Predictive simulation under uncertainties has recently become of major interest. There are basically three classical concepts of doing uncertainty quantification (UQ) in the simulation context: Polynomial Chaos Expansion (PCE), Stochastic Collocation (SC) and Monte-Carlo methods (MC). While PCE has already been successfully employed in SimTech, they suffer Gibb's phenomenon for functions that exhibit local singularities. Moreover, PCE is hardly applicable for higher-dimensional problems as it suffers the curse of dimensionality. MC methods overcome both of these issues, but as they exhibit a low convergence rate, many samples are needed in order to achieve highly accurate results with respect to some quantity of interest. The hierarchical, adaptive sparse grid discretization with localized basis functions promises to deal better with local discontinuities than PCE, needing fewer samples than MC, and, in addition, having an explicit representation of the surrogate model for stochastic collocation in UQ.
This project is based on recent developments for adaptive high-dimensional discretizations and aims to bring these novel ideas to the simulation context. It will provide methods and software that are suited, e.g., for the quantification of uncertainties or the representation of characteristic maps of parametrized simulations. The project targets the analysis and interpretation of parameters of multi-phase and multi-physics simulations in a non-intrusive way. The main emphasis of the methodological developments are efficient and problem-adapted discretizations of high-dimensional function spaces that might incorporate local discontinuities.
Funding
Funded 07/2013 to 07/2016 by the SimTech Cluster of Excellence
People
- Dirk Pflüger
- Fabian Franzelin