×

Reduced basis ANOVA methods for partial differential equations with high-dimensional random inputs. (English) Zbl 1349.65684

Summary: In this paper we present a reduced basis ANOVA approach for partial deferential equations (PDEs) with random inputs. The ANOVA method combined with stochastic collocation methods provides model reduction in high-dimensional parameter space through decomposing high-dimensional inputs into unions of low-dimensional inputs. In this work, to further reduce the computational cost, we investigate spatial low-rank structures in the ANOVA-collocation method, and develop efficient spatial model reduction techniques using hierarchically generated reduced bases. We present a general mathematical framework of the methodology, validate its accuracy and demonstrate its efficiency with numerical experiments.

MSC:

65N75 Probabilistic methods, particle methods, etc. for boundary value problems involving PDEs
62J10 Analysis of variance and covariance (ANOVA)
65C50 Other computational problems in probability (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Xiu, D.; Karniadakis, G., The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput., 24, 619-644 (2002) · Zbl 1014.65004
[2] Ghanem, R.; Spanos, P., Stochastic Finite Elements: A Spectral Approach (2003), Dover Publications: Dover Publications New York
[3] Xiu, D.; Hesthaven, J., High-order collocation methods for differential equations with random inputs, SIAM J. Sci. Comput., 27, 1118-1139 (2005) · Zbl 1091.65006
[4] 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) · Zbl 1176.65137
[5] Xiu, D., Numerical Methods for Stochastic Computations: A Spectral Method Approach (2010), Princeton University Press: Princeton University Press Princeton · Zbl 1210.65002
[6] Elman, H.; Liao, Q., Reduced basis collocation methods for partial differential equations with random coefficients, SIAM/ASA J. Uncertain. Quantificat., 1, 192-217 (2013) · Zbl 1282.35424
[7] Fisher, R., Statistical Methods for Research Workers (1925), Oliver and Boyd: Oliver and Boyd Berlin · JFM 51.0414.08
[8] Cao, Y.; Chen, Z.; Gunzburger, M., ANOVA expansions and efficient sampling methods for parameter dependent nonlinear PDEs, Int. J. Numer. Anal. Model., 6, 256-273 (2009) · Zbl 1161.65336
[9] Winter, C.; Guadagnini, A.; Nychka, D.; Tartakovsky, D., Multivariate sensitivity analysis of saturated flow through simulated highly heterogeneous groundwater aquifers, J. Comput. Phys., 217, 166-175 (2009) · Zbl 1146.86001
[10] Foo, J.; Karniadakis, G., Multi-element probabilistic collocation in high dimensions, J. Comput. Phys., 229, 1536-1557 (2010) · Zbl 1181.65014
[11] Gao, Z.; Hesthaven, J. S., On anova expansions and strategies for choosing the anchor point, Appl. Math. Comput., 217, 3274-3285 (2010) · Zbl 1206.65125
[12] 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) · Zbl 1189.65019
[13] Zhang, Z.; Choi, M.; Karniadakis, G., Anchor points matter in anova decomposition, (Spectral and High Order Methods for Partial Differential Equations. Spectral and High Order Methods for Partial Differential Equations, Lect. Notes Comput. Sci. Eng., vol. 76 (2011)), 347-355 · Zbl 1217.65039
[14] Yang, X.; Choi, M.; Lin, G.; Karniadakis, G. E., Adaptive anova decomposition of stochastic incompressible and compressible flows, J. Comput. Phys., 231, 1587-1614 (2012) · Zbl 1408.76428
[15] Hesthaven, J. S.; Zhang, S., On the use of ANOVA expansions in reduced basis methods for high-dimensional parametric partial differential equations, J. Sci. Comput. (2016), in press · Zbl 1352.65508
[16] Bungartz, H. J.; Griebel, M., Sparse grids, Acta Numer., 13, 147-269 (2004) · Zbl 1118.65388
[17] Ganapathysubramanian, B.; Zabaras, N., Sparse grid collocation schemes for stochastic natural convection problems, J. Comput. Phys., 225, 652-685 (2007) · Zbl 1343.76059
[18] Xiu, D., Efficient collocational approach for parametric uncertainty analysis, Commun. Comput. Phys., 2, 293-309 (2007) · Zbl 1164.65302
[19] Agarwal, N.; Aluru, N. R., A domain adaptive stochastic collocation approach for analysis of MEMS under uncertainties, J. Comput. Phys., 194, 7662-7688 (2009) · Zbl 1391.74293
[20] 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) · Zbl 1161.65006
[21] Wan, X.; Karniadakis, G. E., An adaptive multi-element generalized polynomial chaos method for stochastic differential equations, J. Comput. Phys., 209, 617-642 (2005) · Zbl 1078.65008
[22] Doostan, A.; Owhadi, H., A non-adaptive sparse approximation for PDEs with stochastic inputs, J. Comput. Phys., 230, 3015-3034 (2011) · Zbl 1218.65008
[23] Yan, L.; Guo, L.; Xiu, D., Stochastic collocation algorithms using L1-minimization, Int. J. Uncertain. Quantificat., 3, 279-293 (2012) · Zbl 1291.65024
[24] Karagiannis, G.; Konomi, B.; Lin, G., Mixed shrinkage prior procedure for basis selection and global evaluation of gPC expansions in Bayesian framework: applications to elliptic SPDEs, J. Comput. Phys., 284, 528-546 (2015) · Zbl 1352.65634
[25] Eftang, J.; Patera, A.; Rønquist, E., An “hp” certified reduced basis method for parametrized elliptic partial differential equations, SIAM J. Sci. Comput., 32, 3170-3200 (2010) · Zbl 1228.35097
[26] Peherstorfer, B.; Butnaru, D.; Willcox, K.; Bungartz, H., Localized discrete empirical interpolation method, SIAM J. Sci. Comput., 36, A168-A192 (2014) · Zbl 1290.65080
[27] Albrecht, F.; Haasdonk, B.; Ohlberger, M.; Kaulmann, S., The localized reduced basis multiscale method, (Proceedings of ALGORITMY 2012, Conference on Scientific Computing (2012), Slovak University of Technology in Bratislava, Publishing House of STU: Slovak University of Technology in Bratislava, Publishing House of STU Vysoke Tatry, Podbanske), 393-403 · Zbl 1278.65172
[28] Huynh, D.; Knezevic, D.; Patera, A., A static condensation reduced basis element method: approximation and a posteriori error estimation, Modél. Math. Anal. Numér., 47, 213-251 (2013) · Zbl 1276.65082
[29] Maier, I.; Haasdonk, B., A Dirichlet-Neumann reduced basis method for homogeneous domain decomposition problems, Appl. Numer. Math., 78, 31-48 (2014) · Zbl 1282.65166
[30] Sarkar, A.; Benabbou, N.; Ghanem, R., Domain decomposition of stochastic PDEs: theoretical formulations, Int. J. Numer. Methods Eng., 77, 689-701 (2009) · Zbl 1156.74385
[31] Hadigol, M.; Doostan, A.; Matthies, H. G.; Niekamp, R., Partitioned treatment of uncertainty in coupled domain problems: a separated representation approach, Comput. Methods Appl. Mech. Eng., 274, 103-124 (2014) · Zbl 1296.65158
[32] Chen, Y.; Jakeman, J.; Gittelson, C.; Xiu, D., Local polynomial chaos expansion for linear differential equations with high dimensional random inputs, SIAM J. Sci. Comput., 37, A79-A102 (2015) · Zbl 1330.65189
[33] Alexander, F. J.; Garcia, A. L.; Tartakovsky, D. M., Algorithm refinement for stochastic partial differential equations: II. Correlated systems, J. Comput. Phys., 207, 769-787 (2005) · Zbl 1072.65006
[34] 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) · Zbl 1352.65014
[35] Amaral, S.; Allaire, D.; Willcox, K., A decomposition-based approach to uncertainty analysis of feed-forward multicomponent systems, Int. J. Numer. Methods Eng., 100, 982-1005 (2014) · Zbl 1352.93016
[36] Liao, Q.; Willcox, K., A domain decomposition approach for uncertainty analysis, SIAM J. Sci. Comput., 37, A103-A133 (2015) · Zbl 1327.35464
[37] Ganapathysubramanian, B.; Zabaras, N., Modeling diffusion in random heterogeneous media: data-driven models, stochastic collocation and the variational multiscale method, J. Comput. Phys., 226, 326-353 (2007) · Zbl 1124.65007
[38] Chen, P.; Quarteroni, A.; Rozza, G., Comparison between reduced basis and stochastic collocation methods for elliptic problems, J. Sci. Comput., 59, 187-216 (2014) · Zbl 1301.65007
[39] Chen, P.; Quarteroni, A., A new algorithm for high-dimensional uncertainty quantification based on dimension-adaptive sparse grid approximation and reduced basis methods, J. Comput. Phys., 298, 176-193 (2015) · Zbl 1349.65683
[40] Sobol, I., Theorems and examples on high dimensional model representation, Reliab. Eng. Syst. Saf., 79, 187-193 (2003)
[41] Elman, H.; Forstall, V., Preconditioning techniques for reduced basis methods for parameterized elliptic partial differential equations, SIAM J. Sci. Comput., 37, S177-S194 (2015) · Zbl 1457.65174
[42] Benner, P.; Gugercin, S.; Willcox, K., A survey of projection-based model reduction methods for parametric dynamical systems, SIAM Rev., 57, 483-531 (2015) · Zbl 1339.37089
[43] Sirovich, L., Turbulence and the dynamics of coherent structures, Part I: Coherent structures, Q. Appl. Math., 45, 561-571 (1987) · Zbl 0676.76047
[44] Holmes, P.; Lumley, J. L.; Berkooz, G., Turbulence, Coherent Structures, Dynamical Systems and Symmetry (1996), Cambridge Univ. Press: Cambridge Univ. Press New York · Zbl 0890.76001
[45] Gunzburger, M.; Peterson, J.; Shadid, J., Reduced-order modeling of time-dependent PDEs with multiple parameters in the boundary data, Comput. Methods Appl. Mech. Eng., 196, 1030-1047 (2007) · Zbl 1121.65354
[46] Veroy, K.; Rovas, D.; Patera, A., A posteriori error estimation for reduced-basis approximation of parametrized elliptic coercive partial differential equations: “Convex Inverse” bound conditioners, ESAIM Control Optim. Calc. Var., 8, 1007-1028 (2002) · Zbl 1092.35031
[47] Nguyen, N.; Veroy, K.; Patera, A., Certified real-time solution of parametrized partial differential equations, (Yip, S., Handbook of Materials Modeling (2005), Springer), 1523-1558
[48] Haasdonk, B.; Ohlberger, M., Reduced basis method for finite volume approximations of parametrized linear evolution equations, Modél. Math. Anal. Numér., 42, 277-302 (2008) · Zbl 1388.76177
[49] Boyaval, S.; Bris, C. L.; Lelièvre, T.; Maday, Y.; Nguyen, N.; Patera, A., Reduced basis techniques for stochastic problems, Arch. Comput. Methods Eng., 17, 1-20 (2010) · Zbl 1269.65005
[51] Quarteroni, A.; Manzoni, A.; Negri, F., Reduced Basis Methods for Partial Differential Equations (2016), Springer International Publishing · Zbl 1337.65113
[52] Bui-Thanh, T.; Willcox, K.; Ghattas, O., Model reduction for large-scale systems with high-dimensional parametric input space, SIAM J. Sci. Comput., 30, 3270-3288 (2008) · Zbl 1196.37127
[53] Rozza, G.; Huynh, D.; Patera, A., Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations: application to transport and continuum mechanics, Arch. Comput. Methods Eng., 15, 229-275 (2008) · Zbl 1304.65251
[54] Grepl, M.; Maday, Y.; Nguyen, N.; Patera, A., Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations, Modél. Math. Anal. Numér., 41, 575-605 (2007) · Zbl 1142.65078
[55] Chaturantabut, S.; Sorensen, D., Nonlinear model reduction via discrete empirical interpolation, SIAM J. Sci. Comput., 32, 2737-2764 (2010) · Zbl 1217.65169
[56] Elman, H.; Miller, C.; Phipps, E.; Tuminaro, R., Assessment of collocation and Galerkin approaches to linear diffusion equations with random data, Int. J. Uncertain. Quantificat., 1, 19-34 (2011) · Zbl 1229.65026
[57] Braess, D., Finite Elements (1997), Cambridge University Press: Cambridge University Press London
[58] Elman, H.; Silvester, D.; Wathen, A., Finite Elements and Fast Iterative Solvers (2005), Oxford University Press: Oxford University Press New York
[59] Babuška, 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) · Zbl 1151.65008
[60] Brown, J. L., Mean square truncation error in series expansions of random functions, J. Soc. Ind. Appl. Math., 8, 28-32 (1960) · Zbl 0094.12205
[61] Schwab, C.; Todor, R. A., Karhunen-Loève approximation of random fields by generalized fast multipole methods, J. Comput. Phys., 217, 100-122 (2006) · Zbl 1104.65008
[62] Klimke, A., Sparse grid interpolation toolbox - user’s guide (2007), University of Stuttgart, Tech. Rep. IANS report 2007/017
[63] Tang, K.; Congedo, P. M.; Abgrall, R., Sensitivity analysis using anchored ANOVA expansion and high-order moments computation, Int. J. Numer. Methods Eng., 102, 1554-1584 (2015) · Zbl 1352.62120
[64] Ng, L.; Willcox, K., Multifidelity approaches for optimization under uncertainty, Int. J. Numer. Methods Eng., 100, 746-772 (2014) · Zbl 1352.74230
[65] Zhu, X.; Narayan, A.; Xiu, D., Computational aspects of stochastic collocation with multifidelity models, SIAM/ASA J. Uncertain. Quantificat., 2, 444-463 (2014) · Zbl 1306.65010
[66] Powell, C.; Silvester, D., Preconditioning steady-state Navier-Stokes equations with random data, SIAM J. Sci. Comput., 34, A2482-A2506 (2012) · Zbl 1256.35216
[67] Chen, P.; Quarteroni, A.; Rozza, G., Multilevel and weighted reduced basis method for stochastic optimal control problems constrained by Stokes equations, Numer. Math., 1-36 (2015)
[68] Quarteroni, A.; Rozza, G., Numerical solution of parametrized Navier-Stokes equations by reduced basis methods, Numer. Methods Partial Differ. Equ., 23, 923-948 (2007) · Zbl 1178.76238
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.