Multilevel Adaptive Stochastic Collocation with Dimensionality Reduction
- 443 Downloads
We present a multilevel stochastic collocation (MLSC) with a dimensionality reduction approach to quantify the uncertainty in computationally intensive applications. Standard MLSC typically employs grids with predetermined resolutions. Even more, stochastic dimensionality reduction has not been considered in previous MLSC formulations. In this paper, we design an MLSC approach in terms of adaptive sparse grids for stochastic discretization and compare two sparse grid variants, one with spatial and the other with dimension adaptivity. In addition, while performing the uncertainty propagation, we analyze, based on sensitivity information, whether the stochastic dimensionality can be reduced. We test our approach in two problems. The first one is a linear oscillator with five or six stochastic inputs. The dimensionality is reduced from five to two and from six to three. Furthermore, the dimension-adaptive interpolants proved superior in terms of accuracy and required computational cost. The second test case is a fluid-structure interaction problem with five stochastic inputs, in which we quantify the uncertainty at two instances in the time domain. The dimensionality is reduced from five to two and from five to four.
We thank David Holzmueller for developing the dimension-adaptive interpolation module in SG++, employed in this paper. Moreover, we thankfully acknowledge the financial support of the German Academic Exchange Service (http://daad.de/), of the German Research Foundation through the TUM International Graduate School of Science and Engineering (IGSSE) within the project 10.02 BAYES (http://igsse.tum.de/), and the financial support of the priority program 1648 - Software for Exascale Computing of the German Research Foundation (http://www.sppexa.de).
- 1.H.-J. Bungartz, Finite elements of higher order on sparse grids. Habilitationsschrift, Fakultät für Informatik, Technische Universität München, Shaker Verlag, Aachen, 1998Google Scholar
- 4.H.-J. Bungartz, F. Lindner, B. Gatzhammer, M. Mehl, K. Scheufele, A. Shukaev, B. Uekermann, preCICE – a fully parallel library for multi-physics surface coupling. Comput. Fluids (2016). https://doi.org/10.1016/j.compfluid.2016.04.003
- 8.J. Donea, A. Huerta, J.-P. Ponthot, A. Rodriguez-Ferran, Arbitrary lagrangian-eulerian methods. Encycl. Comput. Mech. 1, 413–437 (2004)Google Scholar
- 10.F. Franzelin, P. Diehl, D. Pflüger, Non-intrusive uncertainty quantification with sparse grids for multivariate peridynamic simulations, in Meshfree Methods for Partial Differential Equations VII, ed. by M. Griebel, M.A. Schweitzer. Lecture Notes in Computational Science and Engineering, vol. 100 (Springer International Publishing, Cham, 2015), pp. 115–143Google Scholar
- 13.M. Griebel, M. Schneider, C. Zenger, A combination technique for the solution of sparse grid problems, in Iterative Methods in Linear Algebra, ed. by P. de Groen, R. Beauwens (IMACS, Elsevier, North Holland, 1992), pp. 263–281Google Scholar
- 15.A. Klimke, Uncertainty modeling using fuzzy arithmetic and sparse grids. PhD thesis, Universität Stuttgart, Shaker Verlag, Aachen, 2006Google Scholar
- 21.N.M. Newmark, A method of computation for structural dynamics. J .Eng. Mech. Div. 85(3), 67–94 (1959)Google Scholar
- 24.B. Peherstorfer, Model order reduction of parametrized systems with sparse grid learning techniques. Dissertation, Department of Informatics, Technische Universität München, 2013Google Scholar
- 33.B. Uekermann, J.C. Cajas, B. Gatzhammer, G. Houzeaux, M. Mehl, M. Vazquez, Towards partitioned fluid-structure interaction on massively parallel systems, in 11th World Congress on Computational Mechanics (WCCM XI), ed. by E. Oñate, J. Oliver, A. Huerta (2014), pp. 1–12Google Scholar
- 37.C. Zenger, Sparse grids, in Parallel Algorithms for Partial Differential Equations, Proceedings of the Sixth GAMM-Seminar, Kiel, 1990, ed. by W. Hackbusch. Notes on Numerical Fluid Mechanics, vol. 31 (Vieweg Verlag, Braunschweig, 1991), pp. 241–251Google Scholar