Engineering with Computers

, Volume 28, Issue 1, pp 83–94 | Cite as

libMoM: a library for stochastic simulations in engineering using statistical moments

Original Article

Abstract

Stochastic simulations are becoming increasingly important in numerous engineering applications. The solution to the governing equations are complicated due to the high-dimensional spaces and the presence of randomness. In this paper we present libMoM (http://libmom.sourceforge.net), a software library to solve various types of Stochastic Differential Equations (SDE) as well as estimate statistical distributions from the moments. The library provides a suite of tools to solve various SDEs using the method of moments (MoM) as well as estimate statistical distributions from the moments using moment matching algorithms. For a large class of problems, MoM provide efficient solutions compared with other stochastic simulation techniques such as Monte Carlo (MC). In the physical sciences, the moments of the distribution are usually the primary quantities of interest. The library enables the solution of moment equations derived from a variety of SDEs, with closure using non-standard Gaussian quadrature. In engineering risk assessment and decision making, statistical distributions are required. The library implements tools for fitting the Generalized Lambda Distribution (GLD) with the given moments. The objectives of this paper are (1) to briefly outline the theory behind moment methods for solving SDEs/estimation of statistical distributions; (2) describe the organization of the software and user interfaces; (3) discuss use of standard software engineering tools for regression testing, aid collaboration, distribution and further development. A number of representative examples of the use of libMoM in various engineering applications are presented and future areas of research are discussed.

Keywords

Stochastic simulations Method of Moments Moment matching Gaussian quadrature 

References

  1. 1.
    van Kampen NG (2007) Stochastic processes in physics and chemistry, 3rd edn. North Holland, AmsterdamGoogle Scholar
  2. 2.
    Ramkrishna D (2000) Population balances: theory and applications to particulate systems in engineering. Academic Press, San DiegoGoogle Scholar
  3. 3.
    Ghanem RG, Spanos PD (2003) Population balances: stochastic finite elements: a spectral approach. Dover Publications, New YorkGoogle Scholar
  4. 4.
    Hulburt HM, Katz S (1964) Some problems in particle technology: a statistical mechanical formulation. Chem Eng Sci 19: 555–574CrossRefGoogle Scholar
  5. 5.
    Pratsinis SE (1988) Simultaneous nucleation, condensation and coagulation in aerosol reactors. J Colloid Interf Sci 2: 416–427CrossRefGoogle Scholar
  6. 6.
    Frenklach M (2002) Method of moments with interpolative closure. Chem Eng Sci 57: 2229–2239CrossRefGoogle Scholar
  7. 7.
    McGraw R (1997) Description of aerosol dynamics by the quadrature method of moments. Aerosol Sci Tech 27: 255–265CrossRefGoogle Scholar
  8. 8.
    Marchisio DL, Fox RO (2005) Solution of population balance equations using the direct quadrature method of moments. J Aerosol Sci 36:43–73CrossRefGoogle Scholar
  9. 9.
    Dorao CA, Jakobsen HA (2006) The quadrature method of moments and its relationship with the method of weighted residuals. Chem Eng Sci 61(23):7795–7804CrossRefGoogle Scholar
  10. 10.
    McGraw R, Wright DL (2003) Chemically resolved aerosol dynamics for internal mixtures by the quadrature method of moments. J Aerosol Sci 34:189–209CrossRefGoogle Scholar
  11. 11.
    Upadhyay RR, Ezekoye OA (2003) Evaluation of the 1-point quadrature approximation in QMOM for combined aerosol growth laws. J Aerosol Sci 34:1665–1683CrossRefGoogle Scholar
  12. 12.
    Upadhyay RR, Ezekoye OA (2005) Smoke buildup and light scattering in a cylindrical cavity above a uniform flow. J Aerosol Sci 36:471–493CrossRefGoogle Scholar
  13. 13.
    Upadhyay RR, Ezekoye OA (2005) Treatment of size-dependent aerosol transport processes using quadrature based moment methods. J Aerosol Sci 37:799–819CrossRefGoogle Scholar
  14. 14.
    Wright DL, McGraw R, Rosner DE (2001) Bivariate extension of the quadrature method of moments for modeling simultaneous coagulation and sintering of particle populations. J Colloid Interf Sci 236:242–251CrossRefGoogle Scholar
  15. 15.
    Terry DA, McGraw R, Rangel RH (2001) Method of moments solutions for a laminar flow reactor model. Aerosol Sci Tech 34(4):353–362Google Scholar
  16. 16.
    Yoon C, McGraw R (2004) Representation of generally mixed multivariate aerosols by the quadrature method of moments: I. Statistical foundation. J Aerosol Sci 35:561–576CrossRefGoogle Scholar
  17. 17.
    Attar PJ, Vedula P (2008) Direct quadrature method of moments solution of the Fokker–Planck equation. J Sound Vib 317(1–2):265–272CrossRefGoogle Scholar
  18. 18.
    Fox RO, Laurent F, Massot M (2008) Numerical simulation of spray coalescence in an Eulerian framework: direct quadrature method of moments and multi-fluid method. J Comput Phys 227(6): 3058–3088CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Upadhyay RR, Ezekoye OA (2008) Treatment of design fire uncertainty using quadrature method of moments. Fire Safety J 43(2):127–139CrossRefGoogle Scholar
  20. 20.
    Xu Y, Vedula P (2009) A quadrature-based method of moments for nonlinear filtering. Automatica 45(5):1291–1298CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Karian ZA, Dudewicz EJ (2000) Fitting statistical distributions: the generalized lambda distribution and generalized bootstrap methods. CRC Press, Boca RatonCrossRefMATHGoogle Scholar
  22. 22.
    Upadhyay RR (2006) Simulation of population balance equations using quadrature based moment methods. Dissertation, University of Texas at AustinGoogle Scholar
  23. 23.
    Gordon RG (1968) Error bounds in equilibrium statistical mechanics. J Math Phys 9:655–663CrossRefMATHGoogle Scholar
  24. 24.
    Dunkl CF, Xu Y (2001) Orthogonal polynomials of several variables. Encyclopedia of mathematics and its applications. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  25. 25.
    Fox RO (2009) Optimal moment sets for multivariate direct quadrature method of moments. Ind Eng Chem Res 48(21):9686–9696CrossRefGoogle Scholar
  26. 26.
    Lakhany A, Mausser H (2000) Estimating the parameters of the generalized lambda distribution. Algo Res Q3(3):47–58Google Scholar
  27. 27.
    Sobol I, Shukman B (1993) Random and quasi random sequences: numerical estimates of uniformity of distribution. Math Comput Model 18(8):39–45CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Nelder J, Mead R (1965) A simplex method for function minimization. Comput J 7:308–313MATHGoogle Scholar
  29. 29.
    Joe S, Kuo FY (2003) Remark on Algorithm 659: implementing Sobol’s quasirandom sequence generator. ACM Trans Math Softw 29: 49-57CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Joe S, Kuo FY (2008) Constructing Sobol sequences with better two-dimensional projections. SIAM J Sci Comput 30: 2635–2654CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Freimer M, Mudholkar GS, Kollia G, Lin CT (1998) A study of the generalized tukey lambda family. Commun Stat A Theor 17:3547–3567CrossRefMathSciNetGoogle Scholar
  32. 32.
    Ramberg JS, Schmeiser BW (1974) An approximate method for generating asymmetric random variables. Commun ACM 17:78–82CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Friedlander SK (2000) Smoke, dust, and haze: fundamentals of aerosol dynamics, 2nd edition. Oxford University Press, OxfordGoogle Scholar
  34. 34.
    Upadhyay RR, Ezekoye OA (2007) Performance based engineering with a bivariate PDF of fire size and vent opening. In: Proceedings of the 5th international seminar on fire and explosion Hazards, Edinburgh, pp 371–380Google Scholar
  35. 35.
    Lambin Ph, Gaspin J-P (1982) Continued-fraction technique for tight binding systems. A generalized moments method. Phys Rev B 26(8):4356–4368CrossRefGoogle Scholar
  36. 36.
    Press WH, Flannery BP, Teukolsky SA, Vetterling WT (1992) Numerical recipes in Fortran 77: the art of scientific computing, 2nd edn. Cambridge University Press, CambridgeGoogle Scholar
  37. 37.
    Kirk BS, Peterson JW, Stogner RH, Carey GF (2006) libMesh: a C++ library for parallel adaptive mesh refinement/coarsening simulations. Eng Comput 22: 237–254CrossRefGoogle Scholar
  38. 38.
    McGraw R (2007) Numerical advection of correlated tracers: preserving particle size/composition moment sequences during transport of aerosol mixtures. J Phys Conf Ser 78:1–5CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.Institute for Computational Engineering and SciencesThe University of Texas at AustinAustinUSA
  2. 2.Department of Mechanical EngineeringThe University of Texas at AustinAustinUSA

Personalised recommendations