OPTIMIZATION TECHNIQUE FOR MESHFREE APPROXIMATION METHODS

Kernel-based methods play a fundamental role in interpolation and approximation, particularly in meshfree techniques, where they enable high-dimensional problem-solving with remarkable flexibility. Despite their advantages, Radial Basis Function (RBF) methods often involve dense and computationally demanding linear systems. The Partition of Unity Method (PUM) mitigates these challenges by decomposing the problem into local interpolants, which are then blended through a weighting scheme. This approach extends naturally to Graph-Based Functions with Partition of Unity Methods (GBF-PUM), where e!ective community detection is crucial for interpolation accuracy. This thesis investigates two key aspects: optimizing hyperparameters in RBF and RBFPUM interpolation using Bayesian Optimization (BO) and developing a topology-driven method for community detection in GBF-PUM interpolation. BO is applied to shape parameter tuning, demonstrating comparable accuracy to the classical Leave-One-Out Cross- Validation (LOOCV) method while significantly reducing computational costs. The study extends BO to RBF-PUM, optimizing both the shape parameter and subdomain radius, o!ering a computationally e"cient alternative to LOOCV. Additionally, in the GBF-PUM framework, a novel community detection method leverages graph topology to define subdomains while ensuring that each contains at least one interpolation node. The findings of this thesis contribute to the fields of interpolation, approximation theory, and graph-based methods by introducing innovative strategies for hyperparameter selection and community detection, enhancing both computational e"ciency and interpolation accuracy. !