Fast and oblivious convolution quadrature

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

Time-Dependent Fluid-Structure Interaction

The problem of determining the manner in which an incoming acoustic wave is scattered by an elastic body immersed in a fluid is one of central importance in detecting and identifying submerged objects. The problem is generally referred to as a fluid-structure interaction and is mathematically formulated as a time-dependent transmission problem. In this paper, we consider a typical fluid-structure interaction problem by using a coupling procedure which reduces the problem to a nonlocal initial-boundary problem in the elastic body with a system of integral equations on the interface between the domains occupied by the elastic body and the fluid. We analyze this nonlocal problem by the Lubich approach via the Laplace transform, an essential feature of which is that it works directly on data in the time domain rather than in the transformed domain. Our results may serve as a mathematical foundation for treating time-dependent fluid-structure interaction problems by convolution quadrature coupling of FEM and BEM

On an explicit representation of the solution of linear stochastic partial differential equations with delays

Based on the analysis of a certain class of linear operators on a Banach space, we provide a closed form expression for the solutions of certain linear partial differential equations with non-autonomous input, time delays and stochastic terms, which takes the form of an infinite series expansion

A new numerical method for obtaining gluon distribution functions G(x,Q2)=xg(x,Q2)G(x,Q^2)=xg(x,Q^2), from the proton structure function F2γp(x,Q2)F_2^{\gamma p}(x,Q^2)

An exact expression for the leading-order (LO) gluon distribution function $G(x,Q^2)=xg(x,Q^2)$ from the DGLAP evolution equation for the proton structure function $F_2^{\gamma p}(x,Q^2)$ for deep inelastic $\gamma^* p$ scattering has recently been obtained [M. M. Block, L. Durand and D. W. McKay, Phys. Rev. D{\bf 79}, 014031, (2009)] for massless quarks, using Laplace transformation techniques. Here, we develop a fast and accurate numerical inverse Laplace transformation algorithm, required to invert the Laplace transforms needed to evaluate $G(x,Q^2)$, and compare it to the exact solution. We obtain accuracies of less than 1 part in 1000 over the entire $x$ and $Q^2$ spectrum. Since no analytic Laplace inversion is possible for next-to-leading order (NLO) and higher orders, this numerical algorithm will enable one to obtain accurate NLO (and NNLO) gluon distributions, using only experimental measurements of $F_2^{\gamma p}(x,Q^2)$.Comment: 9 pages, 2 figure