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In this work, a novel approach for the reliable and efficient numerical integration of the Kuramoto model on graphs is studied. For this purpose, the notion of order parameters is revisited for the classical Kuramoto model describing all-to-all interactions of a set of oscillators. First numerical experiments confirm that the precomputation of certain sums significantly reduces the computational cost for the evaluation of the right-hand side and hence enables the simulation of high-dimensional systems. In order to design numerical integration methods that are favourable in the context of related dynamical systems on network graphs, the concept of localised order parameters is proposed. In addition, the detection of communities for a complex graph and the transformation of the underlying adjacency matrix to block structure is an essential component for further improvement. It is demonstrated that for a submatrix comprising relatively few coefficients equal to zero, the precomputation of sums is advantageous, whereas straightforward summation is appropriate in the complementary case. Concluding theoretical considerations and numerical comparisons show that the strategy of combining effective community detection algorithms with the localisation of order parameters potentially reduces the computation time by several orders of magnitude.
We develop an algorithm for the concurrent (on-the-fly) estimation of parameters for a system of evolutionary dissipative partial differential equations in which the state is partially observed. The intuitive nature of the algorithm makes its extensi
The goal of this paper is twofold. First, we present a unified way of formulating numerical integration problems from both approximation theory and discrepancy theory. Second, we show how techniques, developed in approximation theory, work in proving
Solutions to the stochastic wave equation on the unit sphere are approximated by spectral methods. Strong, weak, and almost sure convergence rates for the proposed numerical schemes are provided and shown to depend only on the smoothness of the drivi
A systematic mathematical framework for the study of numerical algorithms would allow comparisons, facilitate conjugacy arguments, as well as enable the discovery of improved, accelerated, data-driven algorithms. Over the course of the last century,
In this article, we construct and analyse explicit numerical splitting methods for a class of semi-linear stochastic differential equations (SDEs) with additive noise, where the drift is allowed to grow polynomially and satisfies a global one-sided L