Finite Element Methods with Artificial Diffusion for Hamilton-Jacobi-Bellman Equations

In this short note we investigate the numerical performance of the method of artificial diffusion for second-order fully nonlinear Hamilton-Jacobi-Bellman equations. The method was proposed in (M. Jensen and I. Smears, arxiv:1111.5423); where a framework of finite element methods for Hamilton-Jacobi-Bellman equations was studied theoretically. The numerical examples in this note study how the artificial diffusion is activated in regions of degeneracy, the effect of a locally selected diffusion parameter on the observed numerical dissipation and the solution of second-order fully nonlinear equations on irregular geometries.Comment: Enumath 2011, version 2 contains in addition convergence rate

Designing Illumination Lenses and Mirrors by the Numerical Solution of Monge-Amp\`ere Equations

We consider the inverse refractor and the inverse reflector problem. The task is to design a free-form lens or a free-form mirror that, when illuminated by a point light source, produces a given illumination pattern on a target. Both problems can be modeled by strongly nonlinear second-order partial differential equations of Monge-Amp\`ere type. In [Math. Models Methods Appl. Sci. 25 (2015), pp. 803--837, DOI: 10.1142/S0218202515500190] the authors have proposed a B-spline collocation method which has been applied to the inverse reflector problem. Now this approach is extended to the inverse refractor problem. We explain in depth the collocation method and how to handle boundary conditions and constraints. The paper concludes with numerical results of refracting and reflecting optical surfaces and their verification via ray tracing.Comment: 16 pages, 6 figures, 2 tables; Keywords: Inverse refractor problem, inverse reflector problem, elliptic Monge-Amp\`ere equation, B-spline collocation method, Picard-type iteration; OCIS: 000.4430, 080.1753, 080.4225, 080.4228, 080.4298, 100.3190. Minor revision: two typos have been corrected and copyright note has been adde

A viscosity framework for computing Pogorelov solutions of the Monge-Ampere equation

We consider the Monge-Kantorovich optimal transportation problem between two measures, one of which is a weighted sum of Diracs. This problem is traditionally solved using expensive geometric methods. It can also be reformulated as an elliptic partial differential equation known as the Monge-Ampere equation. However, existing numerical methods for this non-linear PDE require the measures to have finite density. We introduce a new formulation that couples the viscosity and Aleksandrov solution definitions and show that it is equivalent to the original problem. Moreover, we describe a local reformulation of the subgradient measure at the Diracs, which makes use of one-sided directional derivatives. This leads to a consistent, monotone discretisation of the equation. Computational results demonstrate the correctness of this scheme when methods designed for conventional viscosity solutions fail

Pseudo transient continuation and time marching methods for Monge-Ampere type equations

We present two numerical methods for the fully nonlinear elliptic Monge-Ampere equation. The first is a pseudo transient continuation method and the second is a pure pseudo time marching method. The methods are proven to converge to a strictly convex solution of a natural discrete variational formulation with $C^1$ conforming approximations. The assumption of existence of a strictly convex solution to the discrete problem is proven for smooth solutions of the continuous problem and supported by numerical evidence for non smooth solutions

Mixed Interior Penalty Discontinuous Galerkin Methods for Fully Nonlinear Second Order Elliptic and Parabolic Equations in High Dimensions

This article is concerned with developing efficient discontinuous Galerkin methods for approximating viscosity (and classical) solutions of fully nonlinear second-order elliptic and parabolic partial differential equations (PDEs) including the Monge–Ampère equation and the Hamilton–Jacobi–Bellman equation. A general framework for constructing interior penalty discontinuous Galerkin (IP-DG) methods for these PDEs is presented. The key idea is to introduce multiple discrete Hessians for the viscosity solution as a means to characterize the behavior of the function. The PDE is rewritten in a mixed form composed of a single nonlinear equation paired with a system of linear equations that defines multiple Hessian approximations. To form the single nonlinear equation, the nonlinear PDE operator is replaced by the projection of a numerical operator into the discontinuous Galerkin test space. The numerical operator uses the multiple Hessian approximations to form a numerical moment which fulfills consistency and g-monotonicity requirements of the framework. The numerical moment will be used to design solvers that will be shown to help the IP-DG methods select the “correct” solution that corresponds to the unique viscosity solution. Numerical experiments are also presented to gauge the effectiveness and accuracy of the proposed mixed IP-DG methods

Finite Element Methods with Artificial Diffusion for Hamilton-Jacobi-Bellman Equations

In this short note we investigate the numerical performance of the method of artificial diffusion for second-order fully nonlinear Hamilton-Jacobi-Bellman equations. The method was proposed in (M. Jensen and I. Smears, arxiv:1111.5423); where a framework of finite element methods for Hamilton-Jacobi-Bellman equations was studied theoretically. The numerical examples in this note study how the artificial diffusion is activated in regions of degeneracy, the effect of a locally selected diffusion parameter on the observed numerical dissipation and the solution of second-order fully nonlinear equations on irregular geometries