TY - GEN
T1 - Parametric and uncertainty computations with tensor product representations
AU - Matthies, Hermann G.
AU - Litvinenko, Alexander
AU - Pajonk, Oliver
AU - Rosić, Bojana V.
AU - Zander, Elmar
PY - 2012
Y1 - 2012
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.
KW - Bayesian updating
KW - low-rank tensor approximation
KW - parametric problems
KW - uncertainty quantification
UR - http://www.scopus.com/inward/record.url?scp=84868322658&partnerID=8YFLogxK
U2 - 10.1007/978-3-642-32677-6_9
DO - 10.1007/978-3-642-32677-6_9
M3 - Conference contribution
AN - SCOPUS:84868322658
SN - 9783642326769
T3 - IFIP Advances in Information and Communication Technology
SP - 139
EP - 150
BT - Uncertainty Quantification in Scientific Computing - 10th IFIP WG 2.5 Working Conference, WoCoUQ 2011, Revised Selected Papers
A2 - Dienstfrey, Andrew M.
A2 - Boisvert, Ronald F.
PB - Springer New York LLC
T2 - 10th IFIP WG 2.5 Working Conference on Uncertainty Quantification in Scientific Computing, WoCoUQ 2011
Y2 - 1 August 2011 through 4 August 2011
ER -