Parametric and uncertainty computations with tensor product representations

Hermann G. Matthies; Alexander Litvinenko; Oliver Pajonk; Bojana V. Rosić; Elmar Zander

doi:10.1007/978-3-642-32677-6_9

Parametric and uncertainty computations with tensor product representations

Hermann G. Matthies^*, Alexander Litvinenko, Oliver Pajonk, Bojana V. Rosić, Elmar Zander

^*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Academic › peer-review

14 Citations (Scopus)

Abstract

Computational uncertainty quantification in a probabilistic setting is a special case of a parametric problem. Parameter dependent state vectors lead via association to a linear operator to analogues of covariance, its spectral decomposition, and the associated Karhunen-Loève expansion. From this one obtains a generalised tensor representation The parameter in question may be a tuple of numbers, a function, a stochastic process, or a random tensor field. The tensor factorisation may be cascaded, leading to tensors of higher degree. When carried on a discretised level, such factorisations in the form of low-rank approximations lead to very sparse representations of the high dimensional quantities involved. Updating of uncertainty for new information is an important part of uncertainty quantification. Formulated in terms or random variables instead of measures, the Bayesian update is a projection and allows the use of the tensor factorisations also in this case.

Original language	English
Title of host publication	Uncertainty Quantification in Scientific Computing
Subtitle of host publication	10th IFIP WG 2.5 Working Conference, WoCoUQ 2011, Revised Selected Papers
Editors	Andrew M. Dienstfrey, Ronald F. Boisvert
Place of Publication	Berlin, Heidelberg
Publisher	Springer
Pages	139-150
Number of pages	12
ISBN (Electronic)	978-3-642-32677-6
ISBN (Print)	978-3-642-32676-9
DOIs	https://doi.org/10.1007/978-3-642-32677-6_9
Publication status	Published - 1 Jan 2012
Externally published	Yes
Event	10th IFIP WG 2.5 Working Conference on Uncertainty Quantification in Scientific Computing, WoCoUQ 2011 - Boulder, United States Duration: 1 Aug 2011 → 4 Aug 2011 Conference number: 10

Publication series

Name	IFIP Advances in Information and Communication Technology
Publisher	Springer
Volume	377
ISSN (Print)	1868-4238
ISSN (Electronic)	1861-2288

Conference

Conference	10th IFIP WG 2.5 Working Conference on Uncertainty Quantification in Scientific Computing, WoCoUQ 2011
Abbreviated title	WoCoUQ
Country/Territory	United States
City	Boulder
Period	1/08/11 → 4/08/11

Keywords

Bayesian updating
low-rank tensor approximation
parametric problems
uncertainty quantification

Access to Document

10.1007/978-3-642-32677-6_9

Cite this

Matthies, H. G., Litvinenko, A., Pajonk, O., Rosić, B. V., & Zander, E. (2012). Parametric and uncertainty computations with tensor product representations. In A. M. Dienstfrey, & R. F. Boisvert (Eds.), Uncertainty Quantification in Scientific Computing: 10th IFIP WG 2.5 Working Conference, WoCoUQ 2011, Revised Selected Papers (pp. 139-150). (IFIP Advances in Information and Communication Technology; Vol. 377). Springer. https://doi.org/10.1007/978-3-642-32677-6_9

Matthies, Hermann G. ; Litvinenko, Alexander ; Pajonk, Oliver ; Rosić, Bojana V. ; Zander, Elmar. / Parametric and uncertainty computations with tensor product representations. Uncertainty Quantification in Scientific Computing: 10th IFIP WG 2.5 Working Conference, WoCoUQ 2011, Revised Selected Papers. editor / Andrew M. Dienstfrey ; Ronald F. Boisvert. Berlin, Heidelberg : Springer, 2012. pp. 139-150 (IFIP Advances in Information and Communication Technology).

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abstract = "Computational uncertainty quantification in a probabilistic setting is a special case of a parametric problem. Parameter dependent state vectors lead via association to a linear operator to analogues of covariance, its spectral decomposition, and the associated Karhunen-Lo{\`e}ve expansion. From this one obtains a generalised tensor representation The parameter in question may be a tuple of numbers, a function, a stochastic process, or a random tensor field. The tensor factorisation may be cascaded, leading to tensors of higher degree. When carried on a discretised level, such factorisations in the form of low-rank approximations lead to very sparse representations of the high dimensional quantities involved. Updating of uncertainty for new information is an important part of uncertainty quantification. Formulated in terms or random variables instead of measures, the Bayesian update is a projection and allows the use of the tensor factorisations also in this case.",

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author = "Matthies, {Hermann G.} and Alexander Litvinenko and Oliver Pajonk and Rosi{\'c}, {Bojana V.} and Elmar Zander",

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Matthies, HG, Litvinenko, A, Pajonk, O, Rosić, BV & Zander, E 2012, Parametric and uncertainty computations with tensor product representations. in AM Dienstfrey & RF Boisvert (eds), Uncertainty Quantification in Scientific Computing: 10th IFIP WG 2.5 Working Conference, WoCoUQ 2011, Revised Selected Papers. IFIP Advances in Information and Communication Technology, vol. 377, Springer, Berlin, Heidelberg, pp. 139-150, 10th IFIP WG 2.5 Working Conference on Uncertainty Quantification in Scientific Computing, WoCoUQ 2011, Boulder, Colorado, United States, 1/08/11. https://doi.org/10.1007/978-3-642-32677-6_9

Parametric and uncertainty computations with tensor product representations. / Matthies, Hermann G.; Litvinenko, Alexander; Pajonk, Oliver; Rosić, Bojana V.; Zander, Elmar.

Uncertainty Quantification in Scientific Computing: 10th IFIP WG 2.5 Working Conference, WoCoUQ 2011, Revised Selected Papers. ed. / Andrew M. Dienstfrey; Ronald F. Boisvert. Berlin, Heidelberg : Springer, 2012. p. 139-150 (IFIP Advances in Information and Communication Technology; Vol. 377).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Academic › peer-review

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T1 - Parametric and uncertainty computations with tensor product representations

AU - Matthies, Hermann G.

AU - Litvinenko, Alexander

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AU - Rosić, Bojana V.

AU - Zander, Elmar

N1 - Conference code: 10

PY - 2012/1/1

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N2 - Computational uncertainty quantification in a probabilistic setting is a special case of a parametric problem. Parameter dependent state vectors lead via association to a linear operator to analogues of covariance, its spectral decomposition, and the associated Karhunen-Loève expansion. From this one obtains a generalised tensor representation The parameter in question may be a tuple of numbers, a function, a stochastic process, or a random tensor field. The tensor factorisation may be cascaded, leading to tensors of higher degree. When carried on a discretised level, such factorisations in the form of low-rank approximations lead to very sparse representations of the high dimensional quantities involved. Updating of uncertainty for new information is an important part of uncertainty quantification. Formulated in terms or random variables instead of measures, the Bayesian update is a projection and allows the use of the tensor factorisations also in this case.

AB - Computational uncertainty quantification in a probabilistic setting is a special case of a parametric problem. Parameter dependent state vectors lead via association to a linear operator to analogues of covariance, its spectral decomposition, and the associated Karhunen-Loève expansion. From this one obtains a generalised tensor representation The parameter in question may be a tuple of numbers, a function, a stochastic process, or a random tensor field. The tensor factorisation may be cascaded, leading to tensors of higher degree. When carried on a discretised level, such factorisations in the form of low-rank approximations lead to very sparse representations of the high dimensional quantities involved. Updating of uncertainty for new information is an important part of uncertainty quantification. Formulated in terms or random variables instead of measures, the Bayesian update is a projection and allows the use of the tensor factorisations also in this case.

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ER -

Matthies HG, Litvinenko A, Pajonk O, Rosić BV, Zander E. Parametric and uncertainty computations with tensor product representations. In Dienstfrey AM, Boisvert RF, editors, Uncertainty Quantification in Scientific Computing: 10th IFIP WG 2.5 Working Conference, WoCoUQ 2011, Revised Selected Papers. Berlin, Heidelberg: Springer. 2012. p. 139-150. (IFIP Advances in Information and Communication Technology). https://doi.org/10.1007/978-3-642-32677-6_9

Parametric and uncertainty computations with tensor product representations

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