Optimal control for reinitialization in finite element level set methods

A new optimal control problem that incorporates the residual of the Eikonal equation into its objective is presented. The formulation of the state equation is based on the level set transport equation but extended by an additional source term, correcting the solution so as to minimize the objective functional. The method enforces the constraint so that the interface cannot be displaced at least in the continuous setting. The system of first‐order optimality conditions is derived, linearized, and solved numerically. The control also prevents numerical instabilities, so that no additional stabilization techniques are required. This approach offers the flexibility to include other desired design criteria into the objective functional. The methodology is evaluated numerically in three different examples and compared with other PDE‐based reinitialization techniques. Copyright © 2016 John Wiley & Sons, Ltd.     

Eikonal approximation to 5D wave equations as geodesic motion in a curved 4D spacetime

We first derive the relation between the eikonal approximation to the Maxwell wave equations in an inhomogeneous anisotropic medium and geodesic motion in a three dimensional Riemannian manifold using a method which identifies the symplectic structure of the corresponding mechanics. We then apply an analogous method to the five dimensional generalization of Maxwell theory required by the gauge invariance of Stueckelbergs covariant classical and quantum dynamics to demonstrate, in the eikonal approximation, the existence of geodesic motion for the flow of mass in a four dimensional pseudo-Riemannian manifold. No motion of the medium is required. These results provide a foundation for the geometrical optics of the five dimensional radiation theory and establish a model in which there is mass flow along geodesics. Finally, we discuss the interesting case of relativistic quantum theory in an anisotropic medium as well. In this case the eikonal approximation to the relativistic quantum mechanical current coincides with the geodesic flow governed by the pseudo-Riemannian metric obtained from the eikonal approximation to solutions of the Stueckelberg-Schrödinger equation. This construction provides a model for an underlying quantum mechanical structure for classical dynamical motion along geodesics on a pseudo-Riemannian manifold. The locally symplectic structure which emerges is that of Stueckelbergs covariant mechanics on this manifold.This revised version was published online in April 2005. The publishing date was inserted.     

On accuracy and efficiency of constrained reinitialization

The reinitialization, which is required to regularize the level set function, can be computationally expensive and hence is a determining factor for the overall efficiency of a level set method. However, it often has a significantly adverse impact on the accuracy of the level set solution. This short note is meant to shed light on the efficiency and accuracy issues of the reinitialization process. Using just one clearly defined level set propagation test case with an analytical solution the solutions obtained using a recently proposed efficient lower‐order constrained reinitialization (CR) scheme and standard low‐ and high‐order reinitialization schemes are juxtaposed to evidence the superiority of the novel CR formulation. It is shown that maintaining the location of the zero level set during the reinitialization is crucial for the accuracy and that the displacement caused by standard high‐order reinitialization schemes clearly outweighs the benefit of the high‐order smoothing of the level set function. Finally, results of a three‐dimensional problem are concisely reported to demonstrate the general applicability of the CR scheme. Copyright © 2009 John Wiley & Sons, Ltd.     

On the Hughes' model for pedestrian flow: The one-dimensional case

In this paper we investigate the mathematical theory of Hughes' model for the flow of pedestrians (cf. Hughes (2002) [17]), consisting of a non-linear conservation law for the density of pedestrians coupled with an eikonal equation for a potential modelling the common sense of the task. For such an approximated system we prove existence and uniqueness of entropy solutions (in one space dimension) in the sense of Kru?kov (1970) [22], in which the boundary conditions are posed following the approach of Bardos et al. (1979) [7]. We use BV estimates on the density ρ and stability estimates on the potential ? in order to prove uniqueness. Furthermore, we analyze the evolution of characteristics for the original Hughes' model in one space dimension and study the behavior of simple solutions, in order to reproduce interesting phenomena related to the formation of shocks and rarefaction waves. The characteristic calculus is supported by numerical simulations.     

Dual pole eikonal ands-channel singularities

Eikonalization of dual pole amplitudes, such as the Veneziano amplitude, is shown to lead to singularities in the impact parameter plane, which, in the eikonal approximation, may be interpreted as branch points in the direct channel angular momentum plane. This result is discussed in the light of dual absorption models.     

A COMPACT UPWIND SECOND ORDER SCHEME FOR THE EIKONAL EQUATION

We present a compact upwind second order scheme for computing the viscosity solution of the Eikonal equation. This new scheme is based on： 1. the numerical observation that classical first order monotone upwind schemes for the Eikonal equation yield numerical upwind gradient which is also first order accurate up to singularities; 2. a remark that partial information on the second derivatives of the solution is known and given in the structure of the Eikonal equation and can be used to reduce the size of the stencil. We implement the second order scheme as a correction to the well known sweeping method but it should be applicable to any first order monotone upwind scheme. Care is needed to choose the appropriate stencils to avoid instabilities.     

On the local semiconcavity of the solutions of the eikonal equation

We show that if the Hamiltonian is locally semiconvex with respect to the state variables and strictly convex with respect to the gradient then every viscosity solution of the eikonal equation is locally semiconcave. Furthermore, in the 1D case, we show that every viscosity solution of the eikonal equation is semiconcave if and only if the Hamiltonian is Lipschitz continuous with respect to the state variable.     

On the System of Hamilton–Jacobi and Transport Equations Arising in Geometrical Optics

Abstract

In the present article, we study the system of eikonal and transport equations arising in geometrical optics. The mathematical analysis is performed by using the suitable notion of solution, i.e., the viscosity solution for the Hamilton–Jacobi equation and the measure solution for the transport equation defined via the generalized Filippov characteristics. We study the stability as well as the geometry of the solution to the system.     

Non-perturbative QCD amplitudes in quenched and eikonal approximations

Even though approximated, strong coupling non-perturbative QCD amplitudes remain very difficult to obtain. In this article, in eikonal and quenched approximations at least, physical insights are presented that rely on the newly-discovered property of effective locality. The present article also provides a more rigorous mathematical basis for the crude approximations used in the previous derivation of the binding potential of quarks and nucleons. Furthermore, the techniques of Random Matrix calculus along with Meijer G-functions are applied to analyze the generic structure of fermionic amplitudes in QCD.     

Improved eikonal approach for charge exchange reactions at intermediate energies

In order to describe charge exchange reactions at intermediate energies,we implemented as a first step the formulation of the normal eikonal approach.The calculated differential cross-sections based on this approach deviated significantly from the conventional DWBA calculations for CE reactions at 140 MeV/nucleon.Thereafter,improvements were made in the application of the eikonal approximation so as to keep a strict three-dimensional form factor.The results obtained with the improved eikonal approach are in good agreement with the DWBA calculations and with the experimental data.Since the improved eikonal approach can be formulated in a microscopic way,it is easy to apply to CE reactions at higher energies,where the phenomenological DWBA is a priori difficult to use due to the lack,in most cases,of the required phenomenological potentials.