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 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 both the new SOE method and 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.
翻译:我们提出一个精确的算法,用于对内核功能进行新型的耗竭总和近似值,并基于SOE开发一个快速的进化二次算法,该算法允许以美元为定单,用于对一个连续的时积分整体进行约同步化的时间步骤计算。SOE方法是由一个半分析性加速扩展总内核(SOE)的组合构建的,以及用于在给定的差错容忍度下最大限度地减少指数数量的模型削减技术。我们使用SOE扩展法对分裂内核的有限部分进行。我们使用SOE扩展法对分裂内核的有限部分进行计算,这样,由于指数内核的加速内核,可以将共振成一个普通的差别方程。我们算法的主要特征是,SOE方法是高效和准确的,对具有可控性上层积极反应的普通内核循环进行计算。我们为新的SOE方法和以SOE为主的进化锥体进行数字分析。我们为新的SOE方法和以SOE为主的进化的变形二次的变形等提供了数字分析,对不同的整体性结果,对不同内等核的完整的精确性表现和拟议的综合性。