Fast and Oblivious Convolution Quadrature

Abstract

Abstract. We give an algorithm to compute N steps of a convolution quadrature approximation to a continuous temporal convolution using only O(N log N ) multiplications and O(log N ) active memory. The method does not require evaluations of the convolution kernel, but instead O(log N ) evaluations of its Laplace transform, which is assumed sectorial. The algorithm can be used for the stable numerical solution with quasi-optimal complexity of linear and nonlinear integral and integrodifferential equations of convolution type. In a numerical example we apply it to solve a subdiffusion equation with transparent boundary conditions.

Numerical solution of exterior Maxwell problems by Galerkin BEM and Runge–Kutta convolution quadrature

“…[23,24]), convolution quadrature based on linear multistep methods has been applied to numerous problems (cf. [25,7,35,34,38,12]); fast numerical implementations were developed in [18,17,20] . For a review on convolution quadrature and its applications we refer to [26,8].…”

A Kernel Compression Scheme for Fractional Differential Equations

“…The algorithm presented in [9] has been used as the foundation for several methods [10,13] and extended to allow variable step size [11]. To begin the discussion we outline the scheme proposed in [9].…”

“…The idea to reduce the costs of evaluating the history term of the fractional integral [8], or more generally [9,10,11] a convolution K * f , by approximating it by a linear combination of solutions to ODEs (1.2) has been explored in the past. In [8], this amounts to the approximation of the kernel w by a linear combination of exponentials (1.3) S(Λ, σ; t) = J j=1…”

Numerical treatment of retarded boundary integral equations by sparse panel clustering

“…A related, interesting variation of the convolution quadrature for convolution kernels whose Laplace transform is sectorial can be found in Schädle et al (2006).…”