Show simple item record

dc.contributor.authorBeck, Joakim
dc.contributor.authorTempone, Raul
dc.contributor.authorNobile, Fabio
dc.contributor.authorTamellini, Lorenzo
dc.date.accessioned2015-08-03T09:59:44Z
dc.date.available2015-08-03T09:59:44Z
dc.date.issued2012-09
dc.identifier.citationBECK, J., TEMPONE, R., NOBILE, F., & TAMELLINI, L. (2012). ON THE OPTIMAL POLYNOMIAL APPROXIMATION OF STOCHASTIC PDES BY GALERKIN AND COLLOCATION METHODS. Mathematical Models and Methods in Applied Sciences, 22(09), 1250023. doi:10.1142/s0218202512500236
dc.identifier.issn02182025
dc.identifier.doi10.1142/S0218202512500236
dc.identifier.urihttp://hdl.handle.net/10754/562295
dc.description.abstractIn this work we focus on the numerical approximation of the solution u of a linear elliptic PDE with stochastic coefficients. The problem is rewritten as a parametric PDE and the functional dependence of the solution on the parameters is approximated by multivariate polynomials. We first consider the stochastic Galerkin method, and rely on sharp estimates for the decay of the Fourier coefficients of the spectral expansion of u on an orthogonal polynomial basis to build a sequence of polynomial subspaces that features better convergence properties, in terms of error versus number of degrees of freedom, than standard choices such as Total Degree or Tensor Product subspaces. We consider then the Stochastic Collocation method, and use the previous estimates to introduce a new class of Sparse Grids, based on the idea of selecting a priori the most profitable hierarchical surpluses, that, again, features better convergence properties compared to standard Smolyak or tensor product grids. Numerical results show the effectiveness of the newly introduced polynomial spaces and sparse grids. © 2012 World Scientific Publishing Company.
dc.description.sponsorshipThe authors would like to recognize the support of the PECOS center at ICES, University of Texas at Austin (Project No. 024550, Center for Predictive Computational Science). Support from the VR project "Effektiva numeriska metoder for stokastiska differentialekvationer med tillampningar" and King Abdullah University of Science and Technology (KAUST) AEA project "Predictability and uncertainty quantification for models of porous media" is also acknowledged. The second and third authors have been supported by the Italian grant FIRB-IDEAS (Project No. RBID08223Z) "Advanced numerical techniques for uncertainty quantification in engineering and life science problems".
dc.publisherWorld Scientific Pub Co Pte Lt
dc.subjectbest M-terms polynomial approximation
dc.subjectelliptic equations
dc.subjectmultivariate polynomial approximation
dc.subjectPDEs with random data
dc.subjectSmolyak approximation
dc.subjectsparse grids
dc.subjectstochastic collocation methods
dc.subjectstochastic Galerkin methods
dc.subjectUncertainty quantification
dc.titleOn the optimal polynomial approximation of stochastic PDEs by galerkin and collocation methods
dc.typeArticle
dc.contributor.departmentApplied Mathematics and Computational Science Program
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
dc.contributor.departmentStochastic Numerics Research Group
dc.identifier.journalMathematical Models and Methods in Applied Sciences
dc.contributor.institutionMOX, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci, 22-20133 Milano, Italy
dc.contributor.institutionCSQI, MATHICSE, Ecole Politechnique Fédérale de Lausanne, CH 1015, Lausanne, Switzerland
kaust.personTempone, Raul
kaust.personBeck, Joakim


This item appears in the following Collection(s)

Show simple item record