Polynomial Chaos Expansion of Random Coefficients and the Solution of Stochastic Partial Differential Equations in the Tensor Train Format

Handle URI:
http://hdl.handle.net/10754/347255
Title:
Polynomial Chaos Expansion of Random Coefficients and the Solution of Stochastic Partial Differential Equations in the Tensor Train Format
Authors:
Dolgov, Sergey; Khoromskij, Boris N.; Litvinenko, Alexander ( 0000-0001-5427-3598 ) ; Matthies, Hermann G.
Abstract:
We apply the tensor train (TT) decomposition to construct the tensor product polynomial chaos expansion (PCE) of a random field, to solve the stochastic elliptic diffusion PDE with the stochastic Galerkin discretization, and to compute some quantities of interest (mean, variance, and exceedance probabilities). We assume that the random diffusion coefficient is given as a smooth transformation of a Gaussian random field. In this case, the PCE is delivered by a complicated formula, which lacks an analytic TT representation. To construct its TT approximation numerically, we develop the new block TT cross algorithm, a method that computes the whole TT decomposition from a few evaluations of the PCE formula. The new method is conceptually similar to the adaptive cross approximation in the TT format but is more efficient when several tensors must be stored in the same TT representation, which is the case for the PCE. In addition, we demonstrate how to assemble the stochastic Galerkin matrix and to compute the solution of the elliptic equation and its postprocessing, staying in the TT format. We compare our technique with the traditional sparse polynomial chaos and the Monte Carlo approaches. In the tensor product polynomial chaos, the polynomial degree is bounded for each random variable independently. This provides higher accuracy than the sparse polynomial set or the Monte Carlo method, but the cardinality of the tensor product set grows exponentially with the number of random variables. However, when the PCE coefficients are implicitly approximated in the TT format, the computations with the full tensor product polynomial set become possible. In the numerical experiments, we confirm that the new methodology is competitive in a wide range of parameters, especially where high accuracy and high polynomial degrees are required.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Citation:
Polynomial Chaos Expansion of Random Coefficients and the Solution of Stochastic Partial Differential Equations in the Tensor Train Format 2015, 3 (1):1109 SIAM/ASA Journal on Uncertainty Quantification
Journal:
SIAM/ASA Journal on Uncertainty Quantification
Issue Date:
3-Nov-2015
DOI:
10.1137/140972536
ARXIV:
arXiv:1503.03210
Type:
Article
ISSN:
2166-2525
Additional Links:
http://epubs.siam.org/doi/10.1137/140972536; http://arxiv.org/abs/1503.03210
Appears in Collections:
Articles; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorDolgov, Sergeyen
dc.contributor.authorKhoromskij, Boris N.en
dc.contributor.authorLitvinenko, Alexanderen
dc.contributor.authorMatthies, Hermann G.en
dc.date.accessioned2017-05-23T08:57:44Z-
dc.date.available2015-03-29T05:56:27Zen
dc.date.available2017-05-23T08:57:44Z-
dc.date.issued2015-11-03en
dc.identifier.citationPolynomial Chaos Expansion of Random Coefficients and the Solution of Stochastic Partial Differential Equations in the Tensor Train Format 2015, 3 (1):1109 SIAM/ASA Journal on Uncertainty Quantificationen
dc.identifier.issn2166-2525en
dc.identifier.doi10.1137/140972536en
dc.identifier.urihttp://hdl.handle.net/10754/347255-
dc.identifier.urihttp://hdl.handle.net/10754/595345en
dc.description.abstractWe apply the tensor train (TT) decomposition to construct the tensor product polynomial chaos expansion (PCE) of a random field, to solve the stochastic elliptic diffusion PDE with the stochastic Galerkin discretization, and to compute some quantities of interest (mean, variance, and exceedance probabilities). We assume that the random diffusion coefficient is given as a smooth transformation of a Gaussian random field. In this case, the PCE is delivered by a complicated formula, which lacks an analytic TT representation. To construct its TT approximation numerically, we develop the new block TT cross algorithm, a method that computes the whole TT decomposition from a few evaluations of the PCE formula. The new method is conceptually similar to the adaptive cross approximation in the TT format but is more efficient when several tensors must be stored in the same TT representation, which is the case for the PCE. In addition, we demonstrate how to assemble the stochastic Galerkin matrix and to compute the solution of the elliptic equation and its postprocessing, staying in the TT format. We compare our technique with the traditional sparse polynomial chaos and the Monte Carlo approaches. In the tensor product polynomial chaos, the polynomial degree is bounded for each random variable independently. This provides higher accuracy than the sparse polynomial set or the Monte Carlo method, but the cardinality of the tensor product set grows exponentially with the number of random variables. However, when the PCE coefficients are implicitly approximated in the TT format, the computations with the full tensor product polynomial set become possible. In the numerical experiments, we confirm that the new methodology is competitive in a wide range of parameters, especially where high accuracy and high polynomial degrees are required.en
dc.language.isoenen
dc.relation.urlhttp://epubs.siam.org/doi/10.1137/140972536en
dc.relation.urlhttp://arxiv.org/abs/1503.03210en
dc.rightsArchived with thanks to SIAM/ASA Journal on Uncertainty Quantificationen
dc.subjectuncertainty quantificationen
dc.subjectpolynomial chaos expansionen
dc.subjectKarhunen-Lo`eve expansionen
dc.subjectstochastic Galerkinen
dc.subjecttensor product methodsen
dc.subjecttensor train formaten
dc.subjectadaptive cross approximationen
dc.subjectblock crossen
dc.titlePolynomial Chaos Expansion of Random Coefficients and the Solution of Stochastic Partial Differential Equations in the Tensor Train Formaten
dc.typeArticleen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.identifier.journalSIAM/ASA Journal on Uncertainty Quantificationen
dc.eprint.versionPublisher's Version/PDFen
dc.contributor.institutionMax-Planck-Institut fur Dynamik komplexer technischer Systeme, Sandtorstr. 1, 39106 Magdeburg, Germanyen
dc.contributor.institutionMax-Planck-Institut fur Mathematik in den Naturwissenschaften, Inselstra├če 22, 04103 Leipzig, Germanyen
dc.contributor.institutionInstitute for Scientific Computing, Technische Universitat Braunschweig, Hans-Sommerstr. 65, Brunswick, Germanyen
dc.contributor.affiliationKing Abdullah University of Science and Technology (KAUST)en
dc.identifier.arxividarXiv:1503.03210en
kaust.authorLitvinenko, Alexanderen

Version History

VersionItem Editor Date Summary
2 10754/347255grenzdm2017-05-23 08:55:22.131Final version published with DOI.
1 10754/347255.1litvina2015-03-29 05:56:27.0
All Items in KAUST are protected by copyright, with all rights reserved, unless otherwise indicated.