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ISSN Печать: 2152-5080
ISSN Онлайн: 2152-5099
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SIMPLEX STOCHASTIC COLLOCATION FOR PIECEWISE SMOOTH FUNCTIONS WITH KINKS
Краткое описание
Most approximation methods in high dimensions exploit smoothness of the function being approximated. These methods provide poor convergence results for nonsmooth functions with kinks. For example, such kinks can arise in the uncertainty quantification of quantities of interest for gas networks. This is due to the regulation of the gas flow, pressure, or temperature. But, one can exploit that, for each sample in the parameter space it is known if a regulator was active or not, which can be obtained from the result of the corresponding numerical solution. This information can be exploited in a stochastic collocation method. We approximate the function separately on each smooth region by polynomial interpolation and obtain an approximation to the kink. Note that we do not need information about the exact location of kinks, but only an indicator assigning each sample point to its smooth region. We obtain a global order of convergence of (p + 1)/d, where p is the degree of the employed polynomials and d the dimension of the parameter space.
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Smith, R.C., Uncertainty Quantification: Theory, Implementation, and Applications, Philadelphia: SIAM, 2014.
-
Jakeman, J., 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.
-
Rushdi, A., Swiler, L., Phipps, E., D'Elia, M., and Ebeida, M., VPS: Voronoi Piecewise Surrogate Models for High-Dimensional Data Fitting, Int. J. Uncertainty Quantif., 7(1):1-21, 2017.
-
Witteveen, J. and Iaccarino, G., Refinement Criteria for Simplex Stochastic Collocation with Local Extremum Diminishing Robustness, SIAM J. Sci. Comput., 34(3):A1522-A1543, 2012.
-
Witteveen, J. and Iaccarino, G., Simplex Stochastic Collocation with Random Sampling and Extrapolation for Nonhypercube Probability Spaces, SIAM J. Sci. Comput., 34(2):814-838, 2012.
-
Witteveen, J. and Iaccarino, G., Simplex Stochastic Collocation with ENO-Type Stencil Selection for Robust Uncertainty Quantification, J. Comput. Phys, 239:1-21, 2013.
-
Colombo, I., Nobile, F., Porta, G., Scotti, A., and Tamellini, L., Uncertainty Quantification of Geochemical and Mechanical Compaction in Layered Sedimentary Basins, Comput. Methods Appl. Mech. Eng., 328:122-146, 2018.
-
Sargsyan, K., Safta, C., Debusschere, B., andNajm, H., Uncertainty Quantification Given Discontinuous Model Response and a Limited Number of Model Runs, SIAMJ. Sci. Comput., 34(1):B44-B64, 2012.
-
Griebel, M., Kuo, F., and Sloan, I., The Smoothing Effect of Integration in Rd and the ANOVA Decomposition, Math. Comput, 82:383-400,2013.
-
Griebel, M., Kuo, F., and Sloan, I., Note on "The Smoothing Effect of Integration in Rd and the ANOVA Decomposition," Math. Comput., 86:1855-1876,2017.
-
Sauer, T. and Xu, Y., A Case Study in Multivariate Lagrange Interpolation, in Approximation Theory, Wavelets and Applications, S. Singh, Ed., Dordrecht, the Netherlands: Springer, pp. 443-452, 1995.
-
Koch, T., Hiller, B., Pfetsch, M., and Schewe, L., Evaluating Gas Network Capacities, Philadelphia: SIAM, 2015.
-
Lurie, M., Modeling of Oil Product and Gas Pipeline Transportation, Weinheim, Germany: Wiley-VCH Verlag, 2008.
-
Schmidt, M., Steinbach, M., and Willert, B., High Detail Stationary Optimization Models for Gas Networks: Validation and Results, Optim. Eng., 17(2):437-472, 2016.
-
Fuchs, B., Numerical Methods for Uncertainty Quantification in Gas Network Simulation, PhD, University of Bonn, 2018.
-
Clees, T., Cassirer, K., Hornung, N., Klaassen, B., Nikitin, I., Nikitina, L., Suter, R., and Torgovitskaia, I., MYNTS: Multi-PhYsics NeTwork Simulator, Proc. of 6th International Conference on Simulation and Modeling Methodologies, Technologies and Applications, pp. 179-186, 2016.
-
Niederreiter, H., Random Number Generation and Quasi-Monte Carlo Methods, Philadelphia: SIAM, 1992.
-
Halton, J., On the Efficiency of Certain Quasi-Random Sequences of Points in Evaluating Multi-Dimensional Integrals, Numer. Math., 2:84-90, 1960.
-
Owen, A., Halton Sequences Avoid the Origin, SIAM Rev, 43(3):487-503,2006.
-
Pfluger, D., Spatially Adaptive Sparse Grids for High-Dimensional Problems, Miinchen, Germany: Verlag Dr. Hut, 2010.
-
Pfluger, D., Spatially Adaptive Refinement, Sparse Grids and Applications, J. Garcke and M. Griebel, Eds., Berlin: Springer, pp. 243-262, 2012.
-
Giovanis, D.G. and Shields, M.D., Variance-Based Simplex Stochastic Collocation with Model Order Reduction for High-Dimensional Systems, Int. J. Numer. Methods Eng., 117(11):1079-1116,2019.
-
Devroye, L., Non-Uniform Random Variate Generation, New York: Springer, 1986.
-
Fuhg Jan N., Fau Amélie, Nackenhorst Udo, State-of-the-Art and Comparative Review of Adaptive Sampling Methods for Kriging, Archives of Computational Methods in Engineering, 28, 4, 2021. Crossref