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

Litvinenko, Alexander and Matthies, HermannG. and El-Moselhy, Tarek A., "Sampling and Low-Rank Tensor Approximation of the Response Surface", Monte Carlo and Quasi-Monte Carlo Methods 2012, Springer Proceedings in Mathematics & Statistics Volume 65, 2013, pp 535-551
Litvinenko, Alexander and Matthies, HermannG. and El-Moselhy, Tarek A.
Low-rank response surface, update of surrogate, PCE, low-rank update, uncertainties in aerodynamics, uncertainty quantification
2013

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.

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