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Uncertainty quantification by geometric characterization of sensitivity spaces. (English) Zbl 1425.65058
Summary: We propose a systematic procedure for both aleatory and epistemic uncertainty quantification of numerical simulations through geometric characteristics of global sensitivity spaces. Two mathematical concepts are used to characterize the geometry of these spaces and to identify possible impacts of variability in data or changes in the models or solution procedures: the dimension of the maximal free generator subspace in vector spaces and the principal angles between subspaces. We show how these characters can be used as indications on the aleatory and epistemic uncertainties. In the case of large dimensional parameter spaces, these characterizations are established for quantile-based extreme scenarios and a multi-point moment-based sensitivity direction permits to propose a directional uncertainty quantification concept for directional extreme scenarios (DES). The approach is non-intrusive and exploits in parallel the elements of existing mono-point gradient-based design platforms. The ingredients of the paper are illustrated on a model problem with the Burgers equation with control and on a constrained aerodynamic performance analysis problem.

65F25 Orthogonalization in numerical linear algebra
62H25 Factor analysis and principal components; correspondence analysis
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[1] AIAA Guide for the Verification and Validation of Computational Fluid Dynamics Simulations, American Institute of Aeronautics and Astronautics, AIAA-G-077, 1998.
[2] Ghanem, R.; Doostan, A., On the construction and analysis of stochastic models: characterization and propagation of the errors associated with limited data, J. of Comput. Phys., 217, 63-81, (2006) · Zbl 1102.65004
[3] Iaccarino, G., (Quantification of Uncertainty in Flow Simulations Using Probabilistic Methods, VKI Lecture Series, (2008))
[4] Xiu, D., Numerical methods for stochastic computations: A spectral method approach, (2010), Princeton University Press · Zbl 1210.65002
[5] Ghanem, R.; Spanos, P., Stochastic finite elements: A spectral approach, (1991), Springer-Verlag New York · Zbl 0722.73080
[6] W.H. Schilders, H.A. Van der Vorst, J. Rommes, Model order reduction: Theory, research aspects and applications, Springer Math in Industry Series, vol. 13, Berlin, 2008. · Zbl 1143.65004
[7] Obinata, G.; Andersonn, B., Model reduction for control system design, (2000), Springer Berlin
[8] Qu, Z., Model order reduction techniques with applications in finite element analysis, (2004), Springer Berlin
[9] Mohammadi, B., Reduced sampling and incomplete sensitivity for low-complexity robust parametric optimization, Internat. J. Numer. Methods Fluids, (2013)
[10] F. Gallard, B. Mohammadi, M. Montagnac, M. Meaux, Robust Parametric Design by Multi-point Optimization. CERFACS Technical Report TR/CFD/12/33. · Zbl 1327.93176
[11] Jorion, Ph., Value at risk: the new benchmark for managing financial risk, (2006), McGraw-Hill
[12] Mohammadi, B., Value at risk for confidence level quantifications in robust engineering optimization, Optimal Control Appl. Methods, 35/2, 179-190, (2014) · Zbl 1290.49067
[13] Mohammadi, B., Principal angles between subspaces and reduced order modeling accuracy in optimization, Struct. Multidiscip. Optim., (2014)
[14] Witteveen, J. A.S.; Bijl, H., Efficient quantification of the effect of uncertainties in advection-diffusion problems using polynomial chaos, Numer. Heat Transfer B, 53/5, 437-465, (2008)
[15] Melchers, R. E., Structural reliability analysis and prediction, (1999), John Wiley and Sons Chichester
[16] Jordan, C., Essay on geometry in \(n\) dimensions, Bull. Soc. Math. France, 3, 103-174, (1875) · JFM 07.0457.01
[17] Gluck, H.; Warner, F., Great circle fibrations of the three-sphere, Duke Math. J., 50, 107-132, (1983) · Zbl 0523.55020
[18] Jiang, S., Angles between Euclidean subspaces, Geom. Dedicata, 63, 2, 113-121, (1996) · Zbl 0860.51008
[19] Shonkwiler, C., Poincare duality angles for Riemannian manifolds with boundary, (2009), Univ. Pennsylvania, (Ph.D. thesis)
[20] Majda, A., The stability of multi-dimensional shock fronts, (Memoire of the A.M.S., vol. 281, (1983), American Math. Soc. Providence) · Zbl 0506.76075
[21] Godlewski, E.; Olazabal, M.; Raviart, P. A., (On the Linearization of Hyperbolic Systems of Conservation Laws. Application to Stability, (1998), Elsevier Paris) · Zbl 0912.35103
[22] Bardos, C.; Pironneau, O., A formalism for the differentiation of conservation laws, C. R. Acad. Sci., Paris I, 453, (2002) · Zbl 1020.35052
[23] Mohammadi, B.; Pironneau, O., Applied shape optimization for fluids, (2009), Oxford Univ. Press London · Zbl 1179.65002
[24] Giles, M. A.; Pierce, NA., Analytic adjoint solutions for the quasi-one-dimensional Euler equations, J. Fluid Mech., 426, 327-345, (2001) · Zbl 0967.76079
[25] Li, W.; Huyse, L.; Padula, S., Robust airfoil optimization to achieve consistent drag reduction over a Mach range, Struct. Multidiscip. Optim., 24/1, 38-50, (2002)
[26] Mohammadi, B.; Pironneau, O., Shape optimization in fluid mechanics, Annu. Rev. Fluid Mech., 36/1, 255-279, (2004) · Zbl 1076.76020
[27] Firl, M.; Wuchner, R.; Bletzinger, K., Regularization of shape optimization problems using fe-based parametrization, Struct. Multidiscip. Optim., 47/4, 507-521, (2013) · Zbl 1274.74268
[28] Mohammadi, B., Fluid dynamics computation with NSC2KE, INRIA report 70005, (1994)
[29] Roe, P. L., Approximate Riemann solvers, parameters vectors and difference schemes, J. Comput. Phys., 43, 357-372, (1981) · Zbl 0474.65066
[30] Van Albada, G. D.; Van Leer, B., Flux vector splitting and Runge Kutta methods for the Euler equations, ICASE, 84/27, (1984)
[31] Griewank, A., Computational derivatives, (2001), Springer New York
[32] Hascoet, L.; Pascual, V., Tapenade 2.1 user’s guide. INRIA technical report RT-300, (2004)
[33] Christianson, B., Reverse accumulation and implicit functions, Optim. Methods Softw., 9/4, 307-322, (1998) · Zbl 0922.65013
[34] Martins, J. R.R. A.; Lambe, A. B., Multidisciplinary design optimization: a survey of architectures, AIAA J., 51/9, 2049-2075, (2013)
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