Analysis and implementation issues for the numerical approximation of parabolic equations with random coefficients

We consider the problem of numerically approximating statistical moments of the solution of a time‐dependent linear parabolic partial differential equation (PDE), whose coefficients and/or forcing terms are spatially correlated random fields. The stochastic coefficients of the PDE are approximated b...

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Veröffentlicht in:	International journal for numerical methods in engineering Jg. 80; H. 6-7; S. 979 - 1006
Hauptverfasser:	Nobile, F., Tempone, Raul
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Chichester, UK John Wiley & Sons, Ltd 05.11.2009
Schlagworte:	Applied mathematics Approximation Convergence Galerkin methods MATEMATIK Mathematical analysis Mathematical models MATHEMATICS Monte Carlo sampling multivariate polynomial approximation Numerical analysis Numerisk analys parabolic equations Partial differential equations PDE's with random data PDEs with random data Point Collocation Random variables Smolyak approximation Smolyak approximation. Point Collocation sparse grids Stochastic Collocation methods Stochastic Galerkin methods Stochastic Galerkin methods. Stochastic Collocation methods Stochasticity Tillämpad matematik
ISSN:	0029-5981, 1097-0207, 1097-0207
Online-Zugang:	Volltext
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Zusammenfassung:	We consider the problem of numerically approximating statistical moments of the solution of a time‐dependent linear parabolic partial differential equation (PDE), whose coefficients and/or forcing terms are spatially correlated random fields. The stochastic coefficients of the PDE are approximated by truncated Karhunen–Loève expansions driven by a finite number of uncorrelated random variables. After approximating the stochastic coefficients, the original stochastic PDE turns into a new deterministic parametric PDE of the same type, the dimension of the parameter set being equal to the number of random variables introduced. After proving that the solution of the parametric PDE problem is analytic with respect to the parameters, we consider global polynomial approximations based on tensor product, total degree or sparse polynomial spaces and constructed by either a Stochastic Galerkin or a Stochastic Collocation approach. We derive convergence rates for the different cases and present numerical results that show how these approaches are a valid alternative to the more traditional Monte Carlo Method for this class of problems. Copyright © 2009 John Wiley & Sons, Ltd.
Bibliographie:	Royal Institute of Technology, Dahlquist Research Fellowship and VR project 'Effective numerical methods for Stochastic Differential Equations with applications' ArticleID:NME2656 UdelaR in Uruguay Center for Predictive Computational Science - No. 024550 istex:B9230E84FB03F3A06BF5FC8311F36B747FCFDB62 ark:/67375/WNG-V3SKVB0J-D SC, Florida State University ObjectType-Article-2 SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 23
ISSN:	0029-5981 1097-0207 1097-0207
DOI:	10.1002/nme.2656