国际不确定性的量化期刊

每年出版 6 

ISSN 打印: 2152-5080

ISSN 在线: 2152-5099

The Impact Factor measures the average number of citations received in a particular year by papers published in the journal during the two preceding years. 2017 Journal Citation Reports (Clarivate Analytics, 2018) IF: 1.7 To calculate the five year Impact Factor, citations are counted in 2017 to the previous five years and divided by the source items published in the previous five years. 2017 Journal Citation Reports (Clarivate Analytics, 2018) 5-Year IF: 1.9 The Immediacy Index is the average number of times an article is cited in the year it is published. The journal Immediacy Index indicates how quickly articles in a journal are cited. Immediacy Index: 0.5 The Eigenfactor score, developed by Jevin West and Carl Bergstrom at the University of Washington, is a rating of the total importance of a scientific journal. Journals are rated according to the number of incoming citations, with citations from highly ranked journals weighted to make a larger contribution to the eigenfactor than those from poorly ranked journals. Eigenfactor: 0.0007 The Journal Citation Indicator (JCI) is a single measurement of the field-normalized citation impact of journals in the Web of Science Core Collection across disciplines. The key words here are that the metric is normalized and cross-disciplinary. JCI: 0.5 SJR: 0.584 SNIP: 0.676 CiteScore™:: 3 H-Index: 25

Indexed in

GLOBAL SENSITIVITY ANALYSIS USING DERIVATIVE-BASED SPARSE POINCARÉ CHAOS EXPANSIONS

卷 13, 册 6, 2023, pp. 57-82
DOI: 10.1615/Int.J.UncertaintyQuantification.2023043593
Get accessGet access

摘要

Variance-based global sensitivity analysis, in particular Sobol' analysis, is widely used for determining the importance of input variables to a computational model. Sobol' indices can be computed cheaply based on spectral methods like polynomial chaos expansions (PCE). Another choice are the recently developed Poincare chaos expansions (PoinCE), whose orthonormal tensor-product basis is generated from the eigenfunctions of one-dimensional Poincaré differential operators. In this paper, we show that the Poincaré basis is the unique orthonormal basis with the property that partial derivatives of the basis again form an orthogonal basis with respect to the same measure as the original basis. This special property makes PoinCE ideally suited for incorporating derivative information into the surrogate modeling process. Assuming that partial derivative evaluations of the computational model are available, we compute spectral expansions in terms of Poincaré basis functions or basis partial derivatives, respectively, by sparse regression. We show on two numerical examples that the derivative-based expansions provide accurate estimates for Sobol' indices, even outperforming PCE in terms of bias and variance. In addition, we derive an analytical expression based on the PoinCE coefficients for a second popular sensitivity index, the derivative-based sensitivity measure (DGSM), and explore its performance as upper bound to the corresponding total Sobol' indices.

参考文献
  1. Smith, R., Uncertainty Quantification, Philadelphia: SIAM, 2014.

  2. Borgonovo, E. and Plischke, E., Sensitivity Analysis: A Review of Recent Advances, Eur. J. Oper. Res., 248:869-887, 2016.

  3. Razavi, S., Jakeman, A., Saltelli, A., Prieur, C., Iooss, B., Borgonovo, E., Plischke, E., Lo Piano, S., Iwanaga, T., Becker, W., Tarantola, S., Guillaume, J., Jakeman, J., Gupta, H., Melillo, N., Rabiti, G., Chabridon, V., Duan, Q., Sun, X., Smith, S., Sheikholeslami, R., Hosseini, N., Asadzadeh, M., Puy, A., Kucherenko, S., and Maier, H., The Future of Sensitivity Analysis: An Essential Discipline for Systems Modelling and Policy Making, Env. Modell. Software, 137:104954, 2021.

  4. Sobol', I., Sensitivity Estimates for Nonlinear Mathematical Models, Math. Modell. Comput. Exp., 1:407-414, 1993.

  5. Efron, B. and Stein, C., The Jackknife Estimate of Variance, Ann. Stat., 9:586-596, 1981.

  6. Homma, T. and Saltelli, A., Importance Measures in Global Sensitivity Analysis of Nonlinear Models, Reliab. Eng. Syst. Saf., 52:1-17, 1996.

  7. Prieur, C. and Tarantola, S., Variance-Based Sensitivity Analysis: Theory and Estimation Algorithms, in Springer Handbook on Uncertainty Quantification, R. Ghanem, D. Higdon, and H. Owhadi, Eds., pp. 1217-1239, Berlin: Springer, 2017.

  8. Fang, K.T., Li, R., and Sudjianto, A., Design and Modeling for Computer Experiments, New York: Chapman & Hall/CRC, 2006.

  9. Le Gratiet, L., Marelli, S., and Sudret, B., Metamodel-Based Sensitivity Analysis: Polynomial Chaos Expansions and Gaussian Processes, in Springer Handbook on Uncertainty Quantification, R. Ghanem, D. Higdon, and H. Owhadi, Eds., pp. 1289-1325, Berlin: Springer, 2017.

  10. Xiu, D. and Karniadakis, G.E., The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations, SIAM J. Sci. Comput., 24(2):619-644, 2002.

  11. Szego, G., Orthogonal Polynomials, Vol. 23, Providence, RI: American Mathematical Society, 1939.

  12. Simon, B., Szego's Theorem and Its Descendants: Spectral Theory for L2 Perturbations of Orthogonal Polynomials, Princeton, NJ: Princeton University Press, 2010.

  13. Sudret, B., Global Sensitivity Analysis Using Polynomial Chaos Expansion, Reliab. Eng. Syst. Saf., 93:964-979, 2008.

  14. Luthen, N., Marelli, S., and Sudret, B., Sparse Polynomial Chaos Expansions: Literature Survey and Benchmark, SIAM/ASA J. Uncertainty Quantif., 9(2):593-649, 2021.

  15. Luthen, N., Marelli, S., and Sudret, B., Automatic Selection of Basis-Adaptive Sparse Polynomial Chaos Expansions for Engineering Applications, Int. J. Uncertainty Quantif., 12(3):49-74, 2022.

  16. Griewank, A. and Walther, A., Evaluating Derivatives: Principles and Techniques of Automatic Differentiation, Philadelphia: SIAM, 2008.

  17. Jakeman, J.D., Eldred, M.S., and Sargsyan, K., Enhancing 1-Minimization Estimates of Polynomial Chaos Expansions Using Basis Selection, J. Comput. Phys., 289:18-34, 2015.

  18. Peng, J., Hampton, J., and Doostan, A., On Polynomial Chaos Expansion via Gradient-Enhanced 1-Minimization, J. Comput. Phys., 310:440-458, 2016.

  19. Roderick, O., Anitescu, M., and Fischer, P., Polynomial Regression Approaches Using Derivative Information for Uncertainty Quantification, Nucl. Sci. Eng., 164:122-139, 2010.

  20. Li, Y., Anitescu, M., Roderick, O., and Hickernell, F., Orthogonal Bases for Polynomial Regression with Derivative Information in Uncertainty Quantification, Int. J. Uncertainty Quantif., 1:297-320, 2011.

  21. Guo, L., Narayan, A., and Zhou, T., A Gradient Enhanced 1-Minimization for Sparse Approximation of Polynomial Chaos Expansions, J. Comput. Phys., 367:49-64, 2018.

  22. Gejadze, I., Malaterre, P.O., and Shutyaev, V., On the Use of Derivatives in the Polynomial Chaos Based Global Sensitivity and Uncertainty Analysis Applied to the Distributed Parameter Models, J. Comput. Phys., 381:218-245, 2019.

  23. Sobol, I. and Gresham, A., On an Alternative Global Sensitivity Estimators, in Proc. of SAMO 1995, pp. 40-42, Belgirate, Italy, 1995.

  24. Sobol', I.M. and Kucherenko, S., Derivative Based Global Sensitivity Measures and Their Link with Global Sensitivity Indices, Math. Comput. Simul., 79(10):3009-3017, 2009.

  25. Kucherenko, S., Rodriguez-Fernandez, M., Pantelides, C., and Shah, N., Monte Carlo Evaluation of Derivative-Based Global Sensitivity Measures, Reliab. Eng. Syst. Saf., 94:1135-1148, 2009.

  26. Sudret, B. and Mai, C.V., Computing Derivative-Based Global Sensitivity Measures Using Polynomial Chaos Expansions, Reliab. Eng. Syst. Saf., 134:241-250, 2015.

  27. Kucherenko, S. and Iooss, B., Derivative-Based Global Sensitivity Measures, in Springer Handbook on Uncertainty Quantification, R. Ghanem, D. Higdon, and H. Owhadi, Eds., pp. 1241-1263, Berlin: Springer, 2017.

  28. Roustant, O., Gamboa, F., and Iooss, B., Parseval Inequalities and Lower Bounds for Variance-Based Sensitivity Indices, Electron. J. Stat., 14:386-412, 2020.

  29. Wiener, N., The Homogeneous Chaos, Am. J. Math., 60:897-936, 1938.

  30. Ghanem, R.G. and Spanos, P., Stochastic Finite Elements-A Spectral Approach, New York: Springer Verlag, 1991.

  31. Ernst, O., Mugler, A., Starkloff, H.J., and Ullmann, E., On the Convergence of Generalized Polynomial Chaos Expansions, ESAIM: Math. Modell. Numer. Anal., 46(02):317-339, 2012.

  32. Roustant, O., Barthe, F., and Iooss, B., Poincare Inequalities on Intervals-Application to Sensitivity Analysis, Electron. J. Stat., 2:3081-3119, 2017.

  33. Bakry, D., Gentil, I., and Ledoux, M., Analysis and Geometry of Markov Diffusion Operators, Vol. 348, Cham, Switzerland: Springer, 2014.

  34. Zettl, A., Sturm-Liouville Theory, Vol. 121, Providence, RI: American Mathematical Society, 2010.

  35. Mikolas, M., Uber Gewisse Eigenschaften Orthogonaler Systeme der Klasse L2 und die Eigenfunktionen Sturm-Liouvillescher Differentialgleichungen, Acta Math. Acad. Sci. Hung., 6:147-190, 1955.

  36. Kwon, K.H. and Lee, D., Orthogonal Functions Satisfying a Second-Order Differential Equation, J. Comput. Appl. Math., 153(1-2):283-293, 2003.

  37. Hoeffding, W., A Class of Statistics with Asymptotically Normal Distributions, Ann. Math. Stat., 19:293-325, 1948.

  38. Antoniadis, A., Analysis of Variance on Function Spaces, Stat. J. Theor. Appl. Stat., 15(1):59-71, 1984.

  39. Sudret, B., Global Sensitivity Analysis Using Polynomial Chaos Expansions, in Proc. 5th Int. Conf. on Comp. Stoch. Mech (CSM5), P. Spanos and G. Deodatis, Eds., Rhodos, Greece, June 21-23, 2006.

  40. Marelli, S. and Sudret, B., UQLab: A Framework for Uncertainty Quantification in Matlab, in Vulnerability, Uncertainty, and Risk (Proc. 2nd Int. Conf. on Vulnerability, Risk Analysis and Management), Liverpool, United Kingdom, pp. 2554-2563, 2014.

  41. Torre, E., Marelli, S., Embrechts, P., and Sudret, B., Data-Driven Polynomial Chaos Expansion for Machine Learning Regression, J. Comput. Phys., 388:601-623, 2019.

  42. Oladyshkin, S. and Nowak, W., Data-Driven Uncertainty Quantification Using the Arbitrary Polynomial Chaos Expansion, Reliab. Eng. Syst. Saf., 106:179-190, 2012.

  43. Bujurke, N., Salimath, C., and Shiralashetti, S., Computation of Eigenvalues and Solutions of Regular Sturm-Liouville Problems Using Haar Wavelets, J. Comput. Appl. Math., 219(1):90-101, 2008.

  44. Sudret, B., Berveiller, M., and Lemaire, M., A Stochastic Finite Element Procedure for Moment and Reliability Analysis, Eur. J. Comput. Mech., 15(7-8):825-866, 2006.

  45. Blatman, G. and Sudret, B., Adaptive Sparse Polynomial Chaos Expansion Based on Least Angle Regression, J. Comput. Phys., 230:2345-2367, 2011.

  46. Chapelle, O., Vapnik, V., and Bengio, Y., Model Selection for Small Sample Regression, Mach. Learn., 48(1):9-23, 2002.

  47. Le Ma?tre, O.P., Reagan, M.T., Najm, H.N., Ghanem, R.G., and Knio, O.M., A Stochastic Projection Method for Fluid Flow: II. Random Process, J. Comput. Phys., 181(1):9-44, 2002.

  48. Matthies, H.G. and Keese, A., Galerkin Methods for Linear and Nonlinear Elliptic Stochastic Partial Differential Equations, Comput. Methods Appl. Mech. Eng., 194(12-16):1295-1331, 2005.

  49. Constantine, P.G., Eldred, M.S., and Phipps, E.T., Sparse Pseudospectral Approximation Method, Comput. Methods Appl. Mech. Eng., 229:1-12, 2012.

  50. Blatman, G. and Sudret, B., Sparse Polynomial Chaos Expansions and Adaptive Stochastic Finite Elements Using a Regression Approach, C. R. Mec., 336(6):518-523, 2008.

  51. Candes, E.J. and Wakin, M.B., An Introduction to Compressive Sampling: A Sensing/Sampling Paradigm That Goes against the Common Knowledge in Data Acquisition, IEEE Signal Process. Mag., 25(2):21-30, 2008.

  52. Kougioumtzoglou, I.A., Petromichelakis, I., and Psaros, A.F., Sparse Representations and Compressive Sampling Approaches in Engineering Mechanics: A Review of Theoretical Concepts and Diverse Applications, Probab. Eng. Mech., 61:103082, 2020.

  53. Efron, B., Hastie, T., Johnstone, I., and Tibshirani, R., Least Angle Regression, Ann. Stat., 32:407-499, 2004.

  54. Marelli, S., Luthen, N., and Sudret, B., UQLab User Manual-Polynomial Chaos Expansions, Tech. Rep., Chair of Risk, Safety and Uncertainty Quantification, ETH Zurich, Switzerland, Report No. UQLab-V1.4-104, 2021.

  55. Candes, E.J. and Plan, Y., A Probabilistic and RIPless Theory of Compressed Sensing, IEEE Trans. Inf. Theory, 57(11):7235-7254, 2011.

  56. McKay, M.D., Beckman, R.J., and Conover, W.J., A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output from a Computer Code, Technometrics, 2:239-245, 1979.

  57. Crestaux, T., Maitre, O.L., and Martinez, J.M., Polynomial Chaos Expansions for Sensitivity Analysis, Reliab. Eng. Syst. Saf., 94:1161-1172, 2009.

  58. Becker, W., Metafunctions for Benchmarking in Sensitivity Analysis, Reliab. Eng. Syst. Saf., 204:107189, 2020.

  59. Iooss, B. and Lemaitre, P., A Review on Global Sensitivity Analysis Methods, in Uncertainty Management in Simulation-Optimization of Complex Systems: Algorithms and Applications, C. Meloni and G. Dellino, Eds., pp. 101-122, Berlin: Springer, 2015.

  60. Blatman, G., Adaptive Sparse Polynomial Chaos Expansions for Uncertainty Propagation and Sensitivity Analysis, PhD, Universite Blaise Pascal, 2009.

  61. Goutal, N., Lacombe, J.M., Zaoui, F., and El-Kadi-Abderrezak, K., MASCARET: A 1D Open-Source Software for Flow Hydrodynamic and Water Quality in Open Channel Networks, in River Flow 2012: Proc. of the Int. Conf. on Fluvial Hydraulics, R. Murillo Munoz, Ed., Vol. 2, pp. 1169-1174, San Jose, Costa Rica, Boca Raton, FL: CRC Press, 2012.

  62. Petit, S., Zaoui, F., Popelin, A.L., Goeury, C., and Goutal, N., Couplage entre Indices a Base de Derivees et Mode Adjoint pour l'Analyse de Sensibilite Globale. Application sur le Code Mascaret, French Open Archive HAL, preprint, from https://hal.science/hal-01373535v1, 2016.

  63. Demangeon, F., Goeury, C., Zaoui, F., Goutal, N., Pascual, V., and Hascoet, L., Algorithmic Differentiation Applied to the Optimal Calibration of a Shallow Water Model, La Houille Blanche Rev. Int. Eau, 102(4):57-65, 2015.

  64. Hascoet, L. and Pascual, V., The Tapenade Automatic Differentiation Tool: Principles, Model and Specification, ACM Trans. Math. Software, 39(3):1-43, 2013.

Begell Digital Portal Begell 数字图书馆 电子图书 期刊 参考文献及会议录 研究收集 订购及政策 Begell House 联系我们 Language English 中文 Русский Português German French Spain