Abstract
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.
Keywords
- Response Surface
- Proper Orthogonal Decomposition
- Quadrature Point
- Sparse Grid
- Polynomial Chaos Expansion
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Acknowledgements
We thank the German Ministry of Economics for the financial support of the project “MUNA—Management and Minimisation of Uncertainties and Errors in Numerical Aerodynamic” within the Luftfahrtforschungsprogramm IV under contract number 20A0604A. We thank also Nathalie Rauschmayr for Fig. 2.
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Litvinenko, A., Matthies, H.G., El-Moselhy, T.A. (2013). Sampling and Low-Rank Tensor Approximation of the Response Surface. In: Dick, J., Kuo, F., Peters, G., Sloan, I. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2012. Springer Proceedings in Mathematics & Statistics, vol 65. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41095-6_27
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DOI: https://doi.org/10.1007/978-3-642-41095-6_27
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