Anisotropic interactions in a first-order aggregation model : a proof of concept

We extend a well-studied ODE model for collective behaviour by considering anisotropic interactions among individuals. Anisotropy is modelled by limited sensorial perception of individuals, that depends on their current direction of motion. Consequently, the first-order model becomes implicit, and new analytical issues, such as non-uniqueness and jump discontinuities in velocities, are being raised. We study the well-posedness of the anisotropic model and discuss its modes of breakdown. To extend solutions beyond breakdown we propose a relaxation system containing a small parameter e, which can be interpreted as a small amount of inertia or response time. We show that the limit e ¿ 0 can be used as a jump criterion to select the physically correct velocities. In smooth regimes, the convergence of the relaxation system as e ¿ 0 is guaranteed by a theorem due to Tikhonov. We illustrate the results with numerical simulations in two dimensions. Keywords: Anisotropy; visual perception; aggregation models; implicit equations; regularization; relaxation time; uniqueness criteria; singular perturbation

From continuum mechanics to SPH particle systems and back : systematic derivation and convergence

In this paper, we employ measure theory to derive from the principle of least action the equation of motion for a continuum with regularized density field. The eventual equation of motion depends on the order in which regularization and the principle of least action are applied. We obtain two different equations, whose discrete counterparts coincide with the scheme used traditionally in the Smoothed Particle Hydrodynamics (SPH) numerical method, and with the equation treated by Di Lisio et al. in 1998, respectively. Additionally, we prove the convergence in the Wasserstein distance of the corresponding measure-valued evolutions, moreover providing the order of convergence of the SPH method. The convergence holds for a general class of force fields, including external and internal conservative forces, friction and non-local interactions. The proof of convergence is illustrated numerically by means of one and two-dimensional examples. Keywords: Smoothed Particle Hydrodynamics, principle of least action, Wasserstein distance, measure-valued equations, convergence rat

Node counting in wireless ad-hoc networks

We study wireless ad-hoc networks consisting of small microprocessors with limited memory, where the wireless communication between the processors can be highly unreliable. For this setting, we propose a number of algorithms to estimate the number of nodes in the network, and the number of direct neighbors of each node. The algorithms are simulated, allowing comparison of their performance

Non-imaging optics for LED-lighting

In this report, several methods are investigated to rapidly compute the light intensity function, either in the far field or on a finite-distance screen, of light emanating from a light fixture with a given shape. Different shapes are considered, namely polygonal and (piecewise) smooth. In the first case, analytic methods are sought to circumvent the use of Monte Carlo methods and ray-tracing with large sample size. In the second case, refinements of the Monte Carlo method (notably using a bootstrap procedure) are devised to minimize the number of samples needed for a good approximation of the intensity function

Mild solutions to a measure-valued mass evolution problem with flux boundary conditions

We investigate the well-posedness and approximation of mild solutions to a class of linear transport equations on the unit interval [0,1][0,1] endowed with a linear discontinuous production term, formulated in the space M([0,1])M([0,1]) of finite Borel measures. Our working technique includes a detailed boundary layer analysis in terms of a semigroup representation of solutions in spaces of measures able to cope with the passage to the singular limit where thickness of the layer vanishes. We obtain not only a suitable concept of solutions to the chosen measure-valued evolution problem, but also derive convergence rates for the approximation procedure and get insight in the structure of flux boundary conditions for the limit problem. Keywords: Measure-valued equations; Flux boundary condition; Mild solutions; Boundary layer asymptotics; Singular limit; Convergence rat

Anisotropic interactions in a first-order aggregation model

We extend a well-studied ODE model for collective behaviour by considering anisotropic interactions among individuals. Anisotropy is modelled by limited sensorial perception of individuals, that depends on their current direction of motion. Consequently, the first-order model becomes implicit, and new analytical issues, such as non-uniqueness and jump discontinuities in velocities, are being raised. We study the well-posedness of the anisotropic model and discuss its modes of breakdown. To extend solutions beyond breakdown we propose a relaxation system containing a small parameter e, which can be interpreted as a small amount of inertia or response time. We show that the limit e ¿ 0 can be used as a jump criterion to select the physically correct velocities. In smooth regimes, the convergence of the relaxation system as e ¿ 0 is guaranteed by a theorem due to Tikhonov. We illustrate the results with numerical simulations in two dimensions. Keywords: Anisotropy; visual perception; aggregation models; implicit equations; regularization; relaxation time; uniqueness criteria; singular perturbation