Resumen de New developments and applications of radial basis functions in interpolation, approximation and data science

Basim Mustafa

• This thesis explores the applications of Radial Basis Functions (RBF) for solving integral and integro-differential equations, and in data science for probability density estimation. We propose a novel method for solving second-kind Volterra integral equations using Wendland-type RBFs, achieving high accuracy and low computational costs with proven convergence results. This method is extended to linear Volterra integro-differential problems, demonstrating its relevance through numerical examples in fields such as quantum physics and population dynamics. Additionally, we develop interpolation and smoothing variational methods in a Generalized Wendland space, supported by convergence results and numerical illustrations.

  Transitioning to data science, we focus on probability density estimation using kernel methods. A three-step strategy adapts to high and low-density regions, employing Gaussian and Birnbaum-Saunders Power-Exponential (BS-PE) kernel estimators with a residual-based error estimator for comprehensive evaluation. Simulation studies and real data underscore the method's applicability in finance, healthcare, environmental science, and machine learning. This thesis aims to advance RBF methodologies and their practical applications, bridging theoretical foundations with real-world challenges.