Abstract
Stokhos (Phipps, Stokhos embedded uncertainty quantification methods. http://trilinos.org/packages/stokhos/, 2015) is a package within Trilinos (Heroux et al., ACM Trans Math Softw 31(3), 2005; Michael et al., Sci Program 20(2):83–88, 2012) that enables embedded or intrusive uncertainty quantification capabilities to C++ codes. It provides tools for implementing stochastic Galerkin methods and embedded sample propagation through the use of template-based generic programming (Pawlowski et al., Sci Program 20:197–219, 2012; Roger et al., Sci Program 20:327–345, 2012) which allows deterministic simulation codes to be easily modified for embedded uncertainty quantification. It provides tools for forming and solving the resulting linear and nonlinear equations these methods generate, leveraging the large-scale linear and nonlinear solver capabilities provided by Trilinos. Furthermore, Stokhos is integrated with the emerging many-core architecture capabilities provided by the Kokkos (Edwards et al., Sci Program 20(2):89–114, 2012; Edwards et al., J Parallel Distrib Comput 74(12):3202–3216, 2014) and Tpetra packages (Baker and Heroux, Sci Program 20(2):115–128, 2012; Hoemmen et al., Tpetra: next-generation distributed linear algebra. http://trilinos.org/packages/tpetra, 2015) within Trilinos, allowing these embedded uncertainty quantification capabilities to be applied in both shared and distributed memory parallel computational environments. Finally, the Stokhos tools have been incorporated into the Albany simulation code (Pawlowski et al., Sci Program 20:327–345, 2012; Salinger et al., Albany multiphysics simulation code. https://github.com/gahansen/Albany, 2015) enabling embedded uncertainty quantification of a wide variety of large-scale PDE-based simulations.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Adams, B.M., Dalbey, K.R., Eldred, M.S., Gay, D.M., Swiler, L.P., Bohnhoff, W.J., Eddy, J.P., Haskell, K., Hough, P.D.: DAKOTA, a multilevel parallel object-oriented framework for design optimization, parameter estimation, Uncertainty Quantification, and Sensitivity Analysis. Sandia National Laboratories, technical report sand2010-2183 edition, May 2010
Baker, C.G., Heroux, M.A.: Tpetra, and the use of generic programming in scientific computing. Sci. Program. 20(2), 115–128 (2012)
Barthelmann, V., Novak, E., Ritter, K.: High dimensional polynomial interpolation on sparse grids. Adv. Comput. Math. 12(4), 273–288 (2000)
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)
Conrad, P.R., Marzouk, Y.M.: Adaptive Smolyak pseudospectral approximations. SIAM J. Sci. Comput. 35(6), A2643–A2670 (2013)
Constantine, P.G., Eldred, M.S., Phipps, E.T.: Sparse pseudospectral approximation method. Comput. Methods Appl. Mech. Eng. 229–232(C), 1–12 (2012)
Debusschere, B.J., Najm, H.N., Pebay, P.P., Knio, O.M., Ghanem, R.G., Le Maitre, O.P.: Numerical challenges in the use of polynomial chaos representations for stochastic processes. SIAM J. Sci. Comput. 26(2), 698–719 (2004)
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)
Edwards, H.C., Trott, C.R., Sunderland, D.: Kokkos: enabling manycore performance portability through polymorphic memory access patterns. J. Parallel Distrib. Comput. 74(12), 3202–3216 (2014)
Gaidamour, J., Hu, J., Siefert, C., Tuminaro, R.: Design considerations for a flexible multigrid preconditioning library. Sci. Program. 20(3), 223–239 (2012)
Ghanem, R., Spanos, P.D.: Polynomial chaos in stochastic finite elements. J. Appl. Mech. 57, 197–202 (1990)
Ghanem, R.G., Spanos, P.D.: Stochastic Finite Elements: A Spectral Approach. Springer, New York (1991)
Griewank, A.: Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation. Number 19 in Frontiers in Applied Mathematics. SIAM, Philadelphia (2000)
Heroux, M.A.: Epetra parallel linear algebra data structures. http://trilinos.org/packages/epetra/ (2015)
Heroux, M.A.: EpetraExt extended epetra utilities. http://trilinos.org/packages/epetraext/ (2015)
Heroux, M.A., Willenbring, J.M.: A new overview of the Trilinos project. Sci. Program. 20(2), 83–88 (2012)
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) (2005). http://trilinos.org/
Hoemmen, M.F., Hu, J.J., Siefert, C.S.: Ifpack2: incomplete factorizations, relaxations, and domain decomposition library. http://trilinos.org/packages/ifpack2 (2015)
Hoemmen, M.F., Thornquist, H.K., Heroux, M.A., Parks, M.: Tpetra: next-generation distributed linear algebra. http://trilinos.org/packages/tpetra (2015)
Hu, J., Prokopenko, A., Siefert, C., Tuminaro, R.: MueLu multigrid framework. http://trilinos.org/packages/muelu (2015)
Le Maitre, O.P., Knio, O.M.: Spectral Methods for Uncertainty Quantification with Applications to Computational Fluid Dynamics. Scientific Computation. Springer, New York (2010)
Novak, E., Ritter, K.: High dimensional integration of smooth functions over cubes. Numerische Mathematik 75, 79–97 (1996)
Øksendal, B.: Stochastic Differential Equations. Springer, Berlin (1998)
Pawlowski, R.P., Kolda, T.G.: NOX object-oriented nonlinear solver package. http://trilinos.org/packages/nox/ (2015)
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)
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)
Phipps, E.T.: Stokhos embedded uncertainty quantification methods. http://trilinos.org/packages/stokhos/ (2015)
Phipps, E.T., Gay, D.M.: Sacado automatic differentiation package. http://trilinos.sandia.gov/packages/sacado/ (2015)
Phipps, E., Pawlowski, R.: Efficient expression templates for operator overloading-based automatic differentiation. In: Forth, S., Hovland, P., Phipps, E., Utke, J., Walther, A. (eds.) Recent Advances in Algorithmic Differentiation. Volume 87 of Lecture Notes in Computational Science and Engineering, pp. 309–319. Springer, Berlin (2012)
Phipps, E., Edwards, H.C., Hu, J., Ostien, J.T.: Exploring emerging manycore architectures for uncertainty quantification through embedded stochastic Galerkin methods. Int. J. Comput. Math. 91(4), 707–729 (2014)
Phipps, E.T., Edwards, H.C., Hu, J.: Exploring heterogeneous multicore architectures for advanced embedded uncertainty quantification. Technical report SAND2014-17875, Sandia National Laboratories, Sept 2014
Powell, C.E., Elman, H.C.: Block-diagonal preconditioning for spectral stochastic finite-element systems. IMA J. Numer. Anal. 29(2), 350–375 (2009)
Rosseel, E., Vandewalle, S.: Iterative solvers for the stochastic finite element method. SIAM J. Sci. Comput. 32(1), 372–397 (2010)
Salinger, A.G.: Piro embedded nonlinear analysis capabilities package. http://trilinos.org/packages/piro/ (2015)
Salinger, A., et al.: Albany multiphysics simulation code. https://github.com/gahansen/Albany (2015)
Smolyak, S.A.: Quadrature and interpolation formulas for tensor products of certain classes of functions. Dokl. Akad. Nauk SSSR 4, 240–243 (1963)
Sousedík, B., Ghanem, R.G., Phipps, E.T.: Hierarchical schur complement preconditioner for the stochastic galerkin finite element methods. Numer. Linear Algebra Appl. 21(1), 136–151 (2014)
Ullmann, E.: A Kronecker product preconditioner for stochastic Galerkin finite element discretizations. SIAM J. Sci. Comput. 32(2), 923–946 (2010)
Wiener, N.: The homogeneous chaos. Am. J. Math. 60, 897–936 (1938)
Xiu, D.B., Karniadakis, G.E.: The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 619–644 (2002)
Acknowledgements
This work was supported by the Advanced Simulation and Computing (ASC) and Laboratory Directed Research and Development (LDRD) programs at Sandia National Laboratories, as well as based upon work supported by the US Department of Energy, Office of Science, Office of Advanced Scientific Computing Research (ASCR). Sandia National Laboratories is a multiprogram laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the US Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing Switzerland
About this entry
Cite this entry
Phipps, E.T., Salinger, A.G. (2017). Embedded Uncertainty Quantification Methods via Stokhos. In: Ghanem, R., Higdon, D., Owhadi, H. (eds) Handbook of Uncertainty Quantification. Springer, Cham. https://doi.org/10.1007/978-3-319-12385-1_55
Download citation
DOI: https://doi.org/10.1007/978-3-319-12385-1_55
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-12384-4
Online ISBN: 978-3-319-12385-1
eBook Packages: Mathematics and StatisticsReference Module Computer Science and Engineering