TY - GEN

T1 - Sampling and Low-Rank Tensor Approximation of the Response Surface

AU - Litvinenko, Alexander

AU - Matthies, Hermann Georg

AU - El-Moselhy, Tarek A.

N1 - KAUST Repository Item: Exported on 2020-10-01

PY - 2013/11/8

Y1 - 2013/11/8

N2 - Most (quasi)-Monte Carlo procedures can be seen as computing some integral over an often high-dimensional domain. If the integrand is expensive to evaluate-we are thinking of a stochastic PDE (SPDE) where the coefficients are random fields and the integrand is some functional of the PDE-solution-there is the desire to keep all the samples for possible later computations of similar integrals. This obviously means a lot of data. To keep the storage demands low, and to allow evaluation of the integrand at points which were not sampled, we construct a low-rank tensor approximation of the integrand over the whole integration domain. This can also be viewed as a representation in some problem-dependent basis which allows a sparse representation. What one obtains is sometimes called a "surrogate" or "proxy" model, or a "response surface". This representation is built step by step or sample by sample, and can already be used for each new sample. In case we are sampling a solution of an SPDE, this allows us to reduce the number of necessary samples, namely in case the solution is already well-represented by the low-rank tensor approximation. This can be easily checked by evaluating the residuum of the PDE with the approximate solution. The procedure will be demonstrated in the computation of a compressible transonic Reynolds-averaged Navier-Strokes flow around an airfoil with random/uncertain data. © Springer-Verlag Berlin Heidelberg 2013.

AB - Most (quasi)-Monte Carlo procedures can be seen as computing some integral over an often high-dimensional domain. If the integrand is expensive to evaluate-we are thinking of a stochastic PDE (SPDE) where the coefficients are random fields and the integrand is some functional of the PDE-solution-there is the desire to keep all the samples for possible later computations of similar integrals. This obviously means a lot of data. To keep the storage demands low, and to allow evaluation of the integrand at points which were not sampled, we construct a low-rank tensor approximation of the integrand over the whole integration domain. This can also be viewed as a representation in some problem-dependent basis which allows a sparse representation. What one obtains is sometimes called a "surrogate" or "proxy" model, or a "response surface". This representation is built step by step or sample by sample, and can already be used for each new sample. In case we are sampling a solution of an SPDE, this allows us to reduce the number of necessary samples, namely in case the solution is already well-represented by the low-rank tensor approximation. This can be easily checked by evaluating the residuum of the PDE with the approximate solution. The procedure will be demonstrated in the computation of a compressible transonic Reynolds-averaged Navier-Strokes flow around an airfoil with random/uncertain data. © Springer-Verlag Berlin Heidelberg 2013.

UR - http://hdl.handle.net/10754/564667

UR - http://link.springer.com/10.1007/978-3-642-41095-6_27

UR - http://www.scopus.com/inward/record.url?scp=84893507842&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-41095-6_27

DO - 10.1007/978-3-642-41095-6_27

M3 - Conference contribution

SN - 9783642410949

SP - 535

EP - 551

BT - Springer Proceedings in Mathematics & Statistics

PB - Springer Nature

ER -