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Multilevel adaptive stochastic collocation with dimensionality reduction. (English) Zbl 1408.65072
Garcke, Jochen (ed.) et al., Sparse grids and applications – Miami 2016. Proceeding of the 4th workshop, SGA, Miami, FL, USA, October 4–7, 2016. Cham: Springer. Lect. Notes Comput. Sci. Eng. 123, 43-68 (2018).
Summary: 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.
For the entire collection see [Zbl 1397.65009].

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35R60 PDEs with randomness, stochastic partial differential equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65M75 Probabilistic methods, particle methods, etc. for initial value and initial-boundary value problems involving PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
78A55 Technical applications of optics and electromagnetic theory
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