Abstract
Polynomial chaos decompositions (PCE) have emerged over the past three decades as a standard among the many tools for uncertainty quantification. They provide a rich mathematical structure that is particularly well suited to enabling probabilistic assessments in situations where interdependencies between physical processes or between spatiotemporal scales of observables constitute credible constraints on system-level predictability. Algorithmic developments exploiting their structural simplicity have permitted the adaptation of PCE to many of the challenges currently facing prediction science. These include requirements for large-scale high-resolution computational simulations implicit in modern applications, non-Gaussian probabilistic models, and non-smooth dependencies and for handling general vector-valued stochastic processes. This chapter presents an overview of polynomial chaos that underscores their relevance to problems of constructing and estimating probabilistic models, propagating them through arbitrarily complex computational representations of underlying physical mechanisms, and updating the models and their predictions as additional constraints become known.
References
Adomian, G.: Stochastic Greenās functions. In: Bellman, R. (ed.) Proceedings of Symposia in Applied Mathematics. VolumeĀ 16: Stochastic Processes in Mathematical Physics and Engineering. American Mathematical Society, Providence (1964)
Adomian, G.: Stochastic Systems. Academic, New York (1983)
Albeverio, S., Daletsky, Y., Kondratiev, Y., Streit, L.: Non-Gaussian infinite dimensional analysis. J. Funct. Anal. 138, 311ā350 (1996)
Arnst, M., Ghanem, R.: Probabilistic equivalence and stochastic model reduction in multiscale analysis. Comput. Methods Appl. Mech. Eng. 197(43ā44), 3584ā3592 (2008)
Arnst, M., Ghanem, R., Phipps, E., Red-Horse, J.: Dimension reduction in stochastic modeling of coupled problems. Int. J. Numer. Methods Eng. 92, 940ā968 (2012)
Arnst, M., Ghanem, R., Phipps, E., Red-Horse, J.: Measure transformation and efficient quadrature in reduced-dimensional stochastic modeling of coupled problems. Int. J. Numer. Methods Eng. 92, 1044ā1080 (2012)
Arnst, M., Ghanem, R., Phipps, E., Red-Horse, J.: Reduced chaos expansions with random coefficients in reduced-dimensional stochastic modeling of coupled problems. Int. J. Numer. Methods Eng. 97(5), 352ā376 (2014)
Arnst, M., Ghanem, R., Soize, C.: Identification of Bayesian posteriors for coefficients of chaos expansions. J. Comput. Phys. 229(9), 3134ā3154 (2010)
BabuÅ”ka, I., Tempone, R., Zouraris, G.E.: Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42(2), 800ā825 (2005)
BabuÅ”ka, I., Tempone, R., Zouraris, G.E.: Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation. Comput. Methods Appl. Mech. Eng. 194(12ā16), 1251ā1294 (2005)
Benaroya, H., Rehak, M.: Finite element methods in probabilistic structural analysis: a selective review. Appl. Mech. Rev. 41(5), 201ā213 (1988)
Berezansky, Y.M.: Infinite-dimensional non-Gaussian analysis and generalized translation operators. Funct. Anal. Appl. 30(4), 269ā272 (1996)
Bharucha-Reid, A.T.: On random operator equations in Banach space. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astr. Phys. 7, 561ā564 (1959)
Billingsley, P.: Probability and Measure. Wiley Interscience, New York (1995)
Bose, A.G.: A theory of nonlinear systems. Technical report 309, Research Laboratory of Electronics, MIT (1956)
Boyce, E.W., Goodwin, B.E.: Random transverse vibration of elastic beams. SIAM J. 12(3), 613ā629 (1964)
Brilliant, M.B.: Theory of the analysis of nonlinear systems. Technical report 345, Research Laboratory of Electronics, MIT (1958)
Cameron, R.H., Martin, W.T.: The orthogonal development of nonlinear funtions in a series of Fourier-Hermite functionals. Ann. Math. 48, 385ā392 (1947)
Chorin, A.: Hermite expansions in Monte-Carlo computation. J. Comput. Phys. 8, 472ā482 (1971)
Cornish, E., Fisher, R.: Moments and cumulants in the specification of distributions. Rev. Int. Stat. Inst. 5(4), 307ā320 (1938)
Das, S., Ghanem, R.: A bounded random matrix approach for stochastic upscaling. SIAM J. Multiscale Model. Simul. 8(1), 296ā325 (2009)
Das, S., Ghanem, R., Finette, S.: Polynomial chaos representation of spatio-temporal random fields from experimental measurements. J. Comput. Phys. 228(23), 8726ā8751 (2009)
Das, S., Ghanem, R., Spall, J.: Sampling distribution for polynomial chaos representation of data: a maximum-entropy and fisher information approach. SIAM J. Sci. Comput. 30(5), 2207ā2234 (2008)
Debusschere, B., Najm, H., Matta, A., Knio, O., Ghanem, R., Le Maitre, O.: Protein labeling reactions in electrochemical microchannel flow: numerical simulation and uncertainty propagation. Phys. Fluids 15(8), 2238ā2250 (2003)
Descelliers, C., Ghanem, R., Soize, C.: Maximum likelihood estimation of stochastic chaos representation from experimental data. Int. J. Numer. Methods Eng. 66(6), 978ā1001 (2006)
Diggle, P., Gratton, R.: Monte Carlo methods of inference for implicit statistical models. J. R. Stat. Soc. Ser. B 46, 193ā227 (1984)
Doostan, A., Ghanem, R., Red-Horse, J.: Stochastic model reduction for chaos representations. Comput. Methods Appl. Mech. Eng. 196, 3951ā3966 (2007)
Ernst, O.G., Ullmann, E.: Stochastic Galerkin matrices. SIAM J. Matrix Anal. Appl. 31(4), 1848ā1872 (2010)
Fisher, R., Cornish, E.: The percentile points of distributions having known cumulants. Technometrics 2(2), 209ā225 (1960)
Ganapathysubramanian, B., Zabaras, N.: Sparse grid collocation methods for stochastic natural convection problems. J. Comput. Phys. 225, 652ā685 (2007)
George, D.A.: Continuous nonlinear systems. Technical report 355, Research Laboratory of Electronics, MIT (1959)
Ghanem, R.: Hybrid stochastic finite elements: coupling of spectral expansions with Monte Carlo simulations. ASME J. Appl. Mech. 65, 1004ā1009 (1998)
Ghanem, R.: Scales of fluctuation and the propagation of uncertainty in random porous media. Water Resour. Res. 34(9), 2123ā2136 (1998)
Ghanem, R., Abras, J.: A general purpose library for stochastic finite element computations. In: Bathe, J. (ed.) Second MIT Conference on Computational Mechanics, Cambridge (2003)
Ghanem, R., Brzkala, V.: Stochastic finite element analysis for randomly layered media. ASCE J. Eng. Mech. 122(4), 361ā369 (1996)
Ghanem, R., Dham, S.: Stochastic finite element analysis for multiphase flow in heterogeneous porous media. Transp. Porous Media 32, 239ā262 (1998)
Ghanem, R., Doostan, A., Red-Horse, J.: A probabilistic construction of model validation. Comput. Methods Appl. Mech. Eng. 197, 2585ā2595 (2008)
Ghanem, R., Red-Horse, J., Benjamin, A., Doostan, A., Yu, A.: Stochastic process model for material properties under incomplete information (AIAA 2007ā1968). In: 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Honolulu, 23ā26 Apr 2007. AIAA (2007)
Ghanem, R., Sarkar, A.: Reduced models for the medium-frequency dynamics of stochastic systems. JASA 113(2), 834ā846 (2003)
Ghanem, R., Spanos, P.: Stochastic Finite Elements: A Spectral Approach. Springer, New York (1991). Revised edition by Dover Publications, (2003)
Ghiocel, D., Ghanem, R.: Stochastic finite element analysis of seismic soil-structure interaction. J. Eng. Mech. 128(1), 66ā77 (2002)
Gikhman, I., Skorohod, A.: The Theory of Stochastic Processes I. Springer, Berlin (1974)
Guilleminot, J., Soize, C., Ghanem, R.: Stochastic representation for anisotropic permeability tensor random fields. Int. J. Numer. Anal. Methods Geomech. 36, 1592ā1608 (2012)
Hart, G.C., Collins, J.D.: The treatment of randomness in finite element modelling. In: SAE Shock and Vibrations Symposium, Los Angeles, pp.Ā 2509ā2519 (1970)
Hasselman, T.K., Hart, G.C.: Modal analysis of random structural systems. ASCE J. Eng. Mech. 98(EM3), 561ā579 (1972)
Hida, T.: White noise analysis and nonlinear filtering problems. Appl. Math. Optim. 2, 82ā89 (1975)
Hida, T., Kuo, H.-H., Potthoff, J., Streit, L.: White Noise: An Infinite Dimensional Calculus. Kluwer Academic Publishers, Dordrecht/Boston (1993)
Imamura, T., Meecham, W.: Wiener-Hermite expansion in model turbulence in the late decay stage. J. Math. Phys. 6(5), 707ā721 (1965)
ItĆ“, K.: Multiple Wiener integrals. J. Math. Soc. Jpn. 3(1), 157ā169 (1951)
ItĆ“, K.: Spectral type of shift transformations of differential process with stationary increments. Trans. Am. Math. Soc. 81, 253ā263 (1956)
Jahedi, A., Ahmadi, G.: Application of Wiener-Hermite expansion to nonstationary random vibration of a Duffing oscillator. ASME J. Appl. Mech. 50, 436ā442 (1983)
Kallianpur, G.: Stochastic Filtering Theory. Springer, New York (1980)
Klein, S., Yasui, S.: Nonlinear systems analysis with non-Gaussian white stimuli: General basis functionals and kernels. IEEE Tran. Inf. Theory IT-25(4), 495ā500 (1979)
Kondratiev, Y., Da Silva, J., Streit, L., Us, G.: Analysis on Poisson and Gamma spaces. Infinite Dimens. Anal. Quantum Probab. Relat. Top. 1(1), 91ā117 (1998)
LĆ©vy, P.: LeƧons dāAnalyses Fonctionelles. Gauthier-Villars, Paris (1922)
Li, R., Ghanem, R.: Adaptive polynomial chaos simulation applied to statistics of extremes in nonlinear random vibration. Probab. Eng. Mech. 13(2), 125ā136 (1998)
Liu, W.K., Besterfield, G., Mani, A.: Probabilistic finite element methods in nonlinear structural dynamics. Comput. Methods Appl. Mech. Eng. 57, 61ā81 (1986)
Lytvynov, E.: Multiple Wiener integrals and non-Gaussian white noise: a Jacobi field approach. Methods Funct. Anal. Topol. 1(1), 61ā85 (1995)
Le Maitre, O., Najm, H., Ghanem, R., Knio, O.: Multi-resolution analysis of Wiener-type uncertainty propagation schemes. J. Comput. Phys. 197(2), 502ā531 (2004)
Le Maitre, O., Reagan, M., Najm, H., Ghanem, R., Knio, O.: A stochastic projection method for fluid flow. II: random process. J. Comput. Phys. 181, 9ā44 (2002)
Matthies, H.G., Keese, A.: Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Eng. 194(12ā16), 1295ā1331 (2005). Special Issue on Computational Methods in Stochastic Mechanics and Reliability Analysis
Meidani, H., Ghanem, R.: Uncertainty quantification for Markov chain models. Chaos 22(4) (2012)
Nakagiri, S., Hisada, T.: Stochastic finite element method applied to structural analysis with uncertain parameters. In: Proceeding of the International Conference on FEM, pp.Ā 206ā211 (1982)
Nakayama, A., Kuwahara, F., Umemoto, T., Hayashi, T.: Heat and fluid flow within an anisotropic porous medium. Trans. ASME 124, 746ā753 (2012)
Ogura, H.: Orthogonal functionals of the Poisson process. IEEE Trans. Inf. Theory IT-18(4), 473ā481 (1972)
Pawlowski, R., Phipps, R., Salinger, A., Owen, S., Ciefert, C., Stalen, A.: Automating embedded analysis capabilities and managing software complexity in multiphysics simulation, Part II: application to partial differential equations. Sci. Program. 20(3), 327ā345 (2012)
Pellissetti, M.F., Ghanem, R.G.: Iterative solution of systems of linear equations arising in the context of stochastic finite elements. Adv. Eng. Softw. 31(8ā9), 607ā616 (2000)
Powell, C.E., Elman, H.C.: Block-diagonal preconditioning for spectral stochastic finite-element systems. IMA J. Numer. Anal. 29(2), 350ā375 (2009)
Pugachev, V., Sinitsyn, I.: Stochastic Systems: Theory and Applications. World Scientific, River Edge (2001)
Red-Horse, J., Ghanem, R.: Elements of a functional analytic approach to probability. Int. J. Numer. Methods Eng. 80(6ā7), 689ā716 (2009)
Reichel, L., Trefethen, L.: Eigenvalues and pseudo-eigenvalues of toeplitz matrices. Linear Algebra Appl. 162, 153ā185 (1992)
Rosenblatt, M.: Remarks on a multivariate transformation. Ann. Math. Stat. 23, 470ā472 (1952)
Rosenblatt, M.: Remarks on some nonparametric estimates of a density function. Ann. Math. Stat. 27, 832ā837 (1956)
Rosseel, E., Vandewalle, S.: Iterative solvers for the stochastic finite element method. SIAM J. Sci. Comput. 32(1), 372ā397 (2010)
Rugh, W.J.: Nonlinear System Theory: The Volterra-Wiener Approach. Johns Hopkins University Press, Baltimore (1981)
Sakamoto, S., Ghanem, R.: Simulation of multi-dimensional non-Gaussian non-stationary random fields. Probab. Eng. Mech. 17(2), 167ā176 (2002)
Sargsyan, K., Najm, H., Ghanem, R.: On the statistical calibration of physical models. Int. J. Chem. Kinet. 47(4), 246ā276 (2015)
Schoutens, W.: Stochastic Processes and Orthogonal Polynomials. Springer, New York (2000)
Segall, A., Kailath, T.: Orthogonal functionals of independent-increment processes. IEEE Trans. Inf. Theory IT-22(3), 287ā298 (1976)
Shinozuka, M., Astill, J.: Random eigenvalue problem in structural mechanics. AIAA J. 10(4), 456ā462 (1972)
Skorohod, A.V.: Random linear operators. Reidel publishing company, Dordrecht (1984)
Sobczyk, K.: Wave Propagation in Random Media. Elsevier, Amsterdam (1985)
Soize, C.: A nonparametric model of random uncertainties for reduced matrix models in structural dynamics. Probab. Eng. Mech. 15(3), 277ā294 (2000)
Soize, C., Ghanem, R.: Physical systems with random uncertainties: chaos representations with arbitrary probability measure. SIAM J. Sci. Comput. 26(2), 395ā410 (2004)
Soize, C., Ghanem, R.: Reduced chaos decomposition with random coefficients of vector-valued random variables and random fields. Comput. Methods Appl. Mech. Eng. 198(21ā26), 1926ā1934 (2009)
Soize, C., Ghanem, R.: Data-driven probability concentration and sampling on manifold. J. Comput. Phys. 321, 242ā258 (2016)
Soong, T.T., Bogdanoff, J.L.: On the natural frequencies of a disordered linear chain of n degrees of freedom. Int. J. Mech. Sci. 5, 237ā265 (1963)
Sousedik, B., Elman, H.: Stochastic Galerkin methods for the steady-state Navier-Stokes equations. J. Comput. Phys. 316, 435ā452 (2016)
Sousedik, B., Ghanem, R.: Truncated hierarchical preconditioning for the stochastic Galerkin FEM. Int. J. Uncertain. Quantif. 4(4), 333ā348 (2014)
Sousedik, B., Ghanem, R., Phipps, E.: Hierarchical schur complement preconditioner for the stochastic Galerkin finite element methods. Numer. Linear Algebra Appl. 21(1), 136ā151 (2014)
Steinwart, I., Scovel, C.: Mercerās theorem on general domains: on the interaction between measures, kernels, and RKHSs. Constr. Approx. 35, 363ā417 (2012)
Stone, M.: The genralized Weierstrass approximation theorem. Math. Mag. 21(4), 167ā184 (1948)
Takemura, A., Takeuchi, K.: Some results on univariate and multivariate Cornish-Fisher expansion: algebraic properties and validity. Sankhy\(\breve{a}\) 50, 111ā136 (1988)
Tan, W., Guttman, I.: On the construction of multi-dimensional orthogonal polynomials. Metron 34, 37ā54 (1976)
Tavare, S., Balding, D., Griffiths, R., Donnelly, P.: Inferring coalescence times from dna sequence data. Genetics 145, 505ā518 (1997)
Thimmisetty, C., Khodabakhshnejad, A., Jabbari, N., Aminzadeh, F., Ghanem, R., Rose, K., Disenhof, C., Bauer, J.: Multiscale stochastic representation in high-dimensional data using Gaussian processes with implicit diffusion metrics. In: Ravela, S., Sandu, A. (eds.) Dynamic Data-Driven Environmental Systems Science. Lecture Notes in Computer Science, vol.Ā 8964. Springer (2015). doi:10.1007/978ā3ā319ā25138ā7_15
Tipireddy, R.: Stochastic Galerkin projections: solvers, basis adaptation and multiscale modeling and reduction. PhD thesis, University of Southern California (2013)
Tipireddy, R., Ghanem, R.: Basis adaptation in homogeneous chaos spaces. J. Comput. Phys. 259, 304ā317 (2014)
Tsilifis, P., Ghanem, R.: Reduced Wiener chaos representation of random fields via basis adaptation and projection. J. Comput. Phys. (2016, submitted)
Volterra, V.: Theory of Functionals and of Integral and Integro-Differential Equations. Blackie & Son, Ltd., Glasgow (1930)
Wan, X., Karniadakis, G.: Multi-element generalized polynomial chaos for arbitrary probability measures. SIAM J. Sci. Comput. 28(3), 901ā928 (2006)
Wiener, N.: Differential space. J. Math. Phys. 2, 131ā174 (1923)
Wiener, N.: The homogeneous chaos. Am. J. Math. 60(4), 897ā936 (1938)
Wintner, A., Wiener, N.: The discrete chaos. Am. J. Math. 65, 279ā298 (1943)
Xiu, D., Karniadakis, G.: The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24, 619ā644 (2002)
Xiu, D., Hesthaven, J.S.: High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27(3), 1118ā1139 (2005)
Yamazaki, F., Shinozuka, M., Dasgupta, G.: Neumann expansion for stochastic finite-element analysis. ASCE J. Eng. Mech. 114(8), 1335ā1354 (1988)
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Ghanem, R., Red-Horse, J. (2016). Polynomial Chaos: Modeling, Estimation, and Approximation. In: Ghanem, R., Higdon, D., Owhadi, H. (eds) Handbook of Uncertainty Quantification. Springer, Cham. https://doi.org/10.1007/978-3-319-11259-6_13-1
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