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, S.; Khoromskij, B.N.; Litvinenko, Alexander ( 0000-0001-5427-3598 ) ; Matthies, H.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, 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. Besides, we demonstrate how to assemble the stochastic Galerkin matrix and to compute the solution of the elliptic equation and its post-processing, 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:
Max-Planck-Institut für Dynamik komplexer technischer Systeme, Sandtorstr. 1, 39106 Magdeburg, Germany; Max-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstraße 22, 04103 Leipzig, Germany.; King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia.; Institute for Scientific Computing, Technische Universit ̈at Braunschweig, Hans- Sommerstr. 65, Brunswick, Germany.
Citation:
@article{dolgov2015polynomial, title={Polynomial Chaos Expansion of random coefficients and the solution of stochastic partial differential equations in the Tensor Train format}, author={Dolgov, Sergey and Khoromskij, Boris N and Litvinenko, Alexander and Matthies, Hermann G}, journal={arXiv preprint arXiv:1503.03210}, year={2015} }
Publisher:
arXiv, http://arxiv.org/abs/1503.03210
Journal:
arXiv
Issue Date:
11-Mar-2015
Type:
Article
Description:
Submitted to SIAM/ASA Journal on Uncertainty Quantification (JUQ)
Sponsors:
SRI Uncertainty Quantification and Extreme Computing Research Centers, King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia
Additional Links:
http://arxiv.org/abs/1503.03210
Appears in Collections:
Articles

Full metadata record

DC FieldValue Language
dc.contributor.authorDolgov, S.en
dc.contributor.authorKhoromskij, B.N.en
dc.contributor.authorLitvinenko, Alexanderen
dc.contributor.authorMatthies, H.G.en
dc.date.accessioned2015-03-29T05:56:27Zen
dc.date.available2015-03-29T05:56:27Zen
dc.date.issued2015-03-11en
dc.identifier.citation@article{dolgov2015polynomial, title={Polynomial Chaos Expansion of random coefficients and the solution of stochastic partial differential equations in the Tensor Train format}, author={Dolgov, Sergey and Khoromskij, Boris N and Litvinenko, Alexander and Matthies, Hermann G}, journal={arXiv preprint arXiv:1503.03210}, year={2015} }en
dc.identifier.urihttp://hdl.handle.net/10754/347255en
dc.descriptionSubmitted to SIAM/ASA Journal on Uncertainty Quantification (JUQ)en
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, 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. Besides, we demonstrate how to assemble the stochastic Galerkin matrix and to compute the solution of the elliptic equation and its post-processing, 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.description.sponsorshipSRI Uncertainty Quantification and Extreme Computing Research Centers, King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabiaen
dc.language.isoenen
dc.publisherarXiv, http://arxiv.org/abs/1503.03210en
dc.relation.urlhttp://arxiv.org/abs/1503.03210en
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.departmentMax-Planck-Institut für Dynamik komplexer technischer Systeme, Sandtorstr. 1, 39106 Magdeburg, Germanyen
dc.contributor.departmentMax-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstraße 22, 04103 Leipzig, Germany.en
dc.contributor.departmentKing Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia.en
dc.contributor.departmentInstitute for Scientific Computing, Technische Universit ̈at Braunschweig, Hans- Sommerstr. 65, Brunswick, Germany.en
dc.identifier.journalarXiven
dc.eprint.versionPre-print, submitted to SIAM/ASA Journal on Uncertainty Quantification (JUQ)en
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