Hamilton-Jacobi Equation Research Articles

On the rate of convergence in homogenization of time-fractional Hamilton–Jacobi equations

Here, we consider periodic homogenization for time-fractional Hamilton–Jacobi equations. By using the perturbed test function method, we establish the convergence, and give estimates on a rate of convergence. A main difficulty is the incompatibility between the function used in the doubling variable method, and the non-locality of the Caputo derivative. Our approach is to provide a lemma to prove the rate of convergence without the doubling variable method with respect to the time variable, which is a key ingredient.

Stability of hypercontractivity, the logarithmic Sobolev inequality, and Talagrand's cost inequality

We provide deficit estimates for Nelson's hypercontractivity inequality, the logarithmic Sobolev inequality, and Talagrand's transportation cost inequality under the restriction that the inputs are semi-log-subharmonic, semi-log-convex, or semi-log-concave. In particular, our result on the logarithmic Sobolev inequality complements a recently obtained result by Eldan, Lehec and Shenfeld concerning a deficit estimate for inputs with small covariance. Similarly, our result on Talagrand's transportation cost inequality complements and, for a large class of semi-log-concave inputs, improves a deficit estimate recently proved by Mikulincer. Our deficit estimates for hypercontractivity will be obtained by using a flow monotonicity scheme built on the Fokker–Planck equation, and our deficit estimates for the logarithmic Sobolev inequality will be derived as a corollary. For Talagrand's inequality, we use an optimal transportation argument. An appealing feature of our framework is robustness and this allows us to derive deficit estimates for the hypercontracivity inequality associated with the Hamilton–Jacobi equation, the Poincaré inequality, and for Beckner's inequality.

Level set function based on conformal geometry theory: An efficient approach for optimization of shell structure with arbitrary Gaussian curvature

Topology optimization based on the level set (LS) theory is of great significance to shell structure design. However, the Gaussian curvature of the shell structure can turn complex in many cases, and the establishing process of the LS function will become extremely cumbersome, which consumes overmuch time to solve the Hamilton-Jacobi(H-J) equation. In order to improve the efficiency of optimization, an LS function based on the conformal geometry theory (LSF-CGT) is proposed in this paper. Firstly, the shell structure with arbitrary Gaussian curvature is conformally mapped into 2D Euclidean space to generate the orthogonal mesh of the preset initial LS function. Then the barycentric interpolation method is used to reveal the transformation relationship between the orthogonal and shell structure mesh. Finally, the LS function of the orthogonal mesh is assigned to the mapped shell structure, and the LS function of the 3D shell structure is subsequently obtained. Three typical shell structures with positive, zero, and negative Gaussian curvature are used to verify the effectiveness of LSF-CGT in topology optimization. Results show that LSF-CGT can effectively reduce the strain energy of shell structures with arbitrary Gaussian curvatures by more than 30%. As a method for optimizing shell structures with arbitrary Gaussian curvatures, LSF-CGT can not only stably and quickly converge to the optimal structure in the 2D domain but also effectively maintain the initial geometric characteristics of the shell structure.

Slow periodic homogenization for Hamilton–jacobi equations

Capuzzo-Dolcetta–Ishii proved that the rate of periodic homogenization for coercive Hamilton–Jacobi equations is . We complement this result by constructing examples of coercive nonconvex Hamiltonians whose rate of periodic homogenization is .

Optimal Trajectories of a UAV Base Station Using Hamilton-Jacobi Equations

We consider the problem of optimizing the trajectory of an Unmanned Aerial Vehicle (UAV). Assuming a traffic intensity map of users to be served, the UAV must travel from a given initial location to a final position within a given duration and serves the traffic on its way. The problem consists in finding the optimal trajectory that minimizes a certain cost depending on the velocity and on the amount of served traffic. We formulate the problem using the framework of Lagrangian mechanics. We derive closed-form formulas for the optimal trajectory when the traffic intensity is quadratic (single-phase) using Hamilton-Jacobi equations. When the traffic intensity is bi-phase, i.e. made of two quadratics, we provide necessary conditions of optimality that allow us to propose a gradient-based algorithm and a new algorithm based on the linear control properties of the quadratic model. These two solutions are of very low complexity because they rely on fast convergence numerical schemes and closed form formulas. These two approaches return a trajectory satisfying the necessary conditions of optimality. At last, we propose a data processing procedure based on a modified K-means algorithm to derive a bi-phase model and an optimal trajectory simulation from real traffic data.

A Dynamical Approach to Lower Gradient Estimates for Viscosity Solutions of Hamilton–Jacobi Equations

A Dynamical Approach to Lower Gradient Estimates for Viscosity Solutions of Hamilton–Jacobi Equations

Deviation of geodesics and particle trajectories in a gravitational wave of the Bianchi type VI universe

For the Bianchi type VI universe, exact solutions of the equation of geodesic deviation in a strong primordial gravitational wave in a privileged coordinate system are obtained. The solutions refer to Shapovalov’s gravitational-wave models of spacetime and allow the existence of complete integrals of the Hamilton–Jacobi equation for test particles. For all the solutions obtained, the analytical form of the tidal acceleration vector in a strong primordial gravitational wave is obtained. An explicit form of the coordinate transformation, an explicit form of the metric of the primordial gravitational wave of the Bianchi type VI universe, and the form of the tidal acceleration vector in the laboratory synchronous coordinate system are obtained. The synchronous coordinate system is associated with a freely falling observer and allows the observer to separate time and spatial coordinates, as well as to synchronize time at different points in space. The presented mathematical approach can be applied both in the general theory of relativity and in modified theories of gravity.

On large solutions for fractional Hamilton–Jacobi equations

We study the existence of large solutions for nonlocal Dirichlet problems posed on a bounded, smooth domain, associated with fully nonlinear elliptic equations of order $2\,s$, with $s\in (1/2,\,1)$, and a coercive gradient term with subcritical power $0< p<2\,s$. Due to the nonlocal nature of the diffusion, new blow-up phenomena arise within the range $0< p<2\,s$, involving a continuum family of solutions and/or solutions blowing-up to $-\infty$ on the boundary. This is in striking difference with the local case studied by Lasry–Lions for the subquadratic case $1< p<2$.

Characteristic curves for Set-Valued Hamilton-Jacobi Equations

The method of characteristics is extended to set-valued Hamilton-Jacobi equations. This problem arises from a calculus of variations’ problem with a multicriteria Lagrangian function: through an embedding into a set-valued framework, a set-valued Hamilton-Jacobi equation is derived, where the Hamiltonian function is the Fenchel conjugate of the Lagrangian function. In this paper a method of characteristics is described and some results are given for the Fenchel conjugate.

Conservation laws and Hamilton–Jacobi equations with space inhomogeneity

Conservation laws with an x-dependent flux and Hamilton–Jacobi equations with an x-dependent Hamiltonian are considered within the same set of assumptions. Uniqueness and stability estimates are obtained only requiring sufficient smoothness of the flux/Hamiltonian. Existence is proved without any convexity assumptions under a mild coercivity hypothesis. The correspondence between the semigroups generated by these equations is fully detailed. With respect to the classical Kružkov approach to conservation laws, we relax the definition of solution and avoid any restriction on the growth of the flux. A key role is played by the construction of sufficiently many entropy stationary solutions in {{textbf{L}}^infty } that provide global bounds in time and space.

Stochastic viscosity approximations of Hamilton–Jacobi equations and variance reduction

We consider the computation of free energy-like quantities for diffusions when resorting to Monte Carlo simulation is necessary, for instance in high dimension. Such stochastic computations typically suffer from high variance, in particular in a low noise regime, because the expectation is dominated by rare trajectories for which the observable reaches large values. Although importance sampling, or tilting of trajectories, is now a standard technique for reducing the variance of such estimators, quantitative criteria for proving that a given control reduces variance are scarce, and often do not apply to practical situations. The goal of this work is to provide a quantitative criterion for assessing whether a given bias reduces variance, and at which scale. We rely for this on a recently introduced notion of stochastic solution for Hamilton–Jacobi–Bellman (HJB) equations. Based on this tool, we introduce the notion of k-stochastic viscosity approximation (SVA) of a HJB equation. We next prove that such approximate solutions are associated with estimators having a relative variance of order k − 1 at log-scale. In particular, a sampling scheme built from a 1-SVA has bounded variance as noise goes to zero. Finally, in order to show that our definition is relevant, we provide examples of stochastic viscosity approximations of order one and two, with a numerical illustration confirming our theoretical findings.

Modified Hawking temperature and entropy of Kerr–de Sitter black hole in Lorentz violation theory

In this paper, we discuss the tunneling of scalar particles near the event horizon of stationary and nonstationary Kerr–de Sitter black hole using Lorentz violation theory in curved space time. The modified form of Hamilton–Jacobi equation is derived from the Klein–Gordon equation by applying Lorentz violation theory. The Hawking temperatures derived from stationary and nonstationary Kerr–de Sitter black holes are modified due to Lorentz violation theory. It is noted that the change in Bekenstein–Hawking entropy and modified Hawking temperatures of stationary and nonstationary Kerr–de Sitter black hole not only depends on the black hole parameters but also on ether-like vectors [Formula: see text].

Navigation System-Based Routing Strategies in Traffic Flows on Networks

Navigation choices play an important role in modeling and forecasting traffic flows on road networks. We introduce a macroscopic differential model coupling a conservation law with a Hamilton–Jacobi equation to, respectively, model the nonlinear transportation process and the strategic choices of users. Furthermore, the model is adapted to the multi-population case, where every population differs in the level of traffic information about the system.

A Nonconstant Gradient Constrained Problem for Nonlinear Monotone Operators

The purpose of the research is the study of a nonconstant gradient constrained problem for nonlinear monotone operators. In particular, we study a stationary variational inequality, defined by a strongly monotone operator, in a convex set of gradient-type constraints. We investigate the relationship between the nonconstant gradient constrained problem and a suitable double obstacle problem, where the obstacles are the viscosity solutions to a Hamilton–Jacobi equation, and we show the equivalence between the two variational problems. To obtain the equivalence, we prove that a suitable constraint qualification condition, Assumption S, is fulfilled at the solution of the double obstacle problem. It allows us to apply a strong duality theory, holding under Assumption S. Then, we also provide the proof of existence of Lagrange multipliers. The elements in question can be not only functions in L2, but also measures.

Implicit contact dynamics and Hamilton-Jacobi theory

In this paper, we introduce implicit Hamiltonian dynamics in the framework of contact geometry in two different ways: first, we introduce classical implicit Hamiltonian dynamics on a contact manifold, followed by evolution Hamiltonian dynamics. In the first case, implicit contact Hamiltonian dynamics is defined as a Legendrian submanifold of a tangent contact space, whilst the implicit evolution dynamic is understood as a Lagrangian submanifold of a certain symplectic space embedded into the tangent contact space. To conclude, we propose a geometric Hamilton-Jacobi theory for both of these formulations.

Shadows of rotating Einstein-Maxwell-dilaton black holes surrounded by a plasma

We consider rotating charged black holes with a scalar dilaton field and surrounded by plasma, with the purpose of studying their shadows. The corresponding metric has been previously obtained in the literature from the static solution by using the Newman-Janis algorithm. Assuming a well known form for the pressureless and nonmagnetized plasma distribution, which is suitable for the separation of the Hamilton-Jacobi equation for light, we derive an expression that determines the shape of the shadow. We present some examples of contours and we analyze their observable properties as functions of the charge and the dilaton coupling. We find that the presence of plasma introduces a dependency on the frequency, with the shadow becoming smaller as the frequency decreases.

A System of Hamilton-Jacobi Equations Characterizing Geodesic Centroidal Tessellations

AbstractWe introduce a class of systems of Hamilton-Jacobi equations characterizing geodesic centroidal tessellations, i.e., tessellations of domains with respect to geodesic distances where generators and centroids coincide. Typical examples are given by geodesic centroidal Voronoi tessellations and geodesic centroidal power diagrams. An appropriate version of the Fast Marching method on unstructured grids allows computing the solution of the Hamilton-Jacobi system and, therefore, the associated tessellations. We propose various numerical examples to illustrate the features of the technique.

From Second-Order Differential Geometry to Stochastic Geometric Mechanics

Classical geometric mechanics, including the study of symmetries, Lagrangian and Hamiltonian mechanics, and the Hamilton–Jacobi theory, are founded on geometric structures such as jets, symplectic and contact ones. In this paper, we shall use a partly forgotten framework of second-order (or stochastic) differential geometry, developed originally by L. Schwartz and P.-A. Meyer, to construct second-order counterparts of those classical structures. These will allow us to study symmetries of stochastic differential equations (SDEs), to establish stochastic Lagrangian and Hamiltonian mechanics and their key relations with second-order Hamilton–Jacobi–Bellman (HJB) equations. Indeed, stochastic prolongation formulae will be derived to study symmetries of SDEs and mixed-order Cartan symmetries. Stochastic Hamilton’s equations will follow from a second-order symplectic structure and canonical transformations will lead to the HJB equation. A stochastic variational problem on Riemannian manifolds will provide a stochastic Euler–Lagrange equation compatible with HJB one and equivalent to the Riemannian version of stochastic Hamilton’s equations. A stochastic Noether’s theorem will also follow. The inspirational example, along the paper, will be the rich dynamical structure of Schrödinger’s problem in optimal transport, where the latter is also regarded as a Euclidean version of hydrodynamical interpretation of quantum mechanics.

Homogenization of some periodic Hamilton-Jacobi equations with defects

We study homogenization for a class of stationary Hamilton-Jacobi equations in which the Hamiltonian is obtained by perturbing near the origin an otherwise periodic Hamiltonian. We prove that the limiting problem consists of a Hamilton-Jacobi equation outside the origin, with the same effective Hamiltonian as in periodic homogenization, supplemented at the origin with an effective Dirichlet condition that keeps track of the perturbation. Various comments and extensions are discussed.

Singularity formation and regularization at multiple times in the viscous Hamilton–Jacobi equation

The Cauchy–Dirichlet problem for the superquadratic viscous Hamilton–Jacobi equation (VHJ) from stochastic control theory, admits a unique, global viscosity solution. Solutions thus exist in the weak sense after appearance of singularity, which occurs through gradient blow-up (GBU) on the boundary. Whereas viscosity solution theory has been extensively applied to many PDEs, there seem to be less results on refined singular behavior of solutions. Although occurrence of two types of GBU, with or without loss of boundary condition (LBC), are known, detailed behavior after GBU has remained open except for a strongly restricted special class of one dimensional solutions. In this paper, in general dimensions, we construct solutions which undergo GBU with LBC arbitrarily many times and then recover regularity, as well as solutions without LBC at first GBU time. In one space dimension, we obtain the complete classification of viscosity solutions at each time, which extends to radial case in higher dimensions. Furthermore we show the existence of solutions exhibiting an arbitrarily given combination of GBU types with/without LBC at multiple times, in which a new type of behavior called “bouncing” is discovered. Global weak solutions of VHJ with multiple times singularity turn out to display larger variety of behaviors than in Fujita equation. We introduce a method based on an arbitrary number of critical parameters, whose continuity requires delicate arguments. Since we do not rely on any known special solution unlike in Fujita equation, our method is expected to apply to other equations. Singular behaviors at multiple times are completely new in the context of VHJ but also of stochastic control theory. Our results imply that for certain spatial distributions of rewards, if a controlled Brownian particle starts near the boundary, then the net gain attains profitable values on different time horizons but not on some intermediate times.