Durchmusterung. Wieners Himmel

Norbert Wieners >Cybernetics> von 1948 geht aus von einem Vergleich zählender, durchmusternder Astronomie mit der neuen, statistikbasierten Meteorologie. Das Buch beginnt mit der ersten Strophe von >Weißt du vieviel Sternlein stehen<. ... Dieses Liedchen ist ein interessantes Thema für die Philosophie und die Geschichte der Wissenschaft, indem es zwei Wissenschaften nebeneinander stellt, die einerseits sich beide mit der Beobachtung des Himmels über uns beschäftigen, andererseits aber beinahe in jeder Beziehung höchst gegensätzlich sind. Die Astronomie ist die älteste der Wissenschaften, während die Meteorologie zu den jüngsten zählt, die erst anfangen, den Namen zu verdienen. ..

Modeling the morphology evolution of organic solar cells

Organic solar cells present a promising alternative for the generation of solar energy at lower material and production costs compared to widely used silicon-based solar cells. The major drawback of organic solar cells currently is a lower rate of energy conversion. Thus many research projects aim to improve the achievable efficiency. In this work a phase field model is used to mathematically describe the morphology evolution of the active layer composed of polymer as electron-donor and fullerene as electron-acceptor. The derivation of a chemical potential term and a surface energy term for the polymer-fullerene solution using the Flory-Huggins theory forms the basis to employ the Cahn-Hilliard equation. After including several specifics of the application in this non-linear partial differential equation of fourth order, an implementation of the model using the FEM solver software FEniCS provides some simulation results that qualitatively match results from the literature

Adaptive rational Krylov methods for exponential Runge--Kutta integrators

We consider the solution of large stiff systems of ordinary differential equations with explicit exponential Runge--Kutta integrators. These problems arise from semi-discretized semi-linear parabolic partial differential equations on continuous domains or on inherently discrete graph domains. A series of results reduces the requirement of computing linear combinations of $\varphi$-functions in exponential integrators to the approximation of the action of a smaller number of matrix exponentials on certain vectors. State-of-the-art computational methods use polynomial Krylov subspaces of adaptive size for this task. They have the drawback that the required Krylov subspace iteration numbers to obtain a desired tolerance increase drastically with the spectral radius of the discrete linear differential operator, e.g., the problem size. We present an approach that leverages rational Krylov subspace methods promising superior approximation qualities. We prove a novel a-posteriori error estimate of rational Krylov approximations to the action of the matrix exponential on vectors for single time points, which allows for an adaptive approach similar to existing polynomial Krylov techniques. We discuss pole selection and the efficient solution of the arising sequences of shifted linear systems by direct and preconditioned iterative solvers. Numerical experiments show that our method outperforms the state of the art for sufficiently large spectral radii of the discrete linear differential operators. The key to this are approximately constant rational Krylov iteration numbers, which enable a near-linear scaling of the runtime with respect to the problem size

A nonlinear spectral core-periphery detection method for multiplex networks

Core-periphery detection aims to separate the nodes of a complex network into two subsets: a core that is densely connected to the entire network and a periphery that is densely connected to the core but sparsely connected internally. The definition of core-periphery structure in multiplex networks that record different types of interactions between the same set of nodes but on different layers is nontrivial since a node may belong to the core in some layers and to the periphery in others. The current state-of-the-art approach relies on linear combinations of individual layer degree vectors whose layer weights need to be chosen a-priori. We propose a nonlinear spectral method for multiplex networks that simultaneously optimizes a node and a layer coreness vector by maximizing a suitable nonconvex homogeneous objective function by an alternating fixed point iteration. We prove global optimality and convergence guarantees for admissible hyper-parameter choices and convergence to local optima for the remaining cases. We derive a quantitative measure for the quality of a given multiplex core-periphery structure that allows the determination of the optimal core size. Numerical experiments on synthetic and real-world networks illustrate that our approach is robust against noisy layers and outperforms baseline methods with respect to a variety of core-periphery quality measures. In particular, all methods based on layer aggregation are improved when used in combination with the novel optimized layer coreness vector weights. As the runtime of our method depends linearly on the number of edges of the network it is scalable to large-scale multiplex networks