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2019 | OriginalPaper | Buchkapitel

Polynomial Chaos and Collocation Methods and Their Range of Applicability

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Abstract

In this chapter the different polynomial chaos and stochastic collocation methodologies used within the UMRIDA project are compared. Guidelines for their use and applicability are formulated.

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Literatur
1.
Zurück zum Zitat Ernst, O.G., Mugler, A., Starkloff, H.J., Ullmann, E.: On the convergence of generalized polynomial chaos expansions. ESAIM. Math. Model. Numer. Anal. 46(2), 317–339 (2012)MathSciNetCrossRef Ernst, O.G., Mugler, A., Starkloff, H.J., Ullmann, E.: On the convergence of generalized polynomial chaos expansions. ESAIM. Math. Model. Numer. Anal. 46(2), 317–339 (2012)MathSciNetCrossRef
2.
Zurück zum Zitat Le Maître, O., Knio, O.: Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics. Springer, Dordrecht (2010) Le Maître, O., Knio, O.: Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics. Springer, Dordrecht (2010)
3.
Zurück zum Zitat Mathelin, L., Hussaini, M., Zang, T.: Stochastic approaches to uncertainty quantification in CFD simulations. Numer. Algorithms 38(1), 209–236 (2005)MathSciNetCrossRef Mathelin, L., Hussaini, M., Zang, T.: Stochastic approaches to uncertainty quantification in CFD simulations. Numer. Algorithms 38(1), 209–236 (2005)MathSciNetCrossRef
4.
Zurück zum Zitat Xiu, D., Hesthaven, J.S.: High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27(3), 1118–1139 (2005)MathSciNetCrossRef Xiu, D., Hesthaven, J.S.: High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27(3), 1118–1139 (2005)MathSciNetCrossRef
5.
Zurück zum Zitat Babuška, I., Nobile, F., Tempone, R.: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 45(3), 1005–1034 (2007)MathSciNetCrossRef Babuška, I., Nobile, F., Tempone, R.: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 45(3), 1005–1034 (2007)MathSciNetCrossRef
6.
Zurück zum Zitat Ganapathysubramanian, B., Zabaras, N.: Sparse grid collocation schemes for stochastic natural convection problems. J. Comput. Phys. 225(1), 652–685 (2007)MathSciNetCrossRef Ganapathysubramanian, B., Zabaras, N.: Sparse grid collocation schemes for stochastic natural convection problems. J. Comput. Phys. 225(1), 652–685 (2007)MathSciNetCrossRef
7.
Zurück zum Zitat Chung, K., Yao, T.: On lattices admitting unique Lagrange interpolations. SIAM J. Numer. Anal. 14(4), 735–743 (1977)MathSciNetCrossRef Chung, K., Yao, T.: On lattices admitting unique Lagrange interpolations. SIAM J. Numer. Anal. 14(4), 735–743 (1977)MathSciNetCrossRef
8.
Zurück zum Zitat Xiu, D.: Efficient collocational approach for parametric uncertainty analysis. Commun. Comput. Phys. 2(2), 293–309 (2007)MathSciNetMATH Xiu, D.: Efficient collocational approach for parametric uncertainty analysis. Commun. Comput. Phys. 2(2), 293–309 (2007)MathSciNetMATH
9.
Zurück zum Zitat Berveiller, M., Sudret, B., Lemaire, M.: Stochastic finite element: a non-intrusive approach by regression. Rev. Européenne de Mécanique Numérique 15(1–3), 81–92 (2006)MATH Berveiller, M., Sudret, B., Lemaire, M.: Stochastic finite element: a non-intrusive approach by regression. Rev. Européenne de Mécanique Numérique 15(1–3), 81–92 (2006)MATH
10.
Zurück zum Zitat Hosder, S., Walters, R.W., Balch, M.: Point-collocation nonintrusive polynomial chaos method for stochastic computational fluid dynamics. AIAA J. 48(12), 2721–2730 (2010)CrossRef Hosder, S., Walters, R.W., Balch, M.: Point-collocation nonintrusive polynomial chaos method for stochastic computational fluid dynamics. AIAA J. 48(12), 2721–2730 (2010)CrossRef
11.
Zurück zum Zitat Gao, Z., Zhou, T.: On the choice of design points for least-square polynomial approximations with application to uncertainty quantification. Commun. Comput. Phys. 16(2), 365–381 (2014)MathSciNetCrossRef Gao, Z., Zhou, T.: On the choice of design points for least-square polynomial approximations with application to uncertainty quantification. Commun. Comput. Phys. 16(2), 365–381 (2014)MathSciNetCrossRef
12.
Zurück zum Zitat Arnst, M., Ponthot, J.P.: An overview of nonintrusive characterization, propagation, and sensitivity analysis of uncertainties in computational mechanics. Int. J. Uncertainty Quantif. 4(5), 387–421 (2014)MathSciNetCrossRef Arnst, M., Ponthot, J.P.: An overview of nonintrusive characterization, propagation, and sensitivity analysis of uncertainties in computational mechanics. Int. J. Uncertainty Quantif. 4(5), 387–421 (2014)MathSciNetCrossRef
13.
Zurück zum Zitat Hosder, S., Walters, R.W., Balch, M.: Efficient sampling for non-intrusive polynomial chaos applications with multiple uncertain input variables. In: 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, AIAA paper 2007-1939, Honolulu HI, 23–26 April 2007 Hosder, S., Walters, R.W., Balch, M.: Efficient sampling for non-intrusive polynomial chaos applications with multiple uncertain input variables. In: 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, AIAA paper 2007-1939, Honolulu HI, 23–26 April 2007
14.
Zurück zum Zitat Raisee, M., Kumar, D., Lacor, C.: A non-intrusive model reduction approach for polynomial chaos expansion using proper orthogonal decomposition. Int. J. Numer. Methods Eng. 103(4), 293–312 (2015)MathSciNetCrossRef Raisee, M., Kumar, D., Lacor, C.: A non-intrusive model reduction approach for polynomial chaos expansion using proper orthogonal decomposition. Int. J. Numer. Methods Eng. 103(4), 293–312 (2015)MathSciNetCrossRef
15.
Zurück zum Zitat Candès, E.J., Romberg, J.K., Tao, T.: Stable signal recovery from incomplete and inaccurate measurements. Commun. Pure Appl. Math. 59(8), 1207–1223 (2006)MathSciNetCrossRef Candès, E.J., Romberg, J.K., Tao, T.: Stable signal recovery from incomplete and inaccurate measurements. Commun. Pure Appl. Math. 59(8), 1207–1223 (2006)MathSciNetCrossRef
17.
Zurück zum Zitat Hampton, J., Doostan, A.: Compressive sampling methods for sparse polynomial chaos expansions. In Ghanem, R.G., Higdon, D., Owhadi, H., (eds.) Handbook of Uncertainty Quantification. Springer, Cham, 29 p (2016) Hampton, J., Doostan, A.: Compressive sampling methods for sparse polynomial chaos expansions. In Ghanem, R.G., Higdon, D., Owhadi, H., (eds.) Handbook of Uncertainty Quantification. Springer, Cham, 29 p (2016)
18.
Zurück zum Zitat Savin, É., Resmini, A., Peter, J.: Sparse polynomial surrogates for aerodynamic computations with random inputs. In: 18th AIAA Non-Deterministic Approaches Conference, AIAA paper 2016-0433, San Diego CA, 4–8 Jan 2016 Savin, É., Resmini, A., Peter, J.: Sparse polynomial surrogates for aerodynamic computations with random inputs. In: 18th AIAA Non-Deterministic Approaches Conference, AIAA paper 2016-0433, San Diego CA, 4–8 Jan 2016
19.
Zurück zum Zitat Chen, S.C., Donoho, D.L., Saunders, M.A.: Atomic decomposition by basis pursuit. SIAM J. Sci. Comput. 20(1), 33–61 (1998)MathSciNetCrossRef Chen, S.C., Donoho, D.L., Saunders, M.A.: Atomic decomposition by basis pursuit. SIAM J. Sci. Comput. 20(1), 33–61 (1998)MathSciNetCrossRef
20.
Zurück zum Zitat Dinescu, C., Smirnov, S., Hirsch, C., Lacor, C.: Assessment of intrusive and non-intrusive non-deterministic CFD methodologies based on polynomial chaos expansions. Int. J. Eng. Syst. Model. Simul. 2(1–2), 87–98 (2010) Dinescu, C., Smirnov, S., Hirsch, C., Lacor, C.: Assessment of intrusive and non-intrusive non-deterministic CFD methodologies based on polynomial chaos expansions. Int. J. Eng. Syst. Model. Simul. 2(1–2), 87–98 (2010)
21.
Zurück zum Zitat Xiu, D.: Fast numerical methods for stochastic computations: A review. Commun. Comput. Phys. 5(2–4), 242–272 (2009)MathSciNetMATH Xiu, D.: Fast numerical methods for stochastic computations: A review. Commun. Comput. Phys. 5(2–4), 242–272 (2009)MathSciNetMATH
22.
Zurück zum Zitat Gottlieb, D., Xiu, D.: Galerkin method for wave equations with uncertain coefficients. Commun. Comput. Phys. 3(2), 505–518 (2008)MathSciNetMATH Gottlieb, D., Xiu, D.: Galerkin method for wave equations with uncertain coefficients. Commun. Comput. Phys. 3(2), 505–518 (2008)MathSciNetMATH
23.
Zurück zum Zitat Poëtte, G., Després, B., Lucor, D.: Uncertainty quantification for systems of conservation laws. J. Comput. Phys. 228(7), 2443–2467 (2009)MathSciNetCrossRef Poëtte, G., Després, B., Lucor, D.: Uncertainty quantification for systems of conservation laws. J. Comput. Phys. 228(7), 2443–2467 (2009)MathSciNetCrossRef
24.
Zurück zum Zitat Tryoen, J., Le Maître, O., Ndjinga, M., Ern, A.: Intrusive Galerkin methods with unwinding for uncertain nonlinear hyperbolic systems. J. Comput. Phys. 229(18), 6485–6511 (2010)MathSciNetCrossRef Tryoen, J., Le Maître, O., Ndjinga, M., Ern, A.: Intrusive Galerkin methods with unwinding for uncertain nonlinear hyperbolic systems. J. Comput. Phys. 229(18), 6485–6511 (2010)MathSciNetCrossRef
25.
Zurück zum Zitat Smolyak, S.: Quadrature and interpolation formulas for tensor products of certain classes of functions. Sov. Math. Dokl. 4, 240–243 (1963)MATH Smolyak, S.: Quadrature and interpolation formulas for tensor products of certain classes of functions. Sov. Math. Dokl. 4, 240–243 (1963)MATH
26.
Zurück zum Zitat Blatman, G., Sudret, B.: Sparse polynomial chaos expansions and adaptive stochastic finite elements using a regression approach. C. R. Méc. 336(6), 518–523 (2008)CrossRef Blatman, G., Sudret, B.: Sparse polynomial chaos expansions and adaptive stochastic finite elements using a regression approach. C. R. Méc. 336(6), 518–523 (2008)CrossRef
27.
Zurück zum Zitat Blatman, G., Sudret, B.: An adaptive algorithm to build sparse polynomial chaos expansions for stochastic finite element analysis. Probab. Eng. Mech. 25(2), 183–197 (2010)CrossRef Blatman, G., Sudret, B.: An adaptive algorithm to build sparse polynomial chaos expansions for stochastic finite element analysis. Probab. Eng. Mech. 25(2), 183–197 (2010)CrossRef
28.
Zurück zum Zitat Blatman, G., Sudret, B.: Adaptive sparse polynomial chaos expansion based on least angle regression. J. Comput. Phys. 230(6), 2345–2367 (2011)MathSciNetCrossRef Blatman, G., Sudret, B.: Adaptive sparse polynomial chaos expansion based on least angle regression. J. Comput. Phys. 230(6), 2345–2367 (2011)MathSciNetCrossRef
29.
Zurück zum Zitat Blatman, G., Sudret, B.: Sparse polynomial chaos expansions of vector-valued response quantities. In: Proceedings 11th International Conference on Structural Safety and Reliability (ICOSSAR13) (2013) Blatman, G., Sudret, B.: Sparse polynomial chaos expansions of vector-valued response quantities. In: Proceedings 11th International Conference on Structural Safety and Reliability (ICOSSAR13) (2013)
30.
Zurück zum Zitat van den Berg, E., Friedlander, M.P.: SPGL1: A Solver For Large-scale Sparse Reconstruction (June 2007) van den Berg, E., Friedlander, M.P.: SPGL1: A Solver For Large-scale Sparse Reconstruction (June 2007)
31.
Zurück zum Zitat Andersen, M., Dahl, J., Vandenberghe, L.: CVXOPT: Python Software For Convex Optimization (2016) Andersen, M., Dahl, J., Vandenberghe, L.: CVXOPT: Python Software For Convex Optimization (2016)
32.
Zurück zum Zitat Hampton, J.: Doostan: compressive sampling of polynomial chaos expansions: convergence analysis and sampling strategies. J. Comput. Phys. 280, 363–386 (2015)MathSciNetCrossRef Hampton, J.: Doostan: compressive sampling of polynomial chaos expansions: convergence analysis and sampling strategies. J. Comput. Phys. 280, 363–386 (2015)MathSciNetCrossRef
33.
Zurück zum Zitat Reid, L., Moore, R.: Performance of single-stage axial-flow transonic compressor with rotor and stator aspect ratios of 1.19 and 1.26, respectively, and with design pressure ratio of 1.82. NASA Technical Paper 1338 (1978) Reid, L., Moore, R.: Performance of single-stage axial-flow transonic compressor with rotor and stator aspect ratios of 1.19 and 1.26, respectively, and with design pressure ratio of 1.82. NASA Technical Paper 1338 (1978)
Metadaten
Titel
Polynomial Chaos and Collocation Methods and Their Range of Applicability
verfasst von
Chris Lacor
Éric Savin
Copyright-Jahr
2019
DOI
https://doi.org/10.1007/978-3-319-77767-2_42

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