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Section B

Exploring emerging manycore architectures for uncertainty quantification through embedded stochastic Galerkin methods

, , &
Pages 707-729 | Received 30 Jan 2013, Accepted 29 Aug 2013, Published online: 05 Dec 2013

References

  • B.M. Adams, K.R. Dalbey, M.S. Eldred, D.M. Gay, L.P. Swiler, W.J. Bohnhoff, J.P. Eddy, K.Haskell, and P.D. Hough, DAKOTA, a multilevel parallel object-oriented framework for design optimization, parameter estimation, uncertainty quantification, and sensitivity analysis, Tech. Rep.sand2010-2183 ed., Sandia National Laboratories, Albuquerque, New Mexico, May 2010.
  • I. Babuska, 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. doi: 10.1137/050645142
  • I. Babuska, R. Tempone, and G. Zouraris, Galerkin finite element approximations of stochastic elliptic partial differential equations, SIAM J. Numer. Anal. 42 (2004), pp. 800–825. doi: 10.1137/S0036142902418680
  • V. Barthelmann, E. Novak, and K. Ritter, High dimensional polynomial interpolation on sparse grids, Adv. Comput. Math. 12 (2000), Portland, Oregon, pp. 273–288. doi: 10.1023/A:1018977404843
  • N. Bell and M. Garland, Implementing sparse matrix–vector multiplication on throughput-oriented processors, in SC ’09: Proceedings of the Conference on High Performance Computing Networking, Storage and Analysis, Portland Oregon, 2009, ACM, New York, pp. 1–11.
  • E. Chow and D. Hysom, Assessing performance of hybrid MPI/OpenMP programs on SMP clusters, Tech. Rep. UCRL-JC-143957, Lawrence Livermore National Laboratory, Livermore, California, 2001.
  • C. Clenshaw and A. Curtis, A method for numerical integration on an automatic computer, Numer. Math. 2 (1960), pp. 197–205. doi: 10.1007/BF01386223
  • P.R. Conrad and Y.M. Marzouk, Adaptive smolyak pseudospectral approximations, ArXiv e-prints (2012). Available at http://arxiv.org/abs/1209.1406
  • P.G. Constantine, M.S. Eldred, and E.T. Phipps, Sparse pseudospectral approximation method, Comput. Methods Appl. Mech. Eng. 229–232 (2012), pp. 1–12. doi: 10.1016/j.cma.2012.03.019
  • B. Debusschere, H. Najm, P. Pebay, O. Knio, R. Ghanem, and O. Le Maitre, Numerical challenges in the use of polynomial chaos representations for stochastic processes, SIAM J. Sci. Comput. 26 (2004), pp. 698–719. doi: 10.1137/S1064827503427741
  • H. Elman, C. Miller, E. Phipps, and R. Tuminaro, Assessment of collocation and galerkin approaches to linear diffusion equations with random data, Int. J. Uncertain. Quantification 1 (2011), pp. 19–33. doi: 10.1615/Int.J.UncertaintyQuantification.v1.i1.20
  • G. Fishman, Monte Carlo Concepts, Algorithms, and Applications, Springer Series in Operations Research, Springer-Verlag, New York, 1996.
  • J. Foo, X. Wan, and G.E. Karniadakis, The multi-element probabilistic collocation method (ME-PCM): Error analysis and applications, J. Comput. Phys. 227 (2008), pp. 9572–9595. doi: 10.1016/j.jcp.2008.07.009
  • M. Gee, C. Siefert, J. Hu, R. Tuminaro, and M. Sala, ML 5.0 smoothed aggregation user's guide, Tech. Rep. SAND2006-2649, Sandia National Laboratories, Albuquerque, New Mexico, 2006.
  • T. Gerstner and M. Griebel, Dimension–adaptive tensor–product quadrature, Computing 71 (2003), pp. 65–87. doi: 10.1007/s00607-003-0015-5
  • R. Ghanem and P.D. Spanos, Polynomial chaos in stochastic finite elements, J. Appl. Mech. 57 (1) (1990), pp. 197–202. doi: 10.1115/1.2888303
  • R.G. Ghanem and P.D. Spanos, Stochastic Finite Elements: A Spectral Approach, Springer-Verlag, New York, 1991.
  • J. Helton and F. Davis, Latin hypercube sampling and the propagation of uncertainty in analyses of complex systems, Reliab. Eng. Syst. Saf. 81 (2003), pp. 23–69. doi: 10.1016/S0951-8320(03)00058-9
  • M. Heroux, R. Bartlett, V. Howle, R. Hoekstra, J. Hu, T. Kolda, R. Lehoucq, K. Long, R. Pawlowski, E. Phipps, A. Salinger, H. Thornquist, R. Tuminaro, J. Willenbring, A. Williams, and K. Stanley, An overview of the Trilinos package, ACM Trans. Math. Softw. 31 (3) (2005), pp. 397–423. Available at http://trilinos.sandia.gov/ doi: 10.1145/1089014.1089021
  • O.P. Le Maitre and O.M. Knio, Spectral Methods for Uncertainty Quantification with Applications to Computational Fluid Dynamics, Scientific Computation, Springer, New York, 2010.
  • P.T. Lin and J.N. Shadid, Towards large-scale multi-socket, multicore parallel simulations: Performance of an MPI-only semiconductor device simulator, J. Comput. Phys. 229 (2010), pp. 6804–6818. doi: 10.1016/j.jcp.2010.05.023
  • X. Ma and N. Zabaras, An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations, J. Comput. Phys. 228 (2009), pp. 3084–3113. doi: 10.1016/j.jcp.2009.01.006
  • X. Ma and N. Zabaras, An adaptive high-dimensional stochastic model representation technique for the solution of stochastic partial differential equations, J. Comput. Phys. 229 (2010), pp. 3884–3915. doi: 10.1016/j.jcp.2010.01.033
  • M.D. McKay, R.J. Beckman, and W.J. Conover, A comparison of three methods for selecting values of input variables in the analysis of output from a computer code, Technometrics 21 (1979), pp. 239–245.
  • N. Metropolis and S. Ulam, The Monte Carlo method, J. Am. Stat. Assoc. 44 (1949), pp. 335–341. doi: 10.1080/01621459.1949.10483310
  • H. Niederreiter, Quasi-Monte Carlo methods and pseudo-random numbers, Bull. Am. Math. Soc. 84 (1978), pp. 957–1041. doi: 10.1090/S0002-9904-1978-14532-7
  • F. Nobile, R. Tempone, and C.G. Webster, A sparse grid stochastic collocation method for partial differential equations with random input data, SIAM J. Num. Anal. 46 (2008), pp. 2309–2345. doi: 10.1137/060663660
  • F. Nobile, R. Tempone, and C.G. Webster, An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data, SIAM J. Num. Anal. 46 (2008), pp. 2411–2442. doi: 10.1137/070680540
  • E. Novak and K. Ritter, High dimensional integration of smooth functions over cubes, Numer. Math. 75 (1996), pp. 79–97. doi: 10.1007/s002110050231
  • NVIDIA CUDA™ C Programming Guide Version; software available at http://www.nvidia.com/object/cuda_home.html, December 2012.
  • B. Øksendal, Stochastic Differential Equations, Springer-Verlag, Heidelberg, 1998.
  • R.P. Pawlowski, E. Phipps, and A.G. Salinger, Automating embedded analysis capabilities and managing software complexity in multiphysics simulation, Part I: Template-based generic programming, Sci. Program. 20 (2012), pp. 197–219.
  • R.P. Pawlowski, E.T. Phipps, A.G. Salinger, S.J. Owen, C.M. Siefert, and M.L. Staten, Automating embedded analysis capabilities and managing software complexity in multiphysics simulation part II: Application to partial differential equations, Sci. Program. 20 (2012), pp. 327–345.
  • M. Pellissetti and R. Ghanem, Iterative solution of systems of linear equations arising in the context of stochastic finite elements, Adv. Eng. Softw. 31 (2000), pp. 607–616. doi: 10.1016/S0965-9978(00)00034-X
  • E.T. Phipps, Stokhos stochastic Galerkin uncertainty quantification methods. Available at http://trilinos.sandia.gov/packages/stokhos/, 2011.
  • C.E. Powell and H. Elman, Block-diagonal preconditioning for spectral stochastic finite-element systems, IMA J Numer. Anal. 29 (2009), pp. 350–375. doi: 10.1093/imanum/drn014
  • M. Reagan, H. Najm, R. Ghanem, and O. Knio, Uncertainty quantification in reacting-flow simulations through non-intrusive spectral projection, Combus. Flame 132 (2003), pp. 545–555. doi: 10.1016/S0010-2180(02)00503-5
  • E. Rosseel and S. Vandewalle, Iterative solvers for the stochastic finite element method, SIAM J. Sci. Comput. 32 (1) (2010), pp. 372–397. doi: 10.1137/080727026
  • S. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions, Dokl. Akad. Nauk SSSR 4 (1963), pp. 240–243.
  • B. Sousedík, R.G. Ghanem, and E.T. Phipps, Hierarchical Schur complement preconditioner for the stochastic Galerkin finite element methods, Numer. Linear Algebra Appl. (2013). Available at http://dx.doi.org/10.1002/nla.1869
  • W. Sun, J.T. Ostien, and A. Salinger, A stabilized assumed deformation gradient finite element formulation for strongly coupled poromechanical simulations at finite strain, Int. J. Numer. Anal. Methods Geomech. 37 (16) (2012), pp. 2755–2788.
  • E. Ullmann, A Kronecker product preconditioner for stochastic Galerkin finite element discretizations, SIAM J. Sci. Comput. 32 (2010), pp. 923–946. doi: 10.1137/080742853
  • C. Van Loan, The ubiquitous Kronecker product, J. Comput. Appl. Math. 123 (2000), pp. 85–100. doi: 10.1016/S0377-0427(00)00393-9
  • X. Wan and G. Karniadakis, An adaptive multi-element generalized polynomial chaos method for stochastic differential equations, J. Comput. Phys. 209 (2005), pp. 617–642. doi: 10.1016/j.jcp.2005.03.023
  • N. Wiener, The homogeneous chaos, Am. J. Math. 60 (1938), pp. 897–936. doi: 10.2307/2371268
  • D. Xiu and J. Hesthaven, High-order collocation methods for differential equations with random inputs, IAM J. Sci. Comput. 27 (2005), pp. 1118–1139.
  • D. Xiu and G. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Scientific Comput. 24 (2002), pp. 619–644. doi: 10.1137/S1064827501387826
  • X. Zhang and X. Qin, Performance prediction and evaluation of parallel processing on a numa multiprocessor, IEEE Trans. Softw. Eng., 17 (1991), pp. 1059–1068. doi: 10.1109/32.99193

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