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Minimal multi-element stochastic collocation for uncertainty quantification of discontinuous functions. (English) Zbl 1311.65158
Summary: We propose a multi-element stochastic collocation method that can be applied in high-dimensional parameter space for functions with discontinuities lying along manifolds of general geometries. The key feature of the method is that the parameter space is decomposed into multiple elements defined by the discontinuities and thus only the minimal number of elements are utilized. On each of the resulting elements the function is smooth and can be approximated using high-order methods with fast convergence properties. The decomposition strategy is in direct contrast to the traditional multi-element approaches which define the sub-domains by repeated splitting of the axes in the parameter space. Such methods are more prone to the curse-of-dimensionality because of the fast growth of the number of elements caused by the axis based splitting. The present method is a two-step approach. Firstly a discontinuity detector is used to partition parameter space into disjoint elements in each of which the function is smooth. The detector uses an efficient combination of the high-order polynomial annihilation technique along with adaptive sparse grids, and this allows resolution of general discontinuities with a smaller number of points when the discontinuity manifold is low-dimensional. After partitioning, an adaptive technique based on the least orthogonal interpolant is used to construct a generalized Polynomial Chaos surrogate on each element. The adaptive technique reuses all information from the partitioning and is variance-suppressing. We present numerous numerical examples that illustrate the accuracy, efficiency, and generality of the method. When compared against standard locally-adaptive sparse grid methods, the present method uses many fewer number of collocation samples and is more accurate.

##### MSC:
 65N75 Probabilistic methods, particle methods, etc. for boundary value problems involving PDEs 65C50 Other computational problems in probability (MSC2010)
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##### References:
 [1] Agarwal, N.; Aluru, N. R., A domain adaptive stochastic collocation approach for analysis of mems under uncertainties, J. Comput. Phys., 228, 20, 7662-7688, (2009) · Zbl 1391.74293 [2] Archibald, R.; Gelb, A.; Saxena, R.; Xiu, D., Discontinuity detection in multivariate space for stochastic simulations, J. Comput. Phys., 228, 7, 2676-2689, (2009) · Zbl 1161.65307 [3] Archibald, R.; Gelb, J.; Yoon, A., Polynomial Fitting for edge detection in irregularly sampled signals and images, SIAM J. Numer. Anal., 43, 1, 259-279, (2005) · Zbl 1093.41009 [4] Archibald, R.; Gelb, J.; Yoon, A., Determining the locations of discontinuities in the derivatives of functions, Appl. Numer. Math., 58, 5, 577-592, (2008) · Zbl 1141.65011 [5] Babuska, I. M.; Nobile, F.; Tempone, R., A stochastic collocation method for elliptic partial differential equations with random input data, SIAM J. Numer. Anal., 45, 3, 1005-1034, (2007) · Zbl 1151.65008 [6] Babuska, I. M.; Tempone, R.; Zouraris, G. E., Galerkin finite element approximations of stochastic elliptic partial differential equations, SIAM J. Numer. Anal., 42, 2, 800-825, (2004) · Zbl 1080.65003 [7] R. Bauer, Band Pass Filters for Determining Shock Locations, PhD thesis, Applied Mathematics, Brown University, 1995. [8] Bieri, M.; Schwab, C., Sparse high order FEM for elliptic SPDEs, Comput. Methods Appl. Mech. Eng., 198, 1149-1170, (2009) · Zbl 1157.65481 [9] C. De Boor, A. Ron, Computational aspects of polynomial interpolation in several variables. Mathematics of Computation, 58(198):705-727, April 1992. ArticleType: primary_article / Full publication date: Apr., 1992 / Copyright -1992 American Mathematical Society. · Zbl 0767.41003 [10] Canny, J., A computational approach to edge detection, IEEE Trans. Pattern Anal. Machine Intell., 8, 679-698, (1986) [11] Chantrasmi, T.; Doostan, A.; Iaccarino, G., Padé-Legendre approximants for uncertainty analysis with discontinuous response surfaces, J. Comput. Phys., 228, 19, 7159G-7180, (2009) · Zbl 1391.76530 [12] Foo, J.; Karniadakis, G. E., Multi-element probabilistic collocation method in high dimensions, J. Comput. Phys., 229, 5, 1536-1557, (2010) · Zbl 1181.65014 [13] Foo, J.; Wan, X.; Karniadakis, G. E., The multi-element probabilistic collocation method (ME-PCM): error analysis and applications, J. Comput. Phys., 227, 22, 9572-9595, (2008) · Zbl 1153.65008 [14] Gardner, T.; Cantor, C.; Collins, J., Construction of a genetic toggle switch in Escherichia coli, Nature, 403, 339-342, (2000) [15] Gelb, A.; Tadmor, E., Adaptive edge detectors for piecewise smooth data based on the minmod limiter, J. Sci. Comput., 28, 2-3, 279-306, (2006) · Zbl 1103.65143 [16] A. Genz, Testing multidimensional integration routines, in: Proceedings of International Conference on Tools, Methods and Languages for Scientific and Engineering Computation, Elsevier North-Holland, Inc., New York, NY, USA, 1984, p. 81G-94. [17] Ghanem, R. G.; Spanos, P. D., Stochastic finite elements: A spectral approach, (1991), Springer-Verlag New York, Inc., New York, NY, USA · Zbl 0722.73080 [18] Griebel, M., Adaptive sparse grid multilevel methods for elliptic PDEs based on finite differences, Computing, 61, 2, 151-179, (1998) · Zbl 0918.65078 [19] J.D. Jakeman, Numerical Methods for the Quantification of Uncertainty in Discontinuous Functions of High Dimension. Ph.D thesis, The Australian National University, November 2010. [20] D Jakeman, J.; Archibald, R.; Xiu, D., Characterization of discontinuities in high-dimensional stochastic problems on adaptive sparse grids, J. Comput. Phys., 230, 10, 3977-3997, (2011) · Zbl 1218.65010 [21] J.D. Jakeman, S.G. Roberts, Local and dimension adaptive stochastic collocation for uncertainty quantification, in: Jochen Garcke, Michael Griebel (Eds.), Sparse Grids and Applications, Lecture Notes in Computational Science and Engineering, vol. 88, Springer, Berlin Heidelberg, 2013, pp. 181-203. [22] Ma, X.; Zabaras, N., An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations, J. Comput. Phys., 228, 3084-3113, (2009) · Zbl 1161.65006 [23] Ma, X.; Zabaras, N., An adaptive high-dimensional stochastic model representation technique for the solution of stochastic partial differential equations, J. Comput. Phys., 229, 10, 3884-3915, (2010) · Zbl 1189.65019 [24] Mathelin, L.; Hussaini, M.; Zang, T., Stochastic approaches to uncertainty quantification in CFD simulations, Numer. Algorithms, 38, 1-3, 209-236, (2005) · Zbl 1130.76062 [25] Narayan, A.; Xiu, D., Stochastic collocation methods on unstructured grids in high dimensions via interpolation, SIAM J. Sci. Comput., 34, 3, A1729-A1752, (2012) · Zbl 1246.65029 [26] Sargsyan, K.; Safta, C.; Debusschere, B.; Najm, H., Uncertainty quantification given discontinuous model response and a limited number of model runs, SIAM J. Sci. Comput., 34, 1, B44-B64, (2012) · Zbl 1237.62035 [27] Sobel, I.; Freeman, H., An isotropic $$3$$ image gradient operator, Machine Vision for Three-Dimensional Scenes, (1990), Academic Press Boston [28] Tatang, M.; Pan, W.; Prinn, R.; McRae, G., An efficient method for parametric uncertainty analysis of numerical geophysical model, J. Geophys. Res., 102, D18, 21925-21932, (1997) [29] Wan, X.; Karniadakis, G. E., An adaptive multi-element generalized polynomial chaos method for stochastic differential equations, J. Comput. Phys., 209, 2, 617-642, (2005) · Zbl 1078.65008 [30] Wan, X.; Karniadakis, G. E., Multi-element generalized polynomial chaos for arbitrary probability measures, SIAM J. Sci. Comput., 28, 3, 901-928, (2006) · Zbl 1128.65009 [31] Xiu, D., Numerical methods for stochastic computations: A spectral method approach, (2010), Princeton University Press · Zbl 1210.65002 [32] Xiu, D.; Hesthaven, J. S., High-order collocation methods for differential equations with random inputs, SIAM J. Sci. Comput., 27, 3, 1118-1139, (2005) · Zbl 1091.65006 [33] Xiu, D.; Karniadakis, G. E., The wiener – askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput., 24, 2, 619-644, (2002) · Zbl 1014.65004
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