Adaptive Methods of Optimal Computational Complexity for Partial Differential Equations with Infinitely Many Parameters

• Applicant: Professor Dr. Markus Bachmayr
• Subject Area: Mathematics
• Term: since 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 501389786

The subject of this project are the construction, analysis and implementation of adaptive stochastic Galerkin finite element methods, with particular focus on a rigorous investigation of convergence and complexity properties. The considered class of methods addresses elliptic partial differential equations that depend on countably many parameters. The results of the project obtained thus far show the substantial effects of combining suitable multilevel expansions of random fields with flexible coefficient-wise mesh adaptation for approximate solutions. A first aim of the continuation of the project is the generalization to further problem classes, such as problems on random domains and vector-valued elliptic problems. Moreover, we consider the modification of the methods developed thus far, which aim at convergence in norm, to goal-oriented adaptivity, where also function-valued target quantities such as expectations of solutions are of interest. Finally, we investigate new adaptive methods for stochastic collocation, where approximations of parameter-dependent solutions are constructed from ones for certain parameter values. Also in this context, making use of multilevel structures in expansions of random fields is of central interest.

• DFG Programme: Research Grants