##
**Efficient reduced basis methods for saddle point problems with applications in groundwater flow.**
*(English)*
Zbl 1398.65282

### MSC:

65N22 | Numerical solution of discretized equations for boundary value problems involving PDEs |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

35R60 | PDEs with randomness, stochastic partial differential equations |

60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |

35Q35 | PDEs in connection with fluid mechanics |

76M10 | Finite element methods applied to problems in fluid mechanics |

76S05 | Flows in porous media; filtration; seepage |

### Keywords:

reduced basis methods; saddle point problems; stochastic collocation; uncertainty quantification
PDF
BibTeX
XML
Cite

\textit{C. J. Newsum} and \textit{C. E. Powell}, SIAM/ASA J. Uncertain. Quantif. 5, 1248--1278 (2017; Zbl 1398.65282)

Full Text:
DOI

### References:

[1] | I. Babuška, F. Nobile, and R. Tempone, A stochastic collocation method for elliptic partial differential equations with random input data, SIAM J. Numer. Anal., 45 (2007), pp. 1005–1034, . · Zbl 1151.65008 |

[2] | I. Babuška, R. Tempone, and G. E. Zouraris, Galerkin finite element approximations of stochastic elliptic partial differential equations, SIAM J. Numer. Anal., 42 (2004), pp. 800–825, . · Zbl 1080.65003 |

[3] | M. Barrault, Y. Maday, N. C. Nguyen, and A. T. Patera, An ï¿½empirical interpolationï¿½ method: Application to efficient reduced-basis discretization of partial differential equations, C. R. Math., 339 (2004), pp. 667–672, . · Zbl 1061.65118 |

[4] | V. Barthelmann, E. Novak, and K. Ritter, High dimensional polynomial interpolation on sparse grids, Adv. Comput. Math., 12 (2000), pp. 273–288, . · Zbl 0944.41001 |

[5] | P. Benner and M. W. Hess, Reduced Basis Modeling for Uncertainty Quantification of Electromagnetic Problems in Stochastically Varying Domains, in Scientific Computing in Electrical Engineering, Springer, New York, 2016, pp. 215–222. · Zbl 1383.78050 |

[6] | M. Benzi, G. H. Golub, and J. Liesen, Numerical solution of saddle point problems, Acta Numer., 14 (2005), pp. 1–137, . · Zbl 1115.65034 |

[7] | A. Bespalov, C. E. Powell, and D. J. Silvester, A priori error analysis of stochastic Galerkin mixed approximations of elliptic PDEs with random data, SIAM J. Numer. Anal., 50 (2012), pp. 2039–2063. · Zbl 1253.35228 |

[8] | P. Binev, A. Cohen, W. Dahmen, R. DeVore, G. Petrova, and P. Wojtaszczyk, Convergence rates for greedy algorithms in reduced basis methods, SIAM J. Math. Anal., 43 (2011), pp. 1457–1472. · Zbl 1229.65193 |

[9] | D. Boffi, F. Brezzi, and M. Fortin, Mixed Finite Element Methods and Applications, Springer, New York, 2013. · Zbl 1277.65092 |

[10] | S. Boyaval, C. Le Bris, T. Leliï¿½vre, Y. Maday, N. Nguyen, and A. Patera, Reduced basis techniques for stochastic problems, Arch. Comput. Methods Eng., 17 (2010), pp. 435–454, . · Zbl 1269.65005 |

[11] | S. C. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods, Vol. 15, Springer, New York, 2008. · Zbl 1135.65042 |

[12] | S. Chaturantabut and D. C. Sorensen, Nonlinear model reduction via discrete empirical interpolation, SIAM J. Sci. Comput., 32 (2010), pp. 2737–2764. · Zbl 1217.65169 |

[13] | P. Chen and A. Quarteroni, A new algorithm for high-dimensional uncertainty quantification based on dimension-adaptive sparse grid approximation and reduced basis methods, J. Comput. Phys., 298 (2015), pp. 176–193. · Zbl 1349.65683 |

[14] | P. Chen, A. Quarteroni, and G. Rozza, A weighted reduced basis method for elliptic partial differential equations with random input data, SIAM J. Numer. Anal., 51 (2013), pp. 3163–3185. · Zbl 1288.65007 |

[15] | P. Chen, A. Quarteroni, and G. Rozza, Comparison between reduced basis and stochastic collocation methods for elliptic problems, J. Sci. Comput., 59 (2014), pp. 187–216, . · Zbl 1301.65007 |

[16] | P. Chen, A. Quarteroni, and G. Rozza, Multilevel and weighted reduced basis method for stochastic optimal control problems constrained by Stokes equations, Numer. Math., 133 (2016), pp. 67–102. · Zbl 1344.93109 |

[17] | P. Chen and C. Schwab, Sparse-grid, reduced-basis bayesian inversion, Comput. Methods Appl. Mech. Engrg., 297 (2015), pp. 84–115. · Zbl 1425.65020 |

[18] | K. Cliffe, I. G. Graham, R. Scheichl, and L. Stals, Parallel computation of flow in heterogeneous media modelled by mixed finite elements, J. Comput. Phys., 164 (2000), pp. 258–282. · Zbl 0995.76044 |

[19] | M. Eigel, C. J. Gittelson, C. Schwab, and E. Zander, Adaptive stochastic Galerkin FEM, Comput. Methods Appl. Mech. Engrg., 270 (2014), pp. 247–269. · Zbl 1296.65157 |

[20] | H. C. Elman and V. Forstall, Preconditioning techniques for reduced basis methods for parameterized elliptic partial differential equations, SIAM J. Sci. Comput., 37 (2015), pp. S177–S194, . · Zbl 1457.65174 |

[21] | H. C. Elman and V. Forstall, Numerical solution of the parameterized steady-state Navier–Stokes equations using Empirical interpolation methods, Comput. Methods Appl. Mech. Engrg., 317 (2017), pp. 380-399. . |

[22] | H. C. Elman and Q. Liao, Reduced basis collocation methods for partial differential equations with random coefficients, SIAM/ASA J. Uncertain. Quantif., 1 (2013), pp. 192–217, . · Zbl 1282.35424 |

[23] | B. Ganis, H. Klie, M. F. Wheeler, T. Wildey, I. Yotov, and D. Zhang, Stochastic collocation and mixed finite elements for flow in porous media, Comput. Methods Appl. Mech. Engrg., 197 (2008), pp. 3547–3559. · Zbl 1194.76242 |

[24] | A.-L. Gerner and K. Veroy, Certified reduced basis methods for parametrized saddle point problems, SIAM J. Sci. Comput., 34 (2012), pp. A2812–A2836. · Zbl 1255.76024 |

[25] | A. D. Gordon and C. E. Powell, On solving stochastic collocation systems with algebraic multigrid, IMA J. Numer. Anal., 32 (2012), pp. 1051–1070. · Zbl 1248.65007 |

[26] | I. G. Graham, R. Scheichl, and E. Ullmann, Mixed finite element analysis of lognormal diffusion and multilevel Monte Carlo methods, Stoch. Partial Differ. Equ. Anal. Comput., 4 (2016), pp. 41–75. · Zbl 1371.65008 |

[27] | A. Greenbaum, Iterative Methods for Solving Linear Systems, Vol. 17, SIAM, Philadelphia, 1997. · Zbl 0883.65022 |

[28] | M. A. Grepl, Y. Maday, N. C. Nguyen, and A. T. Patera, Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations, ESAIM Math. Model. Numer. Anal., 41 (2007), pp. 575–605. · Zbl 1142.65078 |

[29] | M. D. Gunzburger, C. G. Webster, and G. Zhang, Stochastic finite element methods for partial differential equations with random input data, Acta Numer., 23 (2014), pp. 521–650, . · Zbl 1398.65299 |

[30] | B. Haasdonk, K. Urban, and B. Wieland, Reduced basis methods for parameterized partial differential equations with stochastic influences using the Karhunen–Loève expansion, SIAM/ASA J. Uncertain. Quantif., 1 (2013), pp. 79–105. · Zbl 1281.35100 |

[31] | J. S. Hesthaven, G. Rozza, and B. Stamm, Certified Reduced Basis Methods for Parametrized Partial Differential Equations, Springer, New York, 2015. · Zbl 1329.65203 |

[32] | Q. Liao and G. Lin, Reduced basis ANOVA methods for partial differential equations with high-dimensional random inputs, J. Comput. Phys., 317 (2016), pp. 148–164. · Zbl 1349.65684 |

[33] | G. J. Lord, C. E. Powell, and T. Shardlow, An Introduction to Computational Stochastic PDEs, Cambridge University Press, Cambridge, 2014. · Zbl 1327.60011 |

[34] | I. Martini, G. Rozza, and B. Haasdonk, Reduced basis approximation and a-posteriori error estimation for the coupled Stokes–Darcy system, Adv. Comput. Math., 41 (2015), pp. 1131–1157. · Zbl 1336.76021 |

[35] | F. Nobile, R. Tempone, and C. G. Webster, A sparse grid stochastic collocation method for partial differential equations with random input data, SIAM J. Numer. Anal., 46 (2008), pp. 2309–2345, . · Zbl 1176.65137 |

[36] | C. C. Paige and M. A. Saunders, Solution of sparse indefinite systems of linear equations, SIAM J. Numer. Anal., 12 (1975), pp. 617–629. · Zbl 0319.65025 |

[37] | C. E. Powell, D. Silvester, and V. Simoncini, An efficient reduced basis solver for stochastic Galerkin matrix equations, SIAM J. Sci. Comput., 39 (2017), pp. A141–A163, . · Zbl 1381.35257 |

[38] | C. E. Powell and D. J. Silvester, Optimal preconditioning for Raviart–Thomas mixed formulation of second-order elliptic problems, SIAM J. Matrix Anal. Appl., 25 (2003), pp. 718–738. · Zbl 1073.65128 |

[39] | A. Quarteroni, A. Manzoni, and F. Negri, Reduced Basis Methods for Partial Differential Equations: An Introduction, Springer, New York, 2016. · Zbl 1337.65113 |

[40] | B. Riviè, M. F. Wheeler, and C. Baumann, Part II. Discontinuous Galerkin Method Applied to a Single Phase Flow in Porous Media, , 1999. |

[41] | G. Rozza, D. B. P. Huynh, and A. Manzoni, Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: Roles of the INF-SUP stability constants, Numer. Math., 125 (2013), pp. 115–152. · Zbl 1318.76006 |

[42] | G. Rozza and K. Veroy, On the stability of the reduced basis method for Stokes equations in parametrized domains, Comput. Methods Appl. Mech. Engrg., 196 (2007), pp. 1244–1260. · Zbl 1173.76352 |

[43] | T. Rusten and R. Winther, A preconditioned iterative method for saddlepoint problems, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 887–904. · Zbl 0760.65033 |

[44] | D. J. Silvester and C. E. Powell, Potential Incompressible Flow and Iterative Solver Software (PIFISS) version 1.0, 2007, . |

[45] | R. C. Smith, Uncertainty Quantification: Theory, Implementation, and Applications, Vol. 12, SIAM, Philadelphia, 2013. |

[46] | S. A. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions, Dokl. Akad. Nauk SSSR, 4 (1963), pp. 240–243. · Zbl 0202.39901 |

[47] | A. M. Stuart, Inverse Problems: A Bayesian Perspective, Acta Numer., 19 (2010), pp. 451–559, . · Zbl 1242.65142 |

[48] | T. J. Sullivan, Introduction to Uncertainty Quantification, Vol. 63, Springer, New York, 2016. · Zbl 1336.60002 |

[49] | L. N. Trefethen and D. Bau III, Numerical Linear Algebra, Vol. 50, SIAM, Philadelphia, 1997. · Zbl 0874.65013 |

[50] | D. Xiu and J. S. Hesthaven, High-order collocation methods for differential equations with random inputs, SIAM J. Sci. Comput., 27 (2005), pp. 1118–1139, . · Zbl 1091.65006 |

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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.