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.
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References
Adomian, G.: Stochastic Green’s functions. In: Bellman, R. (ed.) Proceedings of Symposia in Applied Mathematics. Volume 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, 311–350 (1996)
Arnst, M., Ghanem, R.: Probabilistic equivalence and stochastic model reduction in multiscale analysis. Comput. Methods Appl. Mech. Eng. 197(43–44), 3584–3592 (2008)
Arnst, M., Ghanem, R., Phipps, E., Red-Horse, J.: Dimension reduction in stochastic modeling of coupled problems. Int. J. Numer. Methods Eng. 92, 940–968 (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, 1044–1080 (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), 352–376 (2014)
Arnst, M., Ghanem, R., Soize, C.: Identification of Bayesian posteriors for coefficients of chaos expansions. J. Comput. Phys. 229(9), 3134–3154 (2010)
Babuška, I., Tempone, R., Zouraris, G.E.: Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42(2), 800–825 (2005)
Babuška, 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(12–16), 1251–1294 (2005)
Benaroya, H., Rehak, M.: Finite element methods in probabilistic structural analysis: a selective review. Appl. Mech. Rev. 41(5), 201–213 (1988)
Berezansky, Y.M.: Infinite-dimensional non-Gaussian analysis and generalized translation operators. Funct. Anal. Appl. 30(4), 269–272 (1996)
Bharucha-Reid, A.T.: On random operator equations in Banach space. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astr. Phys. 7, 561–564 (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), 613–629 (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, 385–392 (1947)
Chorin, A.: Hermite expansions in Monte-Carlo computation. J. Comput. Phys. 8, 472–482 (1971)
Cornish, E., Fisher, R.: Moments and cumulants in the specification of distributions. Rev. Int. Stat. Inst. 5(4), 307–320 (1938)
Das, S., Ghanem, R.: A bounded random matrix approach for stochastic upscaling. SIAM J. Multiscale Model. Simul. 8(1), 296–325 (2009)
Das, S., Ghanem, R., Finette, S.: Polynomial chaos representation of spatio-temporal random fields from experimental measurements. J. Comput. Phys. 228(23), 8726–8751 (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), 2207–2234 (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), 2238–2250 (2003)
Descelliers, C., Ghanem, R., Soize, C.: Maximum likelihood estimation of stochastic chaos representation from experimental data. Int. J. Numer. Methods Eng. 66(6), 978–1001 (2006)
Diggle, P., Gratton, R.: Monte Carlo methods of inference for implicit statistical models. J. R. Stat. Soc. Ser. B 46, 193–227 (1984)
Doostan, A., Ghanem, R., Red-Horse, J.: Stochastic model reduction for chaos representations. Comput. Methods Appl. Mech. Eng. 196, 3951–3966 (2007)
Ernst, O.G., Ullmann, E.: Stochastic Galerkin matrices. SIAM J. Matrix Anal. Appl. 31(4), 1848–1872 (2010)
Fisher, R., Cornish, E.: The percentile points of distributions having known cumulants. Technometrics 2(2), 209–225 (1960)
Ganapathysubramanian, B., Zabaras, N.: Sparse grid collocation methods for stochastic natural convection problems. J. Comput. Phys. 225, 652–685 (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, 1004–1009 (1998)
Ghanem, R.: Scales of fluctuation and the propagation of uncertainty in random porous media. Water Resour. Res. 34(9), 2123–2136 (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), 361–369 (1996)
Ghanem, R., Dham, S.: Stochastic finite element analysis for multiphase flow in heterogeneous porous media. Transp. Porous Media 32, 239–262 (1998)
Ghanem, R., Doostan, A., Red-Horse, J.: A probabilistic construction of model validation. Comput. Methods Appl. Mech. Eng. 197, 2585–2595 (2008)
Ghanem, R., Red-Horse, J., Benjamin, A., Doostan, A., Yu, A.: Stochastic process model for material properties under incomplete information (AIAA 2007–1968). In: 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Honolulu, 23–26 Apr 2007. AIAA (2007)
Ghanem, R., Sarkar, A.: Reduced models for the medium-frequency dynamics of stochastic systems. JASA 113(2), 834–846 (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), 66–77 (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, 1592–1608 (2012)
Hart, G.C., Collins, J.D.: The treatment of randomness in finite element modelling. In: SAE Shock and Vibrations Symposium, Los Angeles, pp. 2509–2519 (1970)
Hasselman, T.K., Hart, G.C.: Modal analysis of random structural systems. ASCE J. Eng. Mech. 98(EM3), 561–579 (1972)
Hida, T.: White noise analysis and nonlinear filtering problems. Appl. Math. Optim. 2, 82–89 (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), 707–721 (1965)
Itô, K.: Multiple Wiener integrals. J. Math. Soc. Jpn. 3(1), 157–169 (1951)
Itô, K.: Spectral type of shift transformations of differential process with stationary increments. Trans. Am. Math. Soc. 81, 253–263 (1956)
Jahedi, A., Ahmadi, G.: Application of Wiener-Hermite expansion to nonstationary random vibration of a Duffing oscillator. ASME J. Appl. Mech. 50, 436–442 (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), 495–500 (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), 91–117 (1998)
Lévy, P.: Leçons d’Analyses 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), 125–136 (1998)
Liu, W.K., Besterfield, G., Mani, A.: Probabilistic finite element methods in nonlinear structural dynamics. Comput. Methods Appl. Mech. Eng. 57, 61–81 (1986)
Lytvynov, E.: Multiple Wiener integrals and non-Gaussian white noise: a Jacobi field approach. Methods Funct. Anal. Topol. 1(1), 61–85 (1995)
Le Maitre, O., Najm, H., Ghanem, R., Knio, O.: Multi-resolution analysis of Wiener-type uncertainty propagation schemes. J. Comput. Phys. 197(2), 502–531 (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, 9–44 (2002)
Matthies, H.G., Keese, A.: Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Eng. 194(12–16), 1295–1331 (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. 206–211 (1982)
Nakayama, A., Kuwahara, F., Umemoto, T., Hayashi, T.: Heat and fluid flow within an anisotropic porous medium. Trans. ASME 124, 746–753 (2012)
Ogura, H.: Orthogonal functionals of the Poisson process. IEEE Trans. Inf. Theory IT-18(4), 473–481 (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), 327–345 (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(8–9), 607–616 (2000)
Powell, C.E., Elman, H.C.: Block-diagonal preconditioning for spectral stochastic finite-element systems. IMA J. Numer. Anal. 29(2), 350–375 (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(6–7), 689–716 (2009)
Reichel, L., Trefethen, L.: Eigenvalues and pseudo-eigenvalues of toeplitz matrices. Linear Algebra Appl. 162, 153–185 (1992)
Rosenblatt, M.: Remarks on a multivariate transformation. Ann. Math. Stat. 23, 470–472 (1952)
Rosenblatt, M.: Remarks on some nonparametric estimates of a density function. Ann. Math. Stat. 27, 832–837 (1956)
Rosseel, E., Vandewalle, S.: Iterative solvers for the stochastic finite element method. SIAM J. Sci. Comput. 32(1), 372–397 (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), 167–176 (2002)
Sargsyan, K., Najm, H., Ghanem, R.: On the statistical calibration of physical models. Int. J. Chem. Kinet. 47(4), 246–276 (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), 287–298 (1976)
Shinozuka, M., Astill, J.: Random eigenvalue problem in structural mechanics. AIAA J. 10(4), 456–462 (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), 277–294 (2000)
Soize, C., Ghanem, R.: Physical systems with random uncertainties: chaos representations with arbitrary probability measure. SIAM J. Sci. Comput. 26(2), 395–410 (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(21–26), 1926–1934 (2009)
Soize, C., Ghanem, R.: Data-driven probability concentration and sampling on manifold. J. Comput. Phys. 321, 242–258 (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, 237–265 (1963)
Sousedik, B., Elman, H.: Stochastic Galerkin methods for the steady-state Navier-Stokes equations. J. Comput. Phys. 316, 435–452 (2016)
Sousedik, B., Ghanem, R.: Truncated hierarchical preconditioning for the stochastic Galerkin FEM. Int. J. Uncertain. Quantif. 4(4), 333–348 (2014)
Sousedik, B., Ghanem, R., Phipps, E.: Hierarchical schur complement preconditioner for the stochastic Galerkin finite element methods. Numer. Linear Algebra Appl. 21(1), 136–151 (2014)
Steinwart, I., Scovel, C.: Mercer’s theorem on general domains: on the interaction between measures, kernels, and RKHSs. Constr. Approx. 35, 363–417 (2012)
Stone, M.: The genralized Weierstrass approximation theorem. Math. Mag. 21(4), 167–184 (1948)
Takemura, A., Takeuchi, K.: Some results on univariate and multivariate Cornish-Fisher expansion: algebraic properties and validity. Sankhy\(\breve{a}\) 50, 111–136 (1988)
Tan, W., Guttman, I.: On the construction of multi-dimensional orthogonal polynomials. Metron 34, 37–54 (1976)
Tavare, S., Balding, D., Griffiths, R., Donnelly, P.: Inferring coalescence times from dna sequence data. Genetics 145, 505–518 (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. 8964. Springer (2015). doi:10.1007/978–3–319–25138–7_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, 304–317 (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), 901–928 (2006)
Wiener, N.: Differential space. J. Math. Phys. 2, 131–174 (1923)
Wiener, N.: The homogeneous chaos. Am. J. Math. 60(4), 897–936 (1938)
Wintner, A., Wiener, N.: The discrete chaos. Am. J. Math. 65, 279–298 (1943)
Xiu, D., Karniadakis, G.: The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24, 619–644 (2002)
Xiu, D., Hesthaven, J.S.: High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27(3), 1118–1139 (2005)
Yamazaki, F., Shinozuka, M., Dasgupta, G.: Neumann expansion for stochastic finite-element analysis. ASCE J. Eng. Mech. 114(8), 1335–1354 (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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