Abstract
Most engineering problems contain a large number of input random variables, and thus their polynomial chaos expansion (PCE) suffers from the curse of dimensionality. This issue can be tackled if the polynomial chaos representation is sparse. In the present paper a novel methodology is presented based on combination of \(\ell _1\)-minimization and multifidelity methods. The proposed method employ the \(\ell _1\)-minimization method to recover important coefficients of PCE using low-fidelity computations. The developed method is applied on a stochastic CFD problem and the results are presented. The transonic RAE2822 airfoil with combined operational and geometrical uncertainties is considered as a test case to examine the performance of the proposed methodology. It is shown that the new method can reproduce accurate results with much lower computational cost than the classical full Polynomial Choas (PC), and \(\ell _1\)-minimization methods. It is observed that the present method is almost 15–20 times faster than the full PC method and 3–4 times faster than the classical \(\ell _1\)-minimization method.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Cook P, Firmin M, McDonald M (1977) Aerofoil RAE 2822-Pressure distributions, and boundary layer and wake measurements. Experimental data base for computer program assessment, AGARD Report AR, p 138
Davenport MA, Duarte MF, Eldar Y, Kutyniok G (2011) Introduction to compressed sensing. In: Kutyniok G, Eldar YC (eds) Compressed sensing, theory and applications. Cambridge University Press, pp 1–64
Davis G, Mallat S, Avellaneda M (1997) Adaptive greedy approximations. Constr. Approximation 13(1):57–98
Doostan A, Owhadi H (2011) A non-adapted sparse approximation of PDEs with stochastic inputs. J Comput Phys 230(8):3015–3034
Fishman G (1996) Monte Carlo: concepts. Algorithms and applications, Springer, New York
Ghanem R, Spanos P (1938) Stochastic finite elementsa spectral approach, 3rd edn. Springer, New York
Hosder S, Walters RW, Perez R (2006) A non-intrusive polynomial chaos method for uncertainty propagation in CFD simulations. In: 44th AIAA aerospace sciences meeting and Exhibit
Lucor D, Karniadakis GE (2004) Adaptive generalized polynomial chaos for nonlinear random oscillators. SIAM J Sci Comput 26(2):720–735
Muthukrishnan S (2005) Data streams: algorithms and applications. Now Publishers Inc
Nair PB, Keane AJ (2002) Stochastic reduced basis methods. AIAA J 40(8):1653–1664
Needell D (2009) Topics in compressed sensing. Ph.D. thesis, University of California
Ng LWT, Eldred M (2012a) Multifidelity uncertainty quantification using non-intrusive polynomial chaos and stochastic collocation. In: 53rd AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics and materials conference 20th AIAA/ASME/AHS adaptive structures conference 14th AIAA, p 1852
Ng LWT, Eldred M (2012b) Multifidelity uncertainty quantification using nonintrusive polynomial chaos and stochastic collocation. In: Proceedings of the 14th AIAA non-deterministic approaches conference, number AIAA-2012-1852, vol 43. Honolulu, HI
Pati YC, Rezaiifar R, Krishnaprasad P (1993) Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition. In: 1993 conference record of the twenty-seventh asilomar conference on signals, systems and computers, 1993, pp 40–44
Salehi S, Raisee M, Cervantes MJ, Ahmad (2017) Efficient uncertainty quantification of stochastic CFD problems using sparse polynomial chaos and compressed sensing. Comput Fluids 154:296 – 321. http://www.sciencedirect.com/science/article/pii/S0045793017302281
Sobol’ I (1967) On the distribution of points in a cube and the approximate evaluation of integrals. USSR Comput Math Math Phys 7(4):86–112
Soize C, Ghanem R (2004) Physical systems with random uncertainties: chaos representations with arbitrary probability measure. SIAM J Sci Comput 26(2):395–410
Wan X, Karniadakis GE (2005) An adaptive multi-element generalized polynomial chaos method for stochastic differential equations. J Comput Phys 209(2):617–642
Wiener N (1938) The homogeneous chaos. Am J Math 60(4):897–936
Witteveen JA, Doostan A, Chantrasmi T, Pecnik R, Iaccarino G (2009) Comparison of stochastic collocation methods for uncertainty quantification of the transonic RAE 2822 airfoil. In: Proceedings of workshop on quantification of CFD uncertainties
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Salehi, S., Raisee, M., Cervantes, M.J., Nourbakhsh, A. (2019). Development of an Efficient Multifidelity Non-intrusive Uncertainty Quantification Method. In: Andrés-Pérez, E., González, L., Periaux, J., Gauger, N., Quagliarella, D., Giannakoglou, K. (eds) Evolutionary and Deterministic Methods for Design Optimization and Control With Applications to Industrial and Societal Problems. Computational Methods in Applied Sciences, vol 49. Springer, Cham. https://doi.org/10.1007/978-3-319-89890-2_31
Download citation
DOI: https://doi.org/10.1007/978-3-319-89890-2_31
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-89889-6
Online ISBN: 978-3-319-89890-2
eBook Packages: EngineeringEngineering (R0)