Weighted balanced truncation method for approximating kernel functions by exponentials

Kernel approximation with exponentials is useful in many problems with convolution quadrature and particle interactions such as integral-differential equations, molecular dynamics, and machine learning. In this paper, we introduce a weighted balanced truncation method that significantly reduces the number of exponential terms required for an accurate representation of the kernel. This method shows great promise in approximating long-range kernels, achieving more than four digits of accuracy improvement for the Ewald splitting and inverse power kernels compared to classical balanced truncation. Numerical results demonstrate the attractive performance of the method and promising features for practical applications.