TY - GEN A1 - Lopez-Fernandez, Maria A1 - Lubich, Christian A1 - Schädle, Achim T1 - Adaptive, Fast and Oblivious Convolution in Evolution Equations with Memory N2 - To approximate convolutions which occur in evolution equations with memory terms, a variable-stepsize algorithm is presented for which advancing $N$ steps requires only $O(N\log N)$ operations and $O(\log N)$ active memory, in place of $O(N^2)$ operations and $O(N)$ memory for a direct implementation. A basic feature of the fast algorithm is the reduction, via contour integral representations, to differential equations which are solved numerically with adaptive step sizes. Rather than the kernel itself, its Laplace transform is used in the algorithm. The algorithm is illustrated on three examples: a blow-up example originating from a Schrödinger equation with concentrated nonlinearity, chemical reactions with inhibited diffusion, and viscoelasticity with a fractional order constitutive law. T3 - ZIB-Report - 06-45 KW - convolution quadrature KW - adaptivity KW - Volterra integral equations KW - numerical inverse Laplace transform KW - anomalous diffusion KW - fractional order visco Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-9399 ER - TY - GEN A1 - Schädle, Achim A1 - Lopez-Fernandez, Maria A1 - Lubich, Christian T1 - Fast and oblivious convolution quadrature N2 - 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 integro-differential equations of convolution type. In a numerical example we apply it to solve a subdiffusion equation with transparent boundary conditions. T3 - ZIB-Report - 05-09 KW - convolution KW - numerical integration KW - Runge-Kutta methods KW - Volterra integral equation KW - anomalous diffusion Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8437 ER -