In this work we analyze the dimension-independent convergence property of an abstract sparse quadrature scheme for numerical integration of functions of high-dimensional parameters with Gaussian measure. Under certain assumptions on the exactness and boundedness of univariate quadrature rules as well as on the regularity assumptions on the parametric functions with respect to the parameters, we prove that the convergence of the sparse quadrature error is independent of the number of the parameter dimensions. Moreover, we propose both an a priori and an a posteriori schemes for the construction of a practical sparse quadrature rule and perform numerical experiments to demonstrate their dimension-independent convergence rates.
Accepté le :
DOI : 10.1051/m2an/2018012
Mots clés : Uncertainty quantification, high-dimensional integration, curse of dimensionality, convergence analysis, Gaussian measure, sparse grids, a priori construction, a posteriori construction
@article{M2AN_2018__52_2_631_0,
author = {Chen, Peng},
title = {Sparse quadrature for high-dimensional integration with {Gaussian} measure},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {631--657},
publisher = {EDP-Sciences},
volume = {52},
number = {2},
year = {2018},
doi = {10.1051/m2an/2018012},
mrnumber = {3834438},
zbl = {06966736},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2018012/}
}
TY - JOUR AU - Chen, Peng TI - Sparse quadrature for high-dimensional integration with Gaussian measure JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 631 EP - 657 VL - 52 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2018012/ DO - 10.1051/m2an/2018012 LA - en ID - M2AN_2018__52_2_631_0 ER -
%0 Journal Article %A Chen, Peng %T Sparse quadrature for high-dimensional integration with Gaussian measure %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 631-657 %V 52 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2018012/ %R 10.1051/m2an/2018012 %G en %F M2AN_2018__52_2_631_0
Chen, Peng. Sparse quadrature for high-dimensional integration with Gaussian measure. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 2, pp. 631-657. doi : 10.1051/m2an/2018012. http://archive.numdam.org/articles/10.1051/m2an/2018012/
[1] and , Handbook of Mathematical Functions. Vol. 55 of Applied Mathematics Series (1966). | MR
[2] , , and , A stochastic collocation method for elliptic partial differential equations with random input data. SIAM Rev. 52 (2010) 317–355. | DOI | MR | Zbl
[3] , , and , Sparse polynomial approximation of parametric elliptic PDEs. Part ii: lognormal coefficients. ESAIM: M2AN 51 (2017) 341–363. | DOI | Numdam | MR | Zbl
[4] , , and , Stochastic spectral Galerkin and collocation methods for PDEs with random coefficients: a numerical comparison, in Spectral and High Order Methods for Partial Differential Equations, edited by and . Springer-Verlag, Berlin (2011) 43–62. | DOI | MR | Zbl
[5] , , and A quasi-optimal sparse grids procedure for groundwater flows, in Spectral and High Order Methods for Partial Differential Equations-ICOSAHOM 2012. Springer (2014) 1–16. | MR | Zbl
[6] and , Sparse grids. Acta Numerica 13 (2004) 147–269. | DOI | MR | Zbl
[7] , Monte Carlo and quasi-Monte Carlo methods. | DOI | MR | Zbl
[8] ,Strong and weak error estimates for elliptic partial differential equations with random coefficients. SIAM J. Numer. Anal. 50 (2012) 216–246. | DOI | MR | Zbl
[9] and , Weighted reduced basis method for stochastic optimal control problems with elliptic PDE constraints. SIAM/ASA J. Uncertain. Quantif . 2 (2014) 364–396. | DOI | MR | Zbl
[10] and , A new algorithm for high-dimensional uncertainty quantification based on dimension-adaptive sparse grid approximation and reduced basis methods. J. Comput. Phys. 298 (2015) 176–193. | DOI | MR | Zbl
[11] and , Sparse-grid, reduced-basis Bayesian inversion. Comput. Methods Appl. Mech. Eng. 297 (2015) 84–115. | DOI | MR | Zbl
[12] and , Sparse-grid, reduced-basis Bayesian inversion: nonaffine-parametric nonlinear equations. J. Comput. Phys. 316 (2016) 470–503. | DOI | MR | Zbl
[13] , and , Hessian-based adaptive sparse quadrature for infinite-dimensional Bayesian inverse problems. Comput. Methods Appl. Mech. Eng. 327 (2017) 147–172. | DOI | MR | Zbl
[14] , and , High-dimensional adaptive sparse polynomial interpolation and applications to parametric pdes. Found. Comput. Math. 14 (2014) 601–633. | DOI | MR | Zbl
[15] , and , Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs. J. Math. Pures Appl. 103 (2015) 400–428. | DOI | MR | Zbl
[16] , and , Convergence rates of best N-term Galerkin approximations for a class of elliptic sPDEs. Found. Comput. Math. 10 (2010) 615–646. | DOI | MR | Zbl
[17] , and , Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDEs. Anal. Appl. 9 (2011) 11–47. | DOI | MR | Zbl
[18] and , Stochastic collocation for elliptic PDEs with random data: the lognormal case, in Sparse Grids and Applications-Munich 2012. Springer (2014) 29–53 | DOI | MR | Zbl
[19] , , and , Convergence of sparse collocation for functions of countably many Gaussian random variables – with application to lognormal elliptic diffusion problems. Preprint arxiv: (2016). | arXiv | MR
[20] and , Fully symmetric interpolatory rules for multiple integrals over infinite regions with Gaussian weight. J. Comput. Appl. Math. 71 (1996) 299–309 | DOI | MR | Zbl
[21] and , Numerical integration using sparse grids. Numer. Algorithms, 18 (1998) 209–232 | DOI | MR | Zbl
[22] and , Dimension–adaptive tensor–product quadrature. Computing 71(2003) 65–87 | DOI | MR | Zbl
[23] and , Stochastic Finite Elements: a Spectral Approach. Dover Civil and Mechanical Engineering. Courier Dover Publication, Springer-Verlag, New York (1991). | DOI | MR | Zbl
[24] , , and , Numerical Methods for Special Functions. SIAM (2007). | MR | Zbl
[25] , Stochastic galerkin discretization of the log-normal isotropic diffusion problem. Math. Model. Methods Appl. Sci. 20 (2010) 237–263. | DOI | MR | Zbl
[26] , , , , and , Quasi-Monte Carlo finite element methods for elliptic PDEs with lognormal random coefficients. Numer. Math. 131 (2015) 1–40. | DOI | MR
[27] and , Dimension-wise integration of high-dimensional functions with applications to finance. J. Complex. 26 2010 455–489. | DOI | MR | Zbl
[28] and , N-term Wiener chaos approximation rates for elliptic PDEs with lognormal gaussian random inputs. Math. Model. Methods Appl. Sci. 24 (2014) 797–826. | DOI | MR | Zbl
[29] , Uncertainty Modeling Using Fuzzy Arithmetic and Sparse Grids. PhD thesis, Universität Stuttgart, Germany (2006). | Zbl
[30] , Nodes and Weights of Quadrature Formulas: sixteen-place tables. Consultants Bureau, NY (1965). | MR | Zbl
[31] , , , and , Multilevel quasi-Monte Carlo methods for lognormal diffusion problems. Math. Comput. 86 (2017) 2827–2860. | DOI | MR | Zbl
[32] and , Introduction: Uncertainty Quantification and Propagation. Springer (2010).
[33] and , Probabilistic collocation method for flow in porous media: comparisons with other stochastic methods. Water Resour. Res. 43 (2007).
[34] and , An efficient, high-order probabilistic collocation method on sparse grids for three-dimensional flow and solute transport in randomly heterogeneous porous media. Adv. Water Res. 32 (2009) 712–722. | DOI
[35] and , An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations. J. Comput. Phys. 228 (2009) 3084–3113. | DOI | MR | Zbl
[36] , Mean convergence of Lagrange interpolation, II. J. Approx. Theory 30 (1980) 263–276. | DOI | MR | Zbl
[37] , and , An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal. 46 (2008) 2411–2442. | DOI | MR | Zbl
[38] , and , A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal. 46 (2008) 2309–2345. | DOI | MR | Zbl
[39] , and , Convergence of quasi-optimal sparse grid approximation of Hilbert-valued functions: application to random elliptic PDEs. Numer. Math. 134 (2016) 343–388. | DOI | MR | Zbl
[40] , , and , An adaptive sparse grid algorithm for elliptic PDEs with lognormal diffusion coefficient, in Sparse Grids and Applications-Stuttgart 2014. Springer (2016) 191–220. | DOI | MR | Zbl
[41] , The optimum addition of points to quadrature formulae. Math. Comput. 22 (1968) 847–856. | DOI | MR | Zbl
[42] , and , Efficient shape optimization for certain and uncertain aerodynamic design. Comput. Fluids 46 (2011) 78–87 | DOI | MR | Zbl
[43] and , Sparse, adaptive Smolyak quadratures for Bayesian inverse problems. Inverse Probl. 29 (2013) 065011. | DOI | MR | Zbl
[44] and , Sparsity in Bayesian inversion of parametric operator equations. Inverse Probl. 30 (2014) 065007. | DOI | MR | Zbl
[45] and , Karhunen–Loève approximation of random fields by generalized fast multipole methods. J. Comput. Phys. 217 (2006) 100–122. | DOI | MR | Zbl
[46] , Uncertainty quantification: theory, implementation, and applications. Vol. 12. SIAM (2013). | MR | Zbl
[47] , Quadrature and interpolation formulas for tensor products of certain classes of functions. Dokl. Akad. Nauk SSSR 4 (1963) 240–243. | MR | Zbl
[48] , Orthogonal Polynomials. Vol. 23. American Mathematical Society (1939). | MR | Zbl
[49] , Numerical Methods for Stochastic Computations: a Spectral Method Approach. Princeton University Press (2010). | MR | Zbl
[50] and , High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27 (2005) 1118–1139. | DOI | MR | Zbl
[51] and , Convergence rates of high dimensional Smolyak quadrature. Technical Report 2017-27, Seminar for Applied Mathematics. ETH Zürich, Switzerland (2017).
Cité par Sources :
