A Kernel-Independent Sum-of-Exponentials Method and Its Application to Convolution Quadrature

We propose an accurate algorithm for a novel sum-of-exponentials (SOE) approximation of kernel functions, and develop a fast algorithm for convolution quadrature based on the SOE, which allows an order $N$ calculation for $N$ time steps of approximating a continuous temporal convolution integral. The SOE method is constructed by a combination of the de la Vall\'ee-Poussin sums for a semi-analytical exponential expansion of a general kernel, and a model reduction technique for the minimization of the number of exponentials under given error tolerance. We employ the SOE expansion for the finite part of the splitting convolution kernel such that the convolution integral can be solved as a system of ordinary differential equations due to the exponential kernels. The remaining part is explicitly approximated by employing the generalized Taylor expansion. The significant features of our algorithm are that the SOE method is efficient and accurate, and works for general kernels with controllable upperbound of positive exponents. We provide numerical analysis for the SOE-based convolution quadrature. Numerical results on different kernels, the convolution integral and integral equations demonstrate attractive performance of both accuracy and efficiency of the proposed method.