Abstract
In this chapter, the authors survey the family of sparse stochastic collocation methods (SCMs) for partial differential equations with random input data. The SCMs under consideration can be viewed as a special case of the generalized stochastic finite element method (Gunzburger et al., Acta Numer 23:521–650, 2014), where the approximation of the solution dependences on the random variables is constructed using Lagrange polynomial interpolation. Relying on the “delta property” of the interpolation scheme, the physical and stochastic degrees of freedom can be decoupled, such that the SCMs have the same nonintrusive property as stochastic sampling methods but feature much faster convergence. To define the interpolation schemes or interpolatory quadrature rules, several approaches have been developed, including global sparse polynomial approximation, for which global polynomial subspaces (e.g., sparse Smolyak spaces (Nobile et al., SIAM J Numer Anal 46:2309–2345, 2008) or quasi-optimal subspaces (Tran et al., Analysis of quasi-optimal polynomial approximations for parameterized PDEs with deterministic and stochastic coefficients. Tech. Rep. ORNL/TM-2015/341, Oak Ridge National Laboratory, 2015)) are used to exploit the inherent regularity of the PDE solution, and local sparse approximation, for which hierarchical polynomial bases (Ma and Zabaras, J Comput Phys 228:3084–3113, 2009; Bungartz and Griebel, Acta Numer 13:1–123, 2004) or wavelet bases (Gunzburger et al., Lect Notes Comput Sci Eng 97:137–170, Springer, 2014) are used to accurately capture irregular behaviors of the PDE solution. All these method classes are surveyed in this chapter, including some novel recent developments. Details about the construction of the various algorithms and about theoretical error estimates of the algorithms are provided.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Babuska, I., Nobile, F., Tempone, R.: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 45, 1005–1034 (2007)
Babuška, I.M., Tempone, R., Zouraris, G.E.: Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42, 800–825 (2004) (electronic)
Babuška, I.M., 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, 1251–1294 (2005)
Barth, A., Lang, A., Schwab, C.: Multilevel Monte Carlo method for parabolic stochastic partial differential equations. BIT Numer. Math. 53, 3–27 (2013)
Beck, J., Nobile, F., Tamellini, L., Tempone, R.: Stochastic spectral Galerkin and collocation methods for PDEs with random coefficients: a numerical comparison. Lect. Notes Comput. Sci. Eng. 76, 43–62 (2011)
Beck, J., Nobile, F., Tamellini, L., Tempone, R.: Convergence of quasi-optimal stochastic Galerkin methods for a class of PDEs with random coefficients. Comput. Math. Appl. 67, 732–751 (2014)
Beck, J., Tempone, R., Nobile, F., Tamellni, L.: On the optimal polynomial approximation of stochastic PDEs by Galerkin and collocation methods. Math. Models Methods Appl. Sci. 22, 1250023 (2012)
Beck, M., Robins, S.: Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra. Springer, New York (2007)
Białas-Cież, L., Calvi, J.-P.: Pseudo Leja sequences. Annali di Matematica Pura ed Applicata 191, 53–75 (2012)
Bieri, M., Andreev, R., Schwab, C.: Sparse tensor discretization of elliptic sPDEs. SIAM J. Sci. Comput. 31, 4281–4304 (2009)
Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer, New York (1994)
Brutman, L.: On the Lebesgue function for polynomial interpolation. SIAM J. Numer. Anal. 15, 694–704 (1978)
Buffa, A., Maday, Y., Patera, A., Prud’homme, C., Turinici, G.: A priori convergence of the greedy algorithm for the parametrized reduced basis method. ESAIM: Math. Model. Numer. Anal. 46, 595–603 (2012)
Bungartz, H.-J., Griebel, M.: Sparse grids. Acta Numer. 13, 1–123 (2004)
Chkifa, A., Cohen, A., DeVore, R., Schwab, C.: Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs. Modél. Math. Anal. Numér. 47, 253–280 (2013)
Chkifa, A., Cohen, A., Schwab, C.: Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs. J. Math. Pures Appl. 103, 400–428 (2015)
Chkifa, M.A.: On the Lebesgue constant of Leja sequences for the complex unit disk and of their real projection. J. Approx. Theory 166, 176–200 (2013)
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, New York (1978)
Clenshaw, C.W., Curtis, A.R.: A method for numerical integration on an automatic computer. Numer. Math. 2, 197–205 (1960)
Cliffe, K.A., Giles, M.B., Scheichl, R., Teckentrup, A.L.: Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients. Comput. Vis. Sci. 14, 3–15 (2011)
Cohen, A., DeVore, R., Schwab, C.: Convergence rates of best n-term Galerkin approximations for a class of elliptic SPDEs. Found. Comput. Math. 10, 615–646 (2010)
Cohen, A., DeVore, R., Schwab, C.: Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDEs. Anal. Appl. 9, 11–47 (2011)
DeVore, R.: Nonlinear approximation. Acta Numer. 7, 51–150 (1998)
DeVore, R.A., Lorentz, G.G.: Constructive approximation. Volume 303 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (1993)
Dexter, N., Webster, C., Zhang, G.: Explicit cost bounds of stochastic Galerkin approximations for parameterized PDEs with random coefficients. ArXiv:1507.05545 (2015)
Dzjadyk, V.K., Ivanov, V.V.: On asymptotics and estimates for the uniform norms of the Lagrange interpolation polynomials corresponding to the Chebyshev nodal points. Anal. Math. 9, 85–97 (1983)
Elman, H., Miller, C.: Stochastic collocation with kernel density estimation. Tech. Rep., Department of Computer Science, University of Maryland (2011)
Elman, H.C., Miller, C.W., Phipps, E.T., Tuminaro, R.S.: Assessment of collocation and Galerkin approaches to linear diffusion equations with random data. Int. J. Uncertain. Quantif. 1, 19–33 (2011)
Fishman, G.: Monte Carlo. Springer Series in Operations Research. Springer, New York (1996)
Frauenfelder, P., Schwab, C., Todor, R.A.: Finite elements for elliptic problems with stochastic coefficients. Comput. Methods Appl. Mech. Eng. 194, 205–228 (2005)
Galindo, D., Jantsch, P., Webster, C.G., Zhang, G.: Accelerating stochastic collocation methods for partial differential equations with random input data. Tech. Rep. ORNL/TM-2015/219, Oak Ridge National Laboratory (2015)
Ganapathysubramanian, B., Zabaras, N.: Sparse grid collocation schemes for stochastic natural convection problems. J. Comput. Phys. 225, 652–685 (2007)
Gentleman, W.M.: Implementing Clenshaw-Curtis quadrature, II computing the cosine transformation. Commun. ACM 15, 343–346 (1972)
Gerstner, T., Griebel, M.: Numerical integration using sparse grids. Numer. Algorithms 18, 209–232 (1998)
Ghanem, R.G., Spanos, P.D.: Stochastic Finite Elements: A Spectral Approach. Springer, New York (1991)
Giles, M.B.: Multilevel Monte Carlo path simulation. Oper. Res. 56, 607–617 (2008)
Griebel, M.: Adaptive sparse grid multilevel methods for elliptic PDEs based on finite differences. Computing 61, 151–179 (1998)
Gruber, P.: Convex and Discrete Geometry. Springer Grundlehren der mathematischen Wissenschaften (2007)
Gunzburger, M., Jantsch, P., Teckentrup, A., Webster, C.G.: A multilevel stochastic collocation method for partial differential equations with random input data. SIAM/ASA J. Uncertainty Quantification 3, 1046–1074 (2015)
Gunzburger, M., Webster, C.G., Zhang, G.: An adaptive wavelet stochastic collocation method for irregular solutions of partial differential equations with random input data. Lect. Notes Comput. Sci. Eng. 97, 137–170. Springer (2014)
Gunzburger, M.D., Webster, C.G., Zhang, G.: Stochastic finite element methods for partial differential equations with random input data. Acta Numer. 23, 521–650 (2014)
Hansen, M., Schwab, C.: Analytic regularity and nonlinear approximation of a class of parametric semilinear elliptic PDEs. Math. Nachr. 286, 832–860 (2013)
Hansen, M., Schwab, C.: Sparse adaptive approximation of high dimensional parametric initial value problems. Vietnam J. Math. 41, 181–215 (2013)
Hoang, V.H., Schwab, C.: Sparse tensor Galerkin discretizations for parametric and random parabolic PDEs – analytic regularity and generalized polynomial chaos approximation. SIAM J. Math. Anal. 45, 3050–3083 (2013)
Jakeman, J.D., Archibald, R., Xiu, D.: Characterization of discontinuities in high-dimensional stochastic problems on adaptive sparse grids. J. Comput. Phys. 230, 3977–3997 (2011)
Kuo, F.Y., Schwab, C., Sloan, I.H.: Quasi-Monte Carlo methods for high-dimensional integration: the standard (weighted Hilbert space) setting and beyond. The ANZIAM J. Aust. N. Z. Ind. Appl. Math. J. 53, 1–37 (2011)
Kuo, F.Y., Schwab, C., Sloan, I.H.: Quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients. SIAM J. Numer. Anal. 50, 3351–3374 (2012)
Li, C.F., Feng, Y.T., Owen, D.R.J., Li, D.F., Davis, I.M.: A Fourier-Karhunen-Loève discretization scheme for stationary random material properties in SFEM. Int. J. Numer. Methods Eng. 73, 1942–1965 (2007)
Loève, M.: Probability Theory. I. Graduate Texts in Mathematics, vol. 45, 4th edn. Springer, New York (1977)
Loève, M.: Probability Theory. II. Graduate Texts in Mathematics, vol. 46, 4th edn. Springer, New York (1978)
Ma, X., Zabaras, N.: An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations. J. Comput. Phys. 228, 3084–3113 (2009)
Ma, X., Zabaras, N.: An adaptive high-dimensional stochastic model representation technique for the solution of stochastic partial differential equations. J. Comput. Phys. 229, 3884–3915 (2010)
Maday, Y., Nguyen, N., Patera, A., Pau, S.: A general multipurpose interpolation procedure: the magic points. Commun. Pure Appl. Anal. 8, 383–404 (2009)
Mathelin, L., Hussaini, M.Y., Zang, T.A.: Stochastic approaches to uncertainty quantification in CFD simulations. Numer. Algorithms 38, 209–236 (2005)
Matthies, H.G., Keese, A.: Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Eng. 194, 1295–1331 (2005)
Milani, R., Quarteroni, A., Rozza, G.: Reduced basis methods in linear elasticity with many parameters. Comput. Methods Appl. Mech. Eng. 197, 4812–4829 (2008)
Nobile, F., Tempone, R., Webster, C.G.: An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal. 46, 2411–2442 (2008)
Nobile, F., Tempone, R., Webster, C.G.: A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal. 46, 2309–2345 (2008)
Sauer, T., Xu, Y.: On multivariate Lagrange interpolation. Math. Comput. 64, 1147–1170 (1995)
Smith, S.J.: Lebesgue constants in polynomial interpolation. Annales Mathematicae et Informaticae. Int. J. Math. Comput. Sci. 33, 109–123 (2006)
Smolyak, S.: Quadrature and interpolation formulas for tensor products of certain classes of functions. Dokl. Akad. Nauk SSSR 4, 240–243 (1963) (English translation)
Stoyanov, M., Webster, C.G.: A gradient-based sampling approach for dimension reduction for partial differential equations with stochastic coefficients. Int. J. Uncertain. Quantif. 5, 49-72 (2015)
Sweldens, W.: The lifting scheme: a custom-design construction of biorthogonal wavelets. Appl. Comput. Harmonic Anal. 3, 186–200 (1996)
Sweldens, W.: The lifting scheme: a construction of second generation wavelets. SIAM J. Math. Anal. 29, 511–546 (1998)
Todor, R.A.: Sparse perturbation algorithms for elliptic PDE’s with stochastic data. Diss. No. 16192, ETH Zurich (2005)
Tran, H., Webster, C.G., Zhang, G.: Analysis of quasi-optimal polynomial approximations for parameterized PDEs with deterministic and stochastic coefficients. Tech. Rep. ORNL/TM-2015/341, Oak Ridge National Laboratory (2015)
Gunzburger, M., Jantsch, P., Teckentrup, A., Webster, C.G.: A multilevel stochastic collocation method for partial differential equations with random input data. Tech. Rep. ORNL/TM-2014/621, Oak Ridge National Laboratory (2014)
Trefethen, L.N.: Is gauss quadrature better than Clenshaw-Curtis? SIAM Rev. 50, 67–87 (2008)
Webster, C.G.: Sparse grid stochastic collocation techniques for the numerical solution of partial differential equations with random input data. PhD thesis, Florida State University (2007)
Wiener, N.: The homogeneous chaos. Am. J. Math. 60, 897–936 (1938)
Xiu, D., Hesthaven, J.S.: High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27, 1118–1139 (2005)
Xiu, D., Karniadakis, G.E.: The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24, 619–644 (2002)
Zhang, G., Gunzburger, M.: Error analysis of a stochastic collocation method for parabolic partial differential equations with random input data. SIAM J. Numer. Anal. 50, 1922–1940 (2012)
Zhang, G., Webster, C., Gunzburger, M., Burkardt, J.: A hyper-spherical adaptive sparse-grid method for high-dimensional discontinuity detection. SIAM J. Numer. Anal. 53, 1508–1536 (2015)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing Switzerland
About this entry
Cite this entry
Gunzburger, M., Webster, C.G., Zhang, G. (2017). Sparse Collocation Methods for Stochastic Interpolation and Quadrature. In: Ghanem, R., Higdon, D., Owhadi, H. (eds) Handbook of Uncertainty Quantification. Springer, Cham. https://doi.org/10.1007/978-3-319-12385-1_29
Download citation
DOI: https://doi.org/10.1007/978-3-319-12385-1_29
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-12384-4
Online ISBN: 978-3-319-12385-1
eBook Packages: Mathematics and StatisticsReference Module Computer Science and Engineering