Mathematical Models and Methods in Applied Sciences

The inverse reflector problem arises in geometrical nonimaging optics: given a light source and a target, the question is how to design a reflecting free-form surface such that a desired light density distribution is generated on the target, e.g. a projected image on a screen. This optical problem can mathematically be understood as a problem of optimal transport and equivalently be expressed by a secondary boundary value problem of the Monge–Ampère equation, which consists of a highly nonlinear partial differential equation of second order and constraints. In our approach the Monge–Ampère equation is numerically solved using a collocation method based on tensor-product B-splines, in which nested iteration techniques are applied to ensure the convergence of the nonlinear solver and to speed up the calculation. In the numerical method special care has to be taken for the constraint: it enters the discrete problem formulation via a Picard-type iteration. Numerical results are presented as well for benchmark problems for the standard Monge–Ampère equation as for the inverse reflector problem for various images. The designed reflector surfaces are validated by a forward simulation using ray tracing.

We apply compensated convex transforms to define a multiscale Hausdorff stable method to extract intersections between smooth compact manifolds represented by their characteristic functions or as point clouds embedded in ℝⁿ. We prove extraction results on intersections of smooth compact manifolds and for points of high curvature. As a result of the Hausdorff–Lipschitz continuity of our transforms, we show that our method is stable against dense sampling of smooth manifolds with noise. Examples of explicitly calculated prototype models for some simple cases are presented, which are also used in the proofs of our main results. Numerical experiments in two- and three-dimensional space, and applications to geometric objects are also shown.

We are interested in flows on general networks and derive a kinetic equation describing general production, social or transportation networks. Corresponding macroscopic transport equations for large time and homogenized behavior are obtained and studied numerically. This work continues a recent discussion [Averaged kinetic models for flows on unstructured networks, Kinetic Related Models 4 (2011) 1081–1096] and provides additionally explicit equilibrium solutions, second-order macroscopic approximations as well as numerical simulations on a large scale homogenized network.

In 1993, it was shown by Geymonat, Müller and Triantafyllidis that, in the setting of linearized elasticity, a Γ-convergence result holds for highly oscillating sequences of elastic energies whose functional coercivity constant in ℝ^N is zero while the corresponding coercivity constant on the torus remains positive. We illustrate the range of applicability of that result by finding sufficient conditions for such a situation to occur. We thereby justify the degenerate laminate construction given by Gutiérrez in 1999. We also demonstrate that the predicted loss of strict strong ellipticity resulting from the construction by Gutiérrez is unique within a "laminate-like" class of microstructures. It will only occur for the specific micro-geometry investigated there. Our results thus confer both a rigorous, and a canonical character to those of Gutiérrez.

The global-in-time existence and uniqueness of bounded weak solutions to a spinorial matrix drift–diffusion model for semiconductors is proved. Developing the electron density matrix in the Pauli basis, the coefficients (charge density and spin-vector density) satisfy a parabolic 4 × 4 cross-diffusion system. The key idea of the existence proof is to work with different variables: the spin-up and spin-down densities as well as the parallel and perpendicular components of the spin-vector density with respect to the precession vector. In these variables, the diffusion matrix becomes diagonal. The proofs of the L^∞ estimates are based on Stampacchia truncation as well as Moser- and Alikakos-type iteration arguments. The monotonicity of the entropy (or free energy) is also proved. Numerical experiments in one-space dimension using a finite-volume discretization indicate that the entropy decays exponentially fast to the equilibrium state.

This paper is concerned with the mathematical analysis of a coupled elliptic–parabolic system modeling the interaction between the propagation of electric potential and subsequent deformation of the cardiac tissue. The problem consists in a reaction–diffusion system governing the dynamics of ionic quantities, intra- and extra-cellular potentials, and the linearized elasticity equations are adopted to describe the motion of an incompressible material. The coupling between muscle contraction, biochemical reactions and electric activity is introduced with a so-called active strain decomposition framework, where the material gradient of deformation is split into an active (electrophysiology-dependent) part and an elastic (passive) one. Under the assumption of linearized elastic behavior and a truncation of the updated nonlinear diffusivities, we prove existence of weak solutions to the underlying coupled reaction–diffusion system and uniqueness of regular solutions. The proof of existence is based on a combination of parabolic regularization, the Faedo–Galerkin method, and the monotonicity-compactness method of Lions. A finite element formulation is also introduced, for which we establish existence of discrete solutions and show convergence to a weak solution of the original problem. We close with a numerical example illustrating the convergence of the method and some features of the model.

We study the statistics of backward clusters in a gas of hard spheres at low density. A backward cluster is defined as the group of particles involved directly or indirectly in the backwards-in-time dynamics of a given tagged sphere. We derive upper and lower bounds on the average size of clusters by using the theory of the homogeneous Boltzmann equation combined with suitable hierarchical expansions. These representations are known in the easier context of Maxwellian molecules (Wild sums). We test our results with a numerical experiment based on molecular dynamics simulations.