In this work we explore the extension of the quasi-optimal sparse grids method proposed in our previous work “On the optimal polynomial approximation of stochastic PDEs by Galerkin and Collocation methods” to a Darcy problem where the permeability is modeled as a lognormal random field. We propose an explicit a-priori/a-posteriori procedure for the construction of such quasi-optimal grid and show its effectiveness on a numerical example. In this approach, the two main ingredients are an estimate of the decay of the Hermite coefficients of the solution and an efficient nested quadrature rule with respect to the Gaussian weight.
|Original language||English (US)|
|Title of host publication||Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2012 - ICOSAHOM conference, Selected papers|
|Editors||Mejdi Azaïez, Jan S. Hesthaven, Henda El Fekih|
|Number of pages||16|
|State||Published - 2014|
|Event||9th International Conference on Spectral and High Order Methods, ICOSAHOM 2012 - Gammarth, Tunisia|
Duration: Jun 25 2012 → Jun 29 2012
|Name||Lecture Notes in Computational Science and Engineering|
|Other||9th International Conference on Spectral and High Order Methods, ICOSAHOM 2012|
|Period||06/25/12 → 06/29/12|
Bibliographical noteFunding Information:
The second and third authors have been supported by the Italian grant FIRB-IDEAS (Project n. RBID08223Z) “Advanced numerical techniques for uncertainty quantification in engineering and life science problems”. Support from the VR project “Effektiva numeriska metoder för stokastiska differentialekvationer med tillämpningar” and King Abdullah University of Science and Technology (KAUST) AEA project “Predictability and Uncertainty Quantification for Models of Porous Media” is also acknowledged. The fourth author is a member of the KAUST SRI Center for Uncertainty Quantification in Computational Science and Engineering.
© Springer International Publishing Switzerland 2014.
ASJC Scopus subject areas
- Modeling and Simulation
- Discrete Mathematics and Combinatorics
- Control and Optimization
- Computational Mathematics