Mathematical Sciences Seminar - Radial basis function approximation methods for solving PDEs
When:
—
Venue:
Birkbeck Main Building, Malet Street
No booking required
For partial differential equations with smooth solutions, radial basis function (RBF) approximation methods are attractive due to the potentially spectral convergence rates, combined with geometrical flexibility. However, in practice, the success is hampered by
ill-conditioning as the problem size grows and as the RBFs are made flatter. Furthermore, the computational cost when direct solution methods are used for the arising full linear systems is prohibitive for large-scale problems.
In this talk, we present a numerical RBF approach that successfully deals with these issues, and is a promising candidate for solving realistic large-scale application problems. First, we discuss how to counteract the conditioning problem by employing the recently developed RBF-QR method, which provides numerically stable evaluations for the small shape parameter range, i.e, nearly flat RBFs, in up to three space dimensions. Then we propose a partition of unity method with local RBF approximants. The locality reduces both memory usage and computational cost compared with the global method. We demonstrate that the RBF-QR algorithm is a key to success for local RBF schemes, and we provide numerical experiments showing spectral convergence with respect to the local problem resolution and algebraic convergence with respect to the partition size. We also discuss how far these results are supported by theory and what the important restrictions are.
Contact name:
Department of Economics, Mathematics and Statistics