Journal of Applied Mathematics and Computing

, Volume 38, Issue 1–2, pp 367–387 | Cite as

Nonparametric density estimation for randomly perturbed elliptic problems III: convergence, computational cost, and generalizations

  • Donald EstepEmail author
  • Michael J. Holst
  • Axel Målqvist


This is the third in a series of three papers on nonparametric density estimation for randomly perturbed elliptic problems. In the previous papers by Estep, Målqvist, and Tavener (SIAM J. Sci. Comput. 31:2935–2959, 2009; Int. J. Numer. Methods Eng. 80:846–867, 2009), we derive an a posteriori error estimate for a computed probability distribution and an efficient adaptive algorithm for propagation of uncertainty into a quantity of interest computed from numerical solutions of an elliptic partial differential equation. We also test the algorithm on various problems including an example relevant to oil reservoir simulation. In this paper, we derive a convergence result for the method based on the assumption that the underlying domain decomposition algorithm converges geometrically. The main ideas of the proof can be applied to a large class of domain decomposition algorithms. We also present several generalizations of the method and an analysis of the computational cost.


A posteriori error analysis Domain decomposition Convergence Elliptic problem Neumann series Nonparametric density estimation Random perturbation Forward sensitivity analysis 

Mathematics Subject Classification (2000)

65N15 65N30 65N55 65C05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adams, R.A.: Sobolev Spaces. Pure and Applied Mathematics, vol. 65. Academic Press, New York (1975) zbMATHGoogle Scholar
  2. 2.
    Rebolla, T.C., Vera, E.C.: Study of a non-overlapping domain decomposition method: Poisson and Stoke problems. Appl. Numer. Math. 48, 169–194 (2004) CrossRefMathSciNetGoogle Scholar
  3. 3.
    Estep, D., Målqvist, A., Tavener, S.: Nonparametric density estimation for elliptic problems with random perturbations I: computational method, a posteriori analysis, and adaptive error control. SIAM J. Sci. Comput. 31, 2935–2959 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Estep, D., Målqvist, A., Tavener, S.: Nonparametric density estimation for elliptic problems with random perturbations II: applications and adaptive modeling. Int. J. Numer. Methods Eng. 80, 846–867 (2009) CrossRefzbMATHGoogle Scholar
  5. 5.
    Guo, W., Hou, L.S.: Generalizations and accelerations of Lions’ nonoverlapping domain decomposition method for linear elliptic PDE. SIAM J. Numer. Anal. 41(6), 2056–2080 (2003) CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Lions, P.L.: On the Schwarz alternating methods III: a variant for nonoverlapping subdomains. In: Chan, T.F., Glowinski, R., Periaux, J., Wildlund, O.B. (eds.) Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, pp. 202–231. SIAM, Philadelphia (1990) Google Scholar
  7. 7.
    Qin, L., Shi, Z., Xu, X.: On the convergence rate of a parallel nonoverlapping domain decomposition method. Sci. China Ser. A, Math. Aug. 51(8), 1461–1478 (2008) CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Qin, L., Xu, X.: On a parallel robin-type nonoverlapping domain decomposition method. SIAM J. Numer. Anal. 44(6), 2539–2558 (2008) CrossRefMathSciNetGoogle Scholar
  9. 9.
    Smith, B., Bjørstad, P., Gropp, W.: Domain Decomposition Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press, Cambridge (1996) zbMATHGoogle Scholar
  10. 10.
    Xu, J., Zou, J.: Some nonoverlapping domain decomposition methods. SIAM Rev. 40(4), 857–914 (1998) CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2011

Authors and Affiliations

  • Donald Estep
    • 1
    Email author
  • Michael J. Holst
    • 2
  • Axel Målqvist
    • 3
  1. 1.Department of StatisticsColorado State UniversityFort CollinsUSA
  2. 2.Departments of Mathematics and PhysicsUC San DiegoLa JollaUSA
  3. 3.Department of Information TechnologyUppsala UniversityUppsalaSweden

Personalised recommendations