Stochastic Algebraic Equations

Chapter
Part of the Springer Series in Reliability Engineering book series (RELIABILITY)

Abstract

Methods are discussed for solving approximately linear algebraic equations with random parameters, referred to as stochastic algebraic equations (SAEs). Section 8.1 defines SAEs and outlines potential difficulties related to the solution of these equations. Section 8.2 presents solutions of SAEs with random entries of arbitrary uncertainty by Monte Carlo simulation, stochastic reduced order models, stochastic Galerkin, stochastic collocation, and reliability methods. SAEs with random entries of small uncertainty are solved in Section 8.3 by Taylor series, perturbation, Neumann series, and equivalent linearization methods.

Keywords

Covariance Assure 

References

  1. 1.
    Babuška IM, Chatzipantelidis P (2002) On solving elliptic stochastic partial differential equations. Comput Methods Appl Mech Eng 191:4093–4122MATHCrossRefGoogle Scholar
  2. 2.
    Babuška IM, Nobile F, Tempone R (2007) A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J Numer Anal 45(3):1005–1034MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Babuška IM, Strouboulis T (2001) The finite element method and its reliability. Clarendon Press, Oxford University Press, New YorkGoogle Scholar
  4. 4.
    Brabenec RL (1990) Introduction to real analysis. PWS-KENT Publishing Company, BostonMATHGoogle Scholar
  5. 5.
    Breitung K (1984) Asymptotic approximation for multinormal integrals. J Eng Mech ASCE 110:357–366CrossRefGoogle Scholar
  6. 6.
    Bryant V (1987) Metric spaces iteration and applications. Cambridge University Press, CambridgeGoogle Scholar
  7. 7.
    Cope JE, Rust BW (1979) Bounds on solutions of linear systems with inaccurate data. SIAM J Numer Anal 16(6):950–963MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Ditlevsen O (1981) Uncertainty modeling with applications to multidimensional civil engineering systems. McGraw-Hill Inc., New YorkMATHGoogle Scholar
  9. 9.
    Field RV, Grigoriu M (2007) Convergence properties of polynomial chaos approximations for \(L^2\) random variables. Technical report SAND2007-1262. Sandia National Laboratories, AlbuquerqueGoogle Scholar
  10. 10.
    Ghanem RG, Spanos PD (1991) Stochastic finite elements: a spectral approach. Springer, New YorkMATHCrossRefGoogle Scholar
  11. 11.
    Grigoriu M (2002) Stochastic calculus. Applications in science and engineering. Birkhäuser, BostonMATHGoogle Scholar
  12. 12.
    Grünbaum FA (1975) The determinant of a random matrix. Bull Am Math Soc 81(2):446–448MATHCrossRefGoogle Scholar
  13. 13.
    Hairer E, Nørsett SP, Warnner G (1993) Solving ordinary differential equations I. Nonstiff problems. Springer, New York (Second revised edition)MATHGoogle Scholar
  14. 14.
    Hildebrandt TH, Schoenberg IJ (1933) On linear functional operations and the moment problem for finite interval in one or several dimensions. Ann Math 34(2):317–328MathSciNetCrossRefGoogle Scholar
  15. 15.
    Hinch EJ (1994) Perturbation methods. Cambridge University Press, CambridgeGoogle Scholar
  16. 16.
    Knight K (2000) Mathematical statistics. Chapman & Hall/CRC, New YorkGoogle Scholar
  17. 17.
    Kuo H-H (2006) Introduction to stochastic integration. Springer, New YorkGoogle Scholar
  18. 18.
    Loeven GIA, Witteveen JAS, Bijl H (2007) Probabilisitc collocation: an efficient non-intrusive approach for arbitrary distributed parametric uncertainties. Reno, NevadaGoogle Scholar
  19. 19.
    Lorentz GG (1986) Bernstein polynomials. Chelsea Publishing Company, New YorkMATHGoogle Scholar
  20. 20.
    Mehta ML (2004) Random matrices. Elsevier, AmsterdamMATHGoogle Scholar
  21. 21.
    Nobile F, Tempone R, Webster CG (2008) An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J Numer Anal 46(5):2411–1442MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Powell MJD (1981) Approximation theory and methods. Cambridge University Press, CambridgeMATHGoogle Scholar
  23. 23.
    Rubinstein R (1981) Simulation and the Monte Carlo method. Wiley, New YorkMATHCrossRefGoogle Scholar
  24. 24.
    Simmonds JG, Mann JE (1986) A first look at perturbation theory. Robert E. Krieger Publishing Company, MalabarMATHGoogle Scholar
  25. 25.
    Södeerling G (2006) The logarithmic norm. History and modern theory. BIT Numer Math 46:631–652CrossRefGoogle Scholar
  26. 26.
    Sudret B, DerKiureghian A (2002) Comparisons of finite element reliability methods. Probab Eng Mech 17:337–348CrossRefGoogle Scholar
  27. 27.
    Tricomi FG (1957) Integral equations. Dover Publications, Inc., New YorkMATHGoogle Scholar
  28. 28.
    Wan X, Karniadakis GE (2005) An adaptive nulti-element generalized polynomial chaos methods for stochastic differential equations. J Comput Phys 209(2):617–642. doi: 10.1016/j.jcp.2005.03.023C822 MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Xiu D, Karniadakis GE (2002) The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J Sci Comput 24(2):619–644. doi: 10.1137/S1064827501387826 MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2012

Authors and Affiliations

  1. 1.Cornell UniversityIthacaUSA

Personalised recommendations