Multilevel sparse grid Lagrange collocation method with positive definite radial basis functions for the solution of elliptic boundary value problems

In this thesis, we introduce the Lagrange collocation method with radial basis functions (LRBF) as a novel approach to solving n-dimensional PDEs. Our method addresses the fundamental challenge – the ‘trade-off principle’ – often associated with standard RBF collocation methods. We aim to maintain the accuracy and convergence of the numerical solution while improving stability and efficiency.

In 1D, we establish its existence and uniqueness for specific differential operators, including the Laplacian operator, and positive definite RBFs. In 2D, we provide some theoretical proofs and validate our claims through numerical experiments.

A pivotal innovation is to perturb the primary matrix, thus defining the perturbed LRBF method (PLRBF). This perturbation enables Cholesky decomposition, reducing the condition number to its square root, leading to the CPLRBF. This allows us to select larger shape parameters without compromising stability and accuracy.

Consequently, we achieve highly accurate solutions early in the process, thus saving time. Surprisingly, the stability of CPLRBF in 2D is not solely influenced by the main matrix but also affected by a second matrix that does not play a key role in 1D. For sufficiently large shape parameters, this second matrix dominates the overall stability, posing limitations to be addressed in future work.

To address stagnation issues arising from using an adaptive shape parameter inversely proportional to the node spacing, we combine PLRBF/CPLRBF with multilevel techniques, resulting in the MuPLRBF/MuCPLRBF. Our findings reveal that MuCPLRBF significantly enhances accuracy early in the process.

In this context, the Multilevel Sparse Grid method yields improvement in terms of accuracy and efficiency for the Gaussian. However, no accuracy improvement was observed for IMQ when using a sparse grid.

Through a series of numerical experiments, we underscore the effectiveness of MuCPLRBF in achieving stability, accuracy, and efficiency in 1D and 2D.