An analysis of polynomial chaos approximations for modeling single-fluid-phase flow in porous medium systems
We examine a variety of polynomial-chaos-motivated approximations to a stochastic form of a steady state groundwater flow model. We consider approaches for truncating the infinite dimensional problem and producing decoupled systems. We discuss conditions under which such decoupling is possible and s...
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| Published in | Journal of computational physics Vol. 226; no. 2; pp. 2175 - 2205 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
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01.10.2007
Elsevier |
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| Abstract | We examine a variety of polynomial-chaos-motivated approximations to a stochastic form of a steady state groundwater flow model. We consider approaches for truncating the infinite dimensional problem and producing decoupled systems. We discuss conditions under which such decoupling is possible and show that to generalize the known decoupling by numerical cubature, it would be necessary to find new multivariate cubature rules. Finally, we use the acceleration of Monte Carlo to compare the quality of polynomial models obtained for all approaches and find that in general the methods considered are more efficient than Monte Carlo for the relatively small domains considered in this work. A curse of dimensionality in the series expansion of the log-normal stochastic random field used to represent hydraulic conductivity provides a significant impediment to efficient approximations for large domains for all methods considered in this work, other than the Monte Carlo method. |
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| AbstractList | We examine a variety of polynomial-chaos-motivated approximations to a stochastic form of a steady state groundwater flow model. We consider approaches for truncating the infinite dimensional problem and producing decoupled systems. We discuss conditions under which such decoupling is possible and show that to generalize the known decoupling by numerical cubature, it would be necessary to find new multivariate cubature rules. Finally, we use the acceleration of Monte Carlo to compare the quality of polynomial models obtained for all approaches and find that in general the methods considered are more efficient than Monte Carlo for the relatively small domains considered in this work. A curse of dimensionality in the series expansion of the log-normal stochastic random field used to represent hydraulic conductivity provides a significant impediment to efficient approximations for large domains for all methods considered in this work, other than the Monte Carlo method. We examine a variety of polynomial-chaos-motivated approximations to a stochastic form of a steady state groundwater flow model. We consider approaches for truncating the infinite dimensional problem and producing decoupled systems. We discuss conditions under which such decoupling is possible and show that to generalize the known decoupling by numerical cubature, it would be necessary to find new multivariate cubature rules. Finally, we use the acceleration of Monte Carlo to compare the quality of polynomial models obtained for all approaches and find that in general the methods considered are more efficient than Monte Carlo for the relatively small domains considered in this work. A curse of dimensionality in the series expansion of the log-normal stochastic random field used to represent hydraulic conductivity provides a significant impediment to efficient approximations for large domains for all methods considered in this work, other than the Monte Carlo method. |
| Author | Rupert, C.P. Miller, C.T. |
| AuthorAffiliation | a Department of Environmental Sciences and Engineering, University of North Carolina, Chapel Hill, NC 27599-7431, USA |
| AuthorAffiliation_xml | – name: a Department of Environmental Sciences and Engineering, University of North Carolina, Chapel Hill, NC 27599-7431, USA |
| Author_xml | – sequence: 1 givenname: C.P. surname: Rupert fullname: Rupert, C.P. email: crupert@email.unc.edu – sequence: 2 givenname: C.T. surname: Miller fullname: Miller, C.T. email: casey_miller@unc.edu |
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| Keywords | Stochastic Monte Carlo Groundwater flow Karhunen–Loève expansion Chaos Karhunen-Loève expansion Dimensionality Polynomial approximation Porous medium Stochastic approximation Calculation methods Random field Monte Carlo methods Decoupling Series expansion Models Modelling Calculation Infinite dimension Single phase flow |
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| Snippet | We examine a variety of polynomial-chaos-motivated approximations to a stochastic form of a steady state groundwater flow model. We consider approaches for... We examine a variety of polynomial-chaos-motivated approximations to a stochastic form of a steady state groundwater flow model. We consider approaches for... |
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| SubjectTerms | APPROXIMATIONS CHAOS THEORY CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS Computational techniques Exact sciences and technology FLOW MODELS GROUND WATER Groundwater flow HYDRAULIC CONDUCTIVITY Karhunen–Loève expansion Mathematical methods in physics Monte Carlo MONTE CARLO METHOD MULTIVARIATE ANALYSIS Physics POLYNOMIALS POROUS MATERIALS RANDOMNESS SERIES EXPANSION SIMULATION STEADY-STATE CONDITIONS Stochastic STOCHASTIC PROCESSES |
| Title | An analysis of polynomial chaos approximations for modeling single-fluid-phase flow in porous medium systems |
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