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International Journal for Uncertainty Quantification
Импакт фактор: 4.911 5-летний Импакт фактор: 3.179 SJR: 1.008 SNIP: 0.983 CiteScore™: 5.2

ISSN Печать: 2152-5080
ISSN Онлайн: 2152-5099

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International Journal for Uncertainty Quantification

DOI: 10.1615/Int.J.UncertaintyQuantification.2019026974
pages 415-437

ROBUST UNCERTAINTY QUANTIFICATION USING RESPONSE SURFACE APPROXIMATIONS OF DISCONTINUOUS FUNCTIONS

Timothy Wildey
Optimization and Uncertainty Quantification Department, Center for Computing Research, Sandia National Laboratories, Albuquerque, NM 87185
A. A. Gorodetsky
University of Michigan, Department of Aerospace Engineering, Ann Arbor, MI 48109
A. Belme
Sorbonne Universités, UPMC Univ Paris 06, UMR 7190, Institut Jean le Rond d'Alembert, F-75005, Paris, France; CNRS, UMR 7190, Institut Jean le Rond d'Alembert, F-75005, Paris, France
John N. Shadid
Computational Mathematics Department, Center for Computing Research, Sandia National Laboratories, Albuquerque, NM 87185, and Department of Mathematics and Statistics, University of New Mexico

Краткое описание

This paper considers response surface approximations for discontinuous quantities of interest. Our objective is not to adaptively characterize the interface defining the discontinuity. Instead, we utilize an epistemic description of the uncertainty in the location of a discontinuity to produce robust bounds on sample-based estimates of probabilistic quantities of interest. We demonstrate that two common machine learning strategies for classification, one based on nearest neighbors (Voronoi cells) and one based on support vector machines, provide reasonable descriptions of the region where the discontinuity may reside. In higher dimensional spaces, we demonstrate that support vector machines are more accurate for discontinuities defined by smooth interfaces. We also show how gradient information, often available via adjoint-based approaches, can be used to define indicators to effectively detect a discontinuity and to decompose the samples into clusters using an unsupervised learning technique. Numerical results demonstrate the epistemic bounds on probabilistic quantities of interest for simplistic models and for a compressible fluid model with a shock-induced discontinuity.

ЛИТЕРАТУРА

  1. 1. Ghanem, R. and Spanos, P., Stochastic Finite Elements: A Spectral Approach, New York: Springer Verlag, 2002.

  2. Xiu, D. and Karniadakis, G., The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations, SIAM J. Sci. Comput, 24:619-644,2002.

  3. Wan, X. and Karniadakis, G., Beyond Wiener-Askey Expansions: Handling Arbitrary PDFs, J. Sci. Comput., 27:455-464, 2006.

  4. Le Maitre, O., Ghanem, R., Knio, O., and Najm, H., Uncertainty Propagation Using Wiener-Haar Expansions, J. Comput. Phys, 197(1):28-57,2004.

  5. Ghanem, R. and Red-Horse, J., Propagation of Probabilistic Uncertainty in Complex Physical Systems Using a Stochastic Finite Element Approach, Phys. D, 133:137-144,1999.

  6. Butler, T., Dawson, C., and Wildey, T., A Posteriori Error Analysis of Stochastic Spectral Methods, SIAM J. Sci. Comput, 33:1267-1291,2011.

  7. Babuska, I., Nobile, F., and Tempone, R., A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data, SIAM J. Numer. Anal, 45(3):1005-1034,2007.

  8. Ganis, B., Klie, H., Wheeler, M.F., Wildey, T., Yotov, I., and Zhang, D., Stochastic Collocation and Mixed Finite Elements for Flow in Porous Media, Comput. Methods Appl. Mech. Eng., 197(43-44):3547-3559,2008.

  9. Gerstner, T. and Griebel, M., Dimension-Adaptive Tensor-Product Quadrature, Comput:., 71(1):65-87,2003.

  10. Hegland, M., Adaptive Sparse Grids, Proc. of 10th Computational Techniques and Applications Conference CTAC-2001, K. Burrage and R.B. Sidje, Eds., Vol. 44, pp. C335-C353,2003.

  11. Bungartz, H.J. and Griebel, M., Sparse Grids, Acta Numer, 13:147-269,2004.

  12. Ma, X. and Zabaras, N., An Adaptive Hierarchical Sparse Grid Collocation Algorithm for the Solution of Stochastic Differential Equations, J. Comput. Phys, 228(8):3084-3113,2009.

  13. Oseledets, I., Tensor-Train Decomposition, SIAM J. Sci. Comput, 33(5):2295-2317,2011.

  14. Doostan, A., Validi, A., and Iaccarino, G., Non-Intrusive Low-Rank Separated Approximation of High-Dimensional Stochastic Models, Comput. Methods Appl. Mech. Eng., 263:42-55,2013.

  15. Gorodetsky, A. and Jakeman, J., Gradient-Based Optimization for Regression in the Functional Tensor-Train Format, J. Comput. Phys, 374:1219-1238,2018.

  16. Gorodetsky, A., Karaman, S., and Marzouk, Y., A Continuous Analogue of the Tensor-Train Decomposition, Comput. Methods Appl. Mech. Eng., 347:59-84,2019.

  17. Giunta, A., Swiler, L., Brown, S., Eldred, M., Richards, M., and Cyr, E., The Surfpack Software Library for Surrogate Modeling of Sparse Irregularly Spaced Multidimensional Data, Proc. of 11th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conf, 2006.

  18. Sacks, J., Schiller, S.B., and Welch, W.J., Designs for Computer Experiments, Technometrics, 31(1):41-47,1989.

  19. Jakeman, J.D., Archibald, R., and Xiu, D., Characterization of Discontinuities in High-Dimensional Stochastic Problems on Adaptive Sparse Grids, J Comput. Phys, 230(10):3977-3997,2011.

  20. Rushdi, A.A., Swiler, L.P., Phipps, E.T., D'Elia, M., and Ebeida, M.S., VPS: Voronoi Piecewise Surrogate Models for High-Dimensional Data Fitting, Int. J. Uncertainty Quantif., 7(1): 1.

  21. Archibald, R., Gelb, A., and Yoon, J., Determining the Locations and Discontinuities in the Derivatives of Functions, Appl. Numer. Math, 58(5):577-592,2008.

  22. Jakeman, J.D., Narayan, A., and Xiu, D., Minimal Multi-Element Stochastic Collocation for Uncertainty Quantification of Discontinuous Functions, J. Comput. Phys, 242:790-808,2013.

  23. Witteveen, J.A. and Iaccarino, G., Subcell Resolution in Simplex Stochastic Collocation for Spatial Discontinuities, J. Comput. Phys, 251:17-52,2013.

  24. Witteveen, J.A. and Iaccarino, G., Simplex Stochastic Collocation with ENO-Type Stencil Selection for Robust Uncertainty Quantification, J. Comput. Phys., 239: 1-21,2013.

  25. Marchuk, G.I., Agoshkov, V.I., and Shutyaev, V.P., Adjoint Equations and Perturbation Algorithms in Nonlinear Problems, Boca Raton, FL: CRC Press, 1996.

  26. Cacuci, D., Sensitivity and Uncertainty Analysis: Theory, Vol. I, New York: Chapman & Hall/CRC, 2003.

  27. Marchuk, G.I., Adjoint Equations and Analysis of Complex Systems, New York: Kluwer, 1995.

  28. Shadid, J.N., Smith, T., Cyr, E.C., Wildey, T., and Pawlowski, R.P., Stabilized FE Simulation of Prototype Thermal-Hydraulics Problems with Integrated Adjoint-Based Capabilities, J. Comput. Phys, 321:321-341,2016.

  29. Dwight, R.P. and Han, Z.H., Efficient Uncertainty Quantification Using Gradient-Enhanced Kriging, Proc. of 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conf. 17th AIAA/ASME/AHS Adaptive Structures Conf. 11th AIAA No, AIAA Paper no. 2276,2009.

  30. Lockwood, B.A. and Anitescu, M., Gradient-Enhanced Universal Kriging for Uncertainty Propagation, Nucl. Sci. Eng., 170:168-195,2012.

  31. Butler, T., Dawson, C., and Wildey, T., Propagation of Uncertainties Using Improved Surrogate Models, SIAM/ASA J. Uncertainty Quantif., 1(1):164-191,2013.

  32. Butler, T., Constantine, P., and Wildey, T., A Posteriori Error Analysis of Parameterized Linear Systems Using Spectral Methods, SIAM. J Matrix Anal. Appl, 33:195-209,2012.

  33. Bryant, C., Prudhomme, S., and Wildey, T., Error Decomposition and Adaptivity for Response Surface Approximations from PDEs with Parametric Uncertainty, SIAM/ASA J. Uncertainty Quantif, 3(1):1020-1045,2015.

  34. Van Langenhove, J.W., Adaptive Control of Deterministic and Stochastic Approximation Errors in Simulations of Compressible Flow, PhD, Universite Pierre et Marie Curie, Paris, France, 2017.

  35. Gorodetsky, A. and Marzouk, Y., Efficient Localization of Discontinuities in Complex Computational Simulations, SIAM J. Sci. Comput, 36(6):A2584-A2610,2014.

  36. Sargsyan, K., Safta, C.,Debusschere, B., andNajm, H., Uncertainty Quantification Given Discontinuous Model Response and a Limited Number of Model Runs, SIAM J. Sci. Comput, 34(1):B44-B64,2012.

  37. Jakeman, J., Eldred, M., and Sargsyan, K., Enhancing-Minimization Estimates of Polynomial Chaos Expansions Using Basis Selection, J. Comput. Phys, 289:18-34,2015.

  38. Blatman, G. and Sudret, B., Adaptive Sparse Polynomial Chaos Expansion based on Least Angle Regression, J. Comput. Phys, 230(6):2345-2367,2011.

  39. Jakeman, J. and Wildey, T., Enhancing Adaptive Sparse Grid Approximations and Improving Refinement Strategies Using Adjoint-Based a Posteriori Error Estimates, J. Comput. Phys., 280:54-71,2015.

  40. Jiang, G.S. and Shu, C.W., Efficient Implementation of Weighted ENO Schemes, J. Comput. Phys, 126(1):202-228,1996.

  41. Shu, C.W., High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems, SIAM Rev., 51(1):82-126,2009.

  42. Colella, P. and Woodward, P.R., The Piecewise Parabolic Method (PPM) for Gas-Dynamical Simulations, J. Comput. Phys, 54(1):174-201,1984.

  43. Chang, C.C. and Lin, C.J., LIBSVM: A Library for Support Vector Machines, ACM Trans. Intelligent Systems Technol, 2(3):27,2011.

  44. Ebeida, M.S., Davidson, A.A., Patney, A., Knupp, P.M., Mitchell, S.A., and Owens, J.D., Efficient Maximal Poisson-Disk Sampling, ACM Trans. Graph, 30(4):49-1-49-12,2011.

  45. Evgeniou, T., Pontil,M., andPoggio, T., RegularizationNetworks and Support Vector Machines, Adv. Comput. Math., 13(1): 1, 2000.

  46. Menier, V., Loseille, A., and Alauzet, F., CFD Validation and Adaptivity for Viscous Flow Simulations, Proc. of 44th AIAA Fluid Dynamics Conference, AIAA Paper No. 2014-2925,2014.


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