Using radial basis functions for control variate integration of high-dimensional functions defined by sparse sample data

Abstract 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 captures large-scale variations effectively but suffers from exponential basis function growth, our RBF approach adapts to both large- and small-scale data variations. For problems with moderately high dimensions ( $$d \le 20$$ ), numerical experiments demonstrate that the RBF-based CVI significantly outperforms the traditional Monte Carlo and Quasi-Monte Carlo methods. Although the polynomial CVI typically preserves the convergence rate of the underlying sampling method, our RBF CVI demonstrates substantially faster convergence rates. Empirical observations show convergence rates between $$\mathcal {O}(N^{-1.5})$$ and $$\mathcal {O}(N^{-2})$$ in multiple test cases. This performance advantage holds for both independent and low-discrepancy sampling scenarios, offering a practical solution for high-dimensional integration problems with sparse data commonly encountered in computational finance, Bayesian inference, and uncertainty quantification.