Using Radial Basis Functions for Control Variate Integration of High-Dimensional Functions Defined by Sparse Sample Data

We introduce a control variate integration (CVI) method using radial basis functions (RBFs) for high-dimensional numerical integration with sparse sample data. Unlike polynomial-based CVI, which effectively captures large-scale variations but suffers from the curse of dimensionality, our RBF approach inherently captures both large- and small-scale data variations. For problems with moderately high dimensions (d ≤ 20), numerical experiments demonstrate that the RBF-based CVI significantly outperforms the traditional Monte Carlo and Quasi-Monte Carlo methods. Although the polynomial CVI typically maintains the same convergence rate as the underlying sampling method, our RBF CVI achieves substantially faster convergence rates, empirically observed between O(N ^{−1.5}) and O(N ^{−2}). This performance advantage holds for both independent and low-discrepancy sampling scenarios, offering a practical solution for high-dimensional integration problems with sparse data.