Nonconvex Hamiltonians Research Articles

Sparse field level set method for non-convex Hamiltonians in 3D plasma etching profile simulations

Level set method [S. Osher, J. Sethian, J. Comput. Phys. 79 (1988) 12] is a highly robust and accurate computational technique for tracking moving interfaces in various application domains. It originates from the idea to view the moving front as a particular level set of a higher dimensional function, so the topological merging and breaking, sharp gradients and cusps can form naturally, and the effects of curvature can be easily incorporated. The resulting equations, describing interface surface evolution, are of Hamilton–Jacobi type and they are solved using techniques developed for hyperbolic equations. In this paper we describe an extension of the sparse field method for solving level set equations in the case of non-convex Hamiltonians, which are common in the simulations of the profile surface evolution during plasma etching and deposition processes. Sparse field method itself, developed by Whitaker [R. Whitaker, Internat. J. Comput. Vision 29 (3) (1998) 203] and broadly used in image processing community, is an alternative to the usual combination of narrow band and fast marching procedures for the computationally effective solving of level set equations. The developed procedure is applied to the simulations of 3D feature profile surface evolution during plasma etching process, that include the effects of ion enhanced chemical etching and physical sputtering, which are the primary causes of the Hamiltonian non-convexity.

Sufficient Conditions for the Existence of Viscosity Solutions for Nonconvex Hamiltonians

We study a sufficient geometric condition for the existence of a $W^{1,\infty}(\Omega)$ viscosity solution of the Hamilton--Jacobi equation $$ \left\{ \begin{array}{@{}r@{\;}c@{\;}lcc} F(Du) & = & 0 & \mbox{in} & \Omega, \\[2pt] u & = & \varphi & \mbox{on} & \partial\Omega, \end{array} \right. $$ where $\Omega \subset {\mathbb R}^n$ and $F:{\mathbb R}^n\to {\mathbb R}$ are not necessarily convex.

Semidiscrete Central-Upwind Schemes for Hyperbolic Conservation Laws and Hamilton--Jacobi Equations

We introduce new Godunov-type semidiscrete central schemes for hyperbolic systems of conservation laws and Hamilton--Jacobi equations. The schemes are based on the use of more precise information about the local speeds of propagation and can be viewed as a generalization of the schemes from [A. Kurganov and E. Tadmor, J. Comput. Phys., 160 (2000), pp. 241--282; A. Kurganov and D. Levy, SIAM J. Sci. Comput., 22 (2000), pp. 1461--1488; A. Kurganov and G. Petrova, A third-order semidiscrete genuinely multidimensional central scheme for hyperbolic conservation laws and related problems, Numer. Math., to appear] and [A. Kurganov and E. Tadmor, J. Comput. Phys., 160 (2000), pp. 720--742]. The main advantages of the proposed central schemes are the high resolution, due to the smaller amount of the numerical dissipation, and the simplicity. There are no Riemann solvers and characteristic decomposition involved, and this makes them a universal tool for a wide variety of applications. At the same time, the developed schemes have an upwind nature, since they respect the directions of wave propagation by measuring the one-sided local speeds. This is why we call them central-upwind schemes. The constructed schemes are applied to various problems, such as the Euler equations of gas dynamics, the Hamilton--Jacobi equations with convex and nonconvex Hamiltonians, and the incompressible Euler and Navier--Stokes equations. The incompressibility condition in the latter equations allows us to treat them both in their conservative and transport form. We apply to these problems the central-upwind schemes, developed separately for each of them, and compute the corresponding numerical solutions.

DISCONTINUOUS SOLUTIONS IN L∞FOR HAMILTON-JACOBI EQUATIONS

An approach is introduced to construct global discontinuous solutions in L∞ for Hamilton-Jacobi equations. This approach allows the initial data only in L∞ and applies to the equations with nonconvex Hamiltonians. The profit functions are introduced to formulate the notion of discontinuous solutions in L∞. The existence of global discontinuous solutions in L∞ is established. These solutions in L∞ coincide with the viscosity solutions and the minimax solutions, provided that the initial data are continuous. A prototypical equation is analyzed to examine the L∞ stability of our L∞ solutions. The analysis also shows that global discontinuous solutions are determined by the topology in which the initial data are approximated.

Subharmonics of nonconvex Hamiltonian systems

In this note we prove the existence of subharmonic solutions of the hamiltonian system \(\dot u = J\nabla H(t,u)\) where H is a nonconvex unbounded hamiltonian with a bounded gradient.

Layering methods for Hamilton–Jacobi equations with nonconvex Hamiltonian

Layering methods for Hamilton–Jacobi equations with nonconvex Hamiltonian

On ${\text{L}}^2 $ Sufficient Conditions and the Gradient Projection Method for Optimal Control Problems

${\text{L}}^2 $-local convergence and active constraint identification theorems are proved for gradient projection iterates in sets of ${\text{L}}^\infty $ functions on $[0,1]$ with range in a polyhedron $U \subset \mathbb{R}^m $. These theorems extend earlier results for $U = [0,\infty ) \subset \mathbb{R}^1 $ and are based on an infinite-dimensional variant of the Karush–Kuhn–Tucker second-order sufficient conditions in polyhedral subsets of $\mathbb{R}^n $. The new sufficient conditions and convergence results proved here are directly applicable to continuous-time optimal control problems with smooth nonconvex nonquadratic objective functions and Hamiltonians that are quadratic in the control input vector u. In particular, these theorems apply to nonconvex nonquadratic regulator problems with control-linear state equations and vector-valued inputs $u(t)$ satisfying unqualified affine inequality constraints at almost all t in $[0,1]$.

On the Gradient Projection Method for Optimal Control Problems with Nonnegative $\mathcal{L}^2 $ Inputs

Local convergence and active constraint identification theorems are proved for gradient-projection iterates in the cone of nonnegative ${\cal L}^2$ functions on $[0,1]$. The theorems are based on recently established infinite-dimensional extensions of the Kuhn-Tucker sufficient conditions and are directly applicable to a large class of continuous-time optimal control problems with smooth nonconvex nonquadratic objective fractions and Hamiltonians that are quadratic in the control input $u$.

An optimal production-mix model

A simple model of a production process for two different products is analyzed. The goal is to minimize production and inventory costs. Conditions for the optimality of simultaneous and separate production are given. Variations of the parameters yield convex and nonconvex Hamiltonians and substantially different solutions. Two examples illustrate the results.

Algorithm for overcoming the curse of dimensionality for certain non-convex Hamilton–Jacobi equations, projections and differential games

Abstract : In this paper, we develop a method for solving a large class of non-convex Hamilton-Jacobi partial differential equations (HJ PDE). The method yields decoupled subproblems, which can be solved in an embarrassingly parallel fashion. The complexity of the resulting algorithm is polynomial in the problem dimension; hence, it overcomes the curse of dimensionality [1, 2]. We extend previous work in[6] and apply the Hopf formula to solve HJ PDE involving non-convex Hamiltonians. We propose an ADMM approach for finding the minimizer associated with the Hopf formula. Some explicit formulae of proximal maps, as well as newly-defined stretch operators, are used in the numerical solutions of ADMM subproblems. Our approach is expected to have wide applications in continuous dynamic games, control theory problems, and elsewhere.