zbMATH — the first resource for mathematics

Surrogate-based ensemble grouping strategies for embedded sampling-based uncertainty quantification. (English) Zbl 1455.62158
D’Elia, Marta (ed.) et al., Quantification of uncertainty: improving efficiency and technology. QUIET. Selected contributions based on the presentations at the international workshop, Trieste, Italy, July 18–21, 2017. Cham: Springer. Lect. Notes Comput. Sci. Eng. 137, 41-66 (2020).
Summary: The embedded ensemble propagation approach introduced in [E. Phipps et al., SIAM J. Sci. Comput. 39, No. 2, C162–C193 (2017; Zbl 1365.65017)] has been demonstrated to be a powerful means of reducing the computational cost of sampling-based uncertainty quantification methods, particularly on emerging computational architectures. A substantial challenge with this method however is ensemble-divergence, whereby different samples within an ensemble choose different code paths. This can reduce the effectiveness of the method and increase computational cost. Therefore grouping samples together to minimize this divergence is paramount in making the method effective for challenging computational simulations. In this work, a new grouping approach based on a surrogate for computational cost built up during the uncertainty propagation is developed and applied to model advection-diffusion problems where computational cost is driven by the number of (preconditioned) linear solver iterations. The approach is developed within the context of locally adaptive stochastic collocation methods, where a surrogate for the number of linear solver iterations, generated from previous levels of the adaptive grid generation, is used to predict iterations for subsequent samples, and group them based on similar numbers of iterations. The effectiveness of the method is demonstrated by applying it to highly anisotropic advection-dominated diffusion problems with a wide variation in solver iterations from sample to sample. It extends the parameter-based grouping approach developed in [M. D’Elia et al., SIAM/ASA J. Uncertain. Quantif. 6, 87–117 (2018; Zbl 1390.60229)] to more general problems without requiring detailed knowledge of how the uncertain parameters affect the simulation’s cost, and is also less intrusive to the simulation code.
For the entire collection see [Zbl 1454.62013].
62L20 Stochastic approximation
62-08 Computational methods for problems pertaining to statistics
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
Full Text: DOI
[1] 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 (2004) · Zbl 1080.65003
[2] Babuška, I., Nobile, F., Tempone, R.: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 45(3), 1005-1034 (2007) · Zbl 1151.65008
[3] Bäck, J., Nobile, F., Tamellini, L., Tempone, R.: Stochastic spectral Galerkin and collocation methods for PDEs with random coefficients: a numerical comparison. In: Spectral and High Order Methods for Partial Differential Equations, pp. 43-62. Springer, Berlin (2011) · Zbl 1216.65004
[4] Baker, C.G., Heroux, M.A.: Tpetra, and the use of generic programming in scientific computing. Sci. Program. 20(2), 115-128 (2012)
[5] Bakr, M.H., Bandler, J.W., Madsen, K., Rayas-Sanchez, J.E., Søndergaard, J.: Space-mapping optimization of microwave circuits exploiting surrogate models. IEEE Trans. Microwave Theory Tech. 48(12), 2297-2306 (2000)
[6] Bandler, J.W., Cheng, Q., Gebre-Mariam, D.H., Madsen, K., Pedersen, F., Sondergaard, J.: Em-based surrogate modeling and design exploiting implicit, frequency and output space mappings. In: Microwave Symposium Digest, 2003 IEEE MTT-S International, vol. 2, pp. 1003-1006. IEEE, Piscataway (2003)
[7] Barth, A., Lang, A.: Multilevel Monte Carlo method with applications to stochastic partial differential equations. Int. J. Comput. Math. 89(18), 2479-2498 (2012) · Zbl 1270.65003
[8] Barth, A., Schwab, C., Zollinger, N.: Multi-level Monte Carlo finite element method for elliptic PDEs with stochastic coefficients. Numer. Math. 119(1), 123-161 (2011) · Zbl 1230.65006
[9] Barth, A., Lang, A., Schwab, C.: Multilevel Monte Carlo method for parabolic stochastic partial differential equations. BIT Numer. Math. 53(1), 3-27 (2013) · Zbl 1272.65009
[10] Bavier, E., Hoemmen, M., Rajamanickam, S., Thornquist, H.: Amesos2 and Belos: direct and iterative solvers for large sparse linear systems. Sci. Program. 20(3), 241-255 (2012)
[11] Bichon, B.J., McFarland, J.M., Mahadevan, S.: Efficient surrogate models for reliability analysis of systems with multiple failure modes. Reliab. Eng. Syst. Saf. 96(10), 1386-1395 (2011)
[12] Breitkopf, P., Coelho, R.F.: Multidisciplinary Design Optimization in Computational Mechanics. Wiley, Hoboken (2013)
[13] Bungartz, H.J., Griebel, M.: Sparse grids. Acta Numer. 13(1), 147-269 (2004) · Zbl 1118.65388
[14] 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(1), 3-15 (2011) · Zbl 1241.65012
[15] Cohen, A., DeVore, R., Schwab, C.: Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDE’s. Anal. Appl. 9(01), 11-47 (2011) · Zbl 1219.35379
[16] Couckuyt, I., Declercq, F., Dhaene, T., Rogier, H., Knockaert, L.: Surrogate-based infill optimization applied to electromagnetic problems. Int. J. RF Microwave Comput. Aided Eng. 20(5), 492-501 (2010)
[17] D’Elia, M., Edwards, H.C., Hu, J., Phipps, E., Rajamanickam, S.: Ensemble grouping strategies for embedded stochastic collocation methods applied to anisotropic diffusion problems. SIAM/ASA J. Uncertain. Quantif. 6, 87 (2017)
[18] Donea, J., Huera, A.: Finite Element Methods for Flow Problems. Wiley, New York (2003)
[19] Doostan, A., Owhadi. H.: A non-adapted sparse approximation of PDEs with stochastic inputs. J. Comput. Phys. 230(8), 3015-3034 (2011) · Zbl 1218.65008
[20] Ebeida, M.S., Mitchell, S.A., Swiler, L.P., Romero, V.J., Rushdi, A.A.: POF-darts: geometric adaptive sampling for probability of failure. Reliab. Eng. Syst. Saf. 155, 64-77 (2016)
[21] Edwards, H.C., Sunderland, D., Porter, V., Amsler, C., Mish, S.: Manycore performance-portability: Kokkos multidimensional array library. Sci. Program. 20(2), 89-114 (2012)
[22] Edwards, H.C., Trott, C.R., Sunderland, D.: Kokkos: enabling manycore performance portability through polymorphic memory access patterns. J. Parallel Distrib. Comput. 74, 3202-3216 (2014)
[23] Fishman, G.S.: Monte Carlo: Concepts, Algorithms, and Applications. Springer Series in Operations Research. Springer, New York (1996)
[24] Forrester, A.I.J., Bressloff, N.W., Keane, A.J.: Optimization using surrogate models and partially converged computational fluid dynamics simulations. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 462, 2177-2204 (2006). The Royal Society · Zbl 1149.76656
[25] Frauenfelder, P., Schwab, C., Todor, R.A.: Finite elements for elliptic problems with stochastic coefficients. Comput. Methods Appl. Mech. Eng. 194(2), 205-228 (2005) · Zbl 1143.65392
[26] Galindo, D., Jantsch, P., Webster, C.G., Zhang, G.: Accelerating stochastic collocation methods for PDEs with random input data. Technical Report TM-2015/219, Oak Ridge National Laboratory (2015) · Zbl 1352.65499
[27] Ganapathysubramanian, B., Zabaras, N.: Sparse grid collocation schemes for stochastic natural convection problems. J. Comput. Phys. 225(1), 652-685 (2007) · Zbl 1343.76059
[28] Ghanem, R.G., Spanos, P.D.: Polynomial chaos in stochastic finite elements. J. Appl. Mech. 57, 197 (1990) · Zbl 0729.73290
[29] Ghanem, R.G., Spanos, P.D.: Stochastic Finite Elements: A Spectral Approach. Springer, New York (1991) · Zbl 0722.73080
[30] Giles, M.B.: Multilevel Monte Carlo path simulation. Oper. Res. 56(3), 607-617 (2008) · Zbl 1167.65316
[31] Griebel, M.: Adaptive sparse grid multilevel methods for elliptic PDEs based on finite differences. Computing 61(2), 151-179 (1998) · Zbl 0918.65078
[32] Gunzburger, M., Webster, C.G., Zhang, G.: An adaptive wavelet stochastic collocation method for irregular solutions of partial differential equations with random input data. In: Sparse Grids and Applications - Munich 2012, pp. 137-170. Springer International Publishing, Cham (2014) · Zbl 1316.65011
[33] 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) · Zbl 1398.65299
[34] Hao, P., Wang, B., Li, G.: Surrogate-based optimum design for stiffened shells with adaptive sampling. AIAA J. 50(11), 2389-2407 (2012)
[35] Helton, J.C., Davis, F.J.: Latin hypercube sampling and the propagation of uncertainty in analyses of complex systems. Reliab. Eng. Syst. Saf. 81, 23-69 (2003)
[36] Heroux, M.A., Willenbring, J.M.: A new overview of the Trilinos project. Sci. Program. 20(2), 83-88 (2012)
[37] Heroux, M.A., Bartlett, R.A., Howle, V.E., Hoekstra, R.J., Hu, J.J., Kolda, T.G., Lehoucq, R.B., Long, K.R., Pawlowski, R.P., Phipps, E.T., Salinger, A.G., Thornquist, H.K., Tuminaro, R.S., Willenbring, J.M., Williams, A.B., Stanley, K.S.: An overview of the Trilinos package. ACM Trans. Math. Softw. 31(3), 397 (2005) · Zbl 1136.65354
[38] Li, J., Xiu, D.: Evaluation of failure probability via surrogate models. J. Comput. Phys. 229(23), 8966-8980 (2010) · Zbl 1204.65010
[39] Li, J., Li, J., Xiu, D.: An efficient surrogate-based method for computing rare failure probability. J. Comput. Phys. 230(24), 8683-8697 (2011) · Zbl 1370.65005
[40] Loève, M.: Probability Theory I. Graduate Texts in Mathematics, 4th edn., vol. 45. Springer, New York (1977)
[41] Loève, M.: Probability Theory II. Graduate Texts in Mathematics, 4th edn., vol. 46. Springer, New York (1978)
[42] Mathelin, L., Gallivan, K.A.: A compressed sensing approach for partial differential equations with random input data. Commun. Comput. Phys. 12(04), 919-954 (2012) · Zbl 1388.65018
[43] McKay, M.D., Beckman, R.J., Conover, W.J.: A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21(2), 239-245 (1979) · Zbl 0415.62011
[44] Metropolis, N., Ulam, S.: The Monte Carlo method. J. Am. Stat. Assoc. 44(247), 335-341 (1949) · Zbl 0033.28807
[45] Niederreiter, H.: Quasi-Monte Carlo methods and pseudo-random numbers. Bull. Amer. Math. Soc 84(6), 957-1041 (1978) · Zbl 0404.65003
[46] 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(5), 2309-2345 (2008) · Zbl 1176.65137
[47] 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(5), 2411-2442 (2008) · Zbl 1176.65007
[48] Pawlowski, R.P., Phipps, E.T., Salinger, A.G.: Automating embedded analysis capabilities and managing software complexity in multiphysics simulation, Part I: Template-based generic programming. Sci. Program. 20, 197-219 (2012)
[49] Pawlowski, R.P., Phipps, E.T., Salinger, A.G., Owen, S.J., Siefert, C.M., Staten, M.L.: Automating embedded analysis capabilities and managing software complexity in multiphysics simulation part II: application to partial differential equations. Sci. Program. 20, 327-345 (2012)
[50] Phipps, E.T.: Stokhos Stochastic Galerkin Uncertainty Quantification Methods (2015). Available online: http://trilinos.org/packages/stokhos/
[51] Phipps, E., D’Elia, M., Edwards, H.C., Hoemmen, M., Hu, J., Rajamanickam, S.: Embedded ensemble propagation for improving performance, portability and scalability of uncertainty quantification on emerging computational architectures. SIAM J. Sci. Comput. 39(2), C162 (2017) · Zbl 1365.65017
[52] Prokopenko, A., Hu, J.J., Wiesner, T.A., Siefert, C.M., Tuminaro, R.S.: MueLu user’s guide 1.0. Technical Report SAND2014-18874, Sandia National Laboratories (2014)
[53] Razavi, S., Tolson, B.A., Burn, D.H.: Review of surrogate modeling in water resources. Water Resour. Res. 48(7), 7401 (2012)
[54] Rikards, R., Abramovich, H., Auzins, J., Korjakins, A., Ozolinsh, O., Kalnins, K., Green, T.: Surrogate models for optimum design of stiffened composite shells. Compos. Struct. 63(2), 243-251 (2004)
[55] Roman, L.J., Sarkis, M.: Stochastic Galerkin method for elliptic SPDEs: a white noise approach. Discrete Contin. Dynam. Syst. B 6(4), 941 (2006) · Zbl 1148.60043
[56] Smolyak, S.A.: Quadrature and interpolation formulas for tensor products of certain classes of functions. Dokl. Akad. Nauk SSSR 4, 240-243 (1963) · Zbl 0202.39901
[57] Stoyanov, M.: Hierarchy-direction selective approach for locally adaptive sparse grids. Technical Report TM-2013/384, Oak Ridge National Laboratory (2013)
[58] Stoyanov, M., Webster, C.G.: A dynamically adaptive sparse grid method for quasi-optimal interpolation of multidimensional analytic functions. Technical Report TM-2015/341, Oak Ridge National Laboratory (2015)
[59] Xiu, D.B., Hesthaven, J.S.: High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27(3), 1118-1139 (2005) · 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.