Embedded Uncertainty Quantification Methods via Stokhos

  • Eric T. Phipps
  • Andrew G. Salinger
Living reference work entry


Stokhos (Phipps, Stokhos embedded uncertainty quantification methods., 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., 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., 2015) enabling embedded uncertainty quantification of a wide variety of large-scale PDE-based simulations.


Stochastic Galerkin methods Embedded sampling methods Polynomial chaos Sparse grids C++ templates Operator overloading Linear solvers Preconditioning Parallel programming Shared memory parallelism Distributed memory parallelism Fine-grained parallelism Multicore architectures 



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.


  1. 1.
    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 2010Google Scholar
  2. 2.
    Baker, C.G., Heroux, M.A.: Tpetra, and the use of generic programming in scientific computing. Sci. Program. 20(2), 115–128 (2012)Google Scholar
  3. 3.
    Barthelmann, V., Novak, E., Ritter, K.: High dimensional polynomial interpolation on sparse grids. Adv. Comput. Math. 12(4), 273–288 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    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)Google Scholar
  5. 5.
    Conrad, P.R., Marzouk, Y.M.: Adaptive Smolyak pseudospectral approximations. SIAM J. Sci. Comput. 35(6), A2643–A2670 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Constantine, P.G., Eldred, M.S., Phipps, E.T.: Sparse pseudospectral approximation method. Comput. Methods Appl. Mech. Eng. 229–232(C), 1–12 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    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)Google Scholar
  9. 9.
    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)CrossRefGoogle Scholar
  10. 10.
    Gaidamour, J., Hu, J., Siefert, C., Tuminaro, R.: Design considerations for a flexible multigrid preconditioning library. Sci. Program. 20(3), 223–239 (2012)Google Scholar
  11. 11.
    Ghanem, R., Spanos, P.D.: Polynomial chaos in stochastic finite elements. J. Appl. Mech. 57, 197–202 (1990)CrossRefzbMATHGoogle Scholar
  12. 12.
    Ghanem, R.G., Spanos, P.D.: Stochastic Finite Elements: A Spectral Approach. Springer, New York (1991)CrossRefzbMATHGoogle Scholar
  13. 13.
    Griewank, A.: Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation. Number 19 in Frontiers in Applied Mathematics. SIAM, Philadelphia (2000)zbMATHGoogle Scholar
  14. 14.
    Heroux, M.A.: Epetra parallel linear algebra data structures. (2015)
  15. 15.
    Heroux, M.A.: EpetraExt extended epetra utilities. (2015)
  16. 16.
    Heroux, M.A., Willenbring, J.M.: A new overview of the Trilinos project. Sci. Program. 20(2), 83–88 (2012)Google Scholar
  17. 17.
    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).
  18. 18.
    Hoemmen, M.F., Hu, J.J., Siefert, C.S.: Ifpack2: incomplete factorizations, relaxations, and domain decomposition library. (2015)
  19. 19.
    Hoemmen, M.F., Thornquist, H.K., Heroux, M.A., Parks, M.: Tpetra: next-generation distributed linear algebra. (2015)
  20. 20.
    Hu, J., Prokopenko, A., Siefert, C., Tuminaro, R.: MueLu multigrid framework. (2015)
  21. 21.
    Le Maitre, O.P., Knio, O.M.: Spectral Methods for Uncertainty Quantification with Applications to Computational Fluid Dynamics. Scientific Computation. Springer, New York (2010)CrossRefzbMATHGoogle Scholar
  22. 22.
    Novak, E., Ritter, K.: High dimensional integration of smooth functions over cubes. Numerische Mathematik 75, 79–97 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Øksendal, B.: Stochastic Differential Equations. Springer, Berlin (1998)CrossRefzbMATHGoogle Scholar
  24. 24.
    Pawlowski, R.P., Kolda, T.G.: NOX object-oriented nonlinear solver package. (2015)
  25. 25.
    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)Google Scholar
  26. 26.
    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)Google Scholar
  27. 27.
    Phipps, E.T.: Stokhos embedded uncertainty quantification methods. (2015)
  28. 28.
    Phipps, E.T., Gay, D.M.: Sacado automatic differentiation package. (2015)
  29. 29.
    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)CrossRefGoogle Scholar
  30. 30.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    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 2014Google Scholar
  32. 32.
    Powell, C.E., Elman, H.C.: Block-diagonal preconditioning for spectral stochastic finite-element systems. IMA J. Numer. Anal. 29(2), 350–375 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Rosseel, E., Vandewalle, S.: Iterative solvers for the stochastic finite element method. SIAM J. Sci. Comput. 32(1), 372–397 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Salinger, A.G.: Piro embedded nonlinear analysis capabilities package. (2015)
  35. 35.
    Salinger, A., et al.: Albany multiphysics simulation code. (2015)
  36. 36.
    Smolyak, S.A.: Quadrature and interpolation formulas for tensor products of certain classes of functions. Dokl. Akad. Nauk SSSR 4, 240–243 (1963)zbMATHGoogle Scholar
  37. 37.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Ullmann, E.: A Kronecker product preconditioner for stochastic Galerkin finite element discretizations. SIAM J. Sci. Comput. 32(2), 923–946 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Wiener, N.: The homogeneous chaos. Am. J. Math. 60, 897–936 (1938)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Xiu, D.B., Karniadakis, G.E.: The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 619–644 (2002)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Sandia National LaboratoriesCenter for Computing ResearchAlbuquerqueUSA
  2. 2.Sandia National LaboratoriesCenter for Computing ResearchAlbuquerqueUSA

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