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 objective of this project is the construction and analysis of adaptive stochastic Galerkin methods with asymptotically optimal computational complexity for elliptic partial differential equations that depend on countably many parameters. Problems of this type are of importance in uncertainty quantification, where the random solution is deterministically approximated by product polynomials in the parametric variables. Our focus is on the development of new methods with spatial discretisation by adaptive finite elements that by a suitable multiscale parameterisation of diffusion coefficients achieve significantly improved optimality guarantees. On this basis, we also consider goal-oriented adaptive methods that adjust discretisations to certain output quantities derived from solutions, as well as the transfer of the main concepts to stochastic collocation methods, which use only evaluations of approximate solutions for certain parameter values.

• DFG Programme: Research Grants