We study properties of an attractive-repulsive energy functional based on power-kernels, which can be used for halftoning of images. In the first part of this work, using a variational framework for probability measures, we examine existence and behavior of minimizers to the functional and to a regularization of it by a total variation term. Moreover, we introduce particle approximations to the functional and to its regularized version and prove their consistency in terms of Gamma-convergence, which we additionally illustrate by numerical examples. In the second part, we consider the gradient flow of the functional in the 2-Wasserstein space and prove statements about its asymptotic behavior for large times, for which we employ the pseudo-inverse technique for probability measures in 1D. Depending on the parameter range, this includes existence of a subsequence converging to a steady state or even convergence of the whole trajectory to a limit which we can specify explicitly. For both parts of the work, a key ingredient is the generalized Fourier transform, which allows us to verify the conditional positive definiteness of the interaction kernel for coinciding attractive and repulsive exponents.
With respect to our particular problem and the static setting, see @cite_28 for efficient optimization algorithms to find local minima of ( E ) and @cite_6 for the relationship of minimizers of ( E ) and the error of quadrature formulas, also highlighting the connection between those minimizers and the problem of optimal quantization of measures (see @cite_23 @cite_32 ). As for the gradient flow, see for example @cite_3 for the analysis of symmetric steady states for the gradient flow of interaction functionals similar to ( W ), but being composed of the sum of an attractive and a repulsive power function.
@cite_28 @cite_32 @cite_3 @cite_6 @cite_23
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Motivated by a recent halftoning method which is based on electrostatic principles, we analyze a halftoning framework where one minimizes a functional consisting of the difference of two convex functions. One describes attracting forces caused by the image's gray values; the other one enforces repulsion between points. In one dimension, the minimizers of our functional can be computed analytically and have the following desired properties: The points are pairwise distinct, lie within the image frame, and can be placed at grid points. In the two-dimensional setting, we prove some useful properties of our functional, such as its coercivity, and propose computing a minimizer by a forward-backward splitting algorithm. We suggest computing the special sums occurring in each iteration step of our dithering algorithm by a fast summation technique based on the fast Fourier transform at nonequispaced knots, which requires only @math arithmetic operations for @math points. Finally, we present numerical results showing the excellent performance of our dithering method. In this paper, we investigate nonlocal interaction equations with repulsive-attractive radial potentials. Such equations describe the evolution of a continuum density of particles in which they repulse each other in the short range and attract each other in the long range. We prove that under some conditions on the potential, radially symmetric solutions converge exponentially fast in some transport distance toward a spherical shell stationary state. Otherwise we prove that it is not possible for a radially symmetric solution to converge weakly toward the spherical shell stationary state. We also investigate under which condition it is possible for a non-radially symmetric solution to converge toward a singular stationary state supported on a general hypersurface. Finally we provide a detailed analysis of the specific case of the repulsive-attractive power law potential as well as numerical results. We point out the the conditions of radial ins stability are sharp. Moving groups of animals, including fish, ungulates, birds and honeybee swarms seem able to take complex decisions in the absence of signalling mechanisms, and when group members cannot establish who has or has not got information. A numerical simulation shows how such groups can make accurate consensus decisions, and that the larger the group, the smaller the proportion of informed individuals needed to guide the group. A very small proportion of informed individuals is sufficient for near maximal accuracy. This has implications for our understanding of the evolution of information transfer in groups, and also suggests a new design protocol for the guidance of grouping robots. Cover photo, by Phillip Colla Natural History Photography ( http: www.OceanLight.com ), shows schooling jack mackerel. Minimum sums of moments or, equivalently, distortion of optimum quantizers play an important role in several branches of mathematics. Fejes Toth's inequality for sums of moments in the plane and Zador's asymptotic formula for minimum distortion in Euclidean d-space are the first precise pertinent results in dimension d⩾2. In this article these results are generalized in the form of asymptotic formulae for minimum sums of moments, resp. distortion of optimum quantizers on Riemannian d-manifolds and normed d-spaces. In addition, we provide geometric and analytic information on the structure of optimum configurations. Our results are then used to obtain information on (i) the minimum distortion of high-resolution vector quantization and optimum quantizers, (ii) the error of best approximation of probability measures by discrete measures and support sets of best approximating discrete measures, (iii) the minimum error of numerical integration formulae for classes of Holder continuous functions and optimum sets of nodes, (iv) best volume approximation of convex bodies by circumscribed convex polytopes and the form of best approximating polytopes, and (v) the minimum isoperimetric quotient of convex polytopes in Minkowski spaces and the form of the minimizing polytopes.