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

Legendre-Gauss-Lobatto grids and associated nested dyadic grids

Legendre-Gauss-Lobatto (LGL) grids play a pivotal role in nodal spectral methods for the numerical solution of partial differential equations. They not only provide efficient high-order quadrature rules, but give also rise to norm equivalences that could eventually lead to efficient preconditioning techniques in high-order methods. Unfortunately, a serious obstruction to fully exploiting the potential of such concepts is the fact that LGL grids of different degree are not nested. This affects, on the one hand, the choice and analysis of suitable auxiliary spaces, when applying the auxiliary space method as a principal preconditioning paradigm, and, on the other hand, the efficient solution of the auxiliary problems. As a central remedy, we consider certain nested hierarchies of dyadic grids of locally comparable mesh size, that are in a certain sense properly associated with the LGL grids. Their actual suitability requires a subtle analysis of such grids which, in turn, relies on a number of refined properties of LGL grids. The central objective of this paper is to derive just these properties. This requires first revisiting properties of close relatives to LGL grids which are subsequently used to develop a refined analysis of LGL grids. These results allow us then to derive the relevant properties of the associated dyadic grids.Comment: 35 pages, 7 figures, 2 tables, 2 algorithms; Keywords: Legendre-Gauss-Lobatto grid, dyadic grid, graded grid, nested grid

Solving the Monge-Amp\`ere Equations for the Inverse Reflector Problem

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-Amp\`ere equation, which consists of a highly nonlinear partial differential equation of second order and constraints. In our approach the Monge-Amp\`ere 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-Amp\`ere equation as for the inverse reflector problem for various images. The designed reflector surfaces are validated by a forward simulation using ray tracing.Comment: 28 pages, 8 figures, 2 tables; Keywords: Inverse reflector problem, elliptic Monge-Amp\`ere equation, B-spline collocation method, Picard-type iteration; Minor revision: reference [59] to a recent preprint has been added and a few typos have been correcte

Robust preconditioners for hphp-discontinuous Galerkin discretizations for elliptic problems

Discontinuous Galerkin (DG) methods offer a very powerful discretization tool because they not only are suited for a wide range of partial differential equations but also because they support as well local mesh refinement as varying polynomial degrees. The linear systems of equations arising from DG schemes for elliptic boundary value problems quickly become ill-conditioned such that they cannot be solved efficiently. Existing preconditioning concepts usually restrict the flexibility of DG methods by imposing strong conditions on the meshes or the distribution of the polynomial degrees such that the full capabilities of DG methods remain unused. Moreover, the condition numbers of the resulting linear systems of equations grow with the polynomial degrees employed in the discretization. This thesis aims at the construction, analysis and implementation of a preconditioner for spectral DG discretizations of elliptic boundary value problems that is fully robust in the arbitrary large and locally varying polynomial degree, i.e. under mild grading conditions the condition numbers of the preconditioned system stay uniformly bounded independent of the mesh size and the polynomial degrees. The key ingredients to achieve full robustness in the polynomial degree are Legendre-Gauss-Lobatto (LGL) grids in combination with certain equivalences that couple the norms of nodal spectral element and low order finite element functions. Since the family of LGL grids lacks some properties, e.g. they are not nested, we construct a family of nested dyadic companion grids and investigate their properties. The proposed preconditioner is composed of three stages, where in each step a different obstruction is attacked and overcome and the auxiliary space method serves as conceptual platform for the design of the preconditioners. In the first stage the spectral DG formulation is preconditioned by a corresponding conforming formulation. The second stage serves to precondition the high-order formulation on anisotropic LGL meshes to a finite-element formulation on an anisotropic dyadic companion mesh. In the third stage a multilevel preconditioner is provided that exploits the multilevel hierarchy inherent to the dyadic grids by spaces of appropriate multiwavelets. The theoretical constructions are complemented by quantitative numerical experiments, which provide some insight in the condition numbers and their dependencies on the parameters. American Mathematical Society Subject Classification (MSC2010):33C45, 34C10, 65N35, 65N55, 65N30, 65N22, 65F10, 65F0

Robust Preconditioners for DG–Discretizations with Arbitrary Polynomial Degrees

Discontinuous Galerkin (DG) methods offer an enormous flexibility regarding local grid refinement and variation of polynomial degrees for a variety of different problem classes. With a focus on diffusion problems, we consider DG discretizations for elliptic boundary value problems, in particular the efficient solution of the linear systems of equations that arise from the Symmetric Interior Penalty DG method. We announce a multi-stage preconditioner which produces uniformly bounded condition numbers and aims at supporting the full flexibility of DG methods under mild grading conditions. The constructions and proofs are detailed in an upcoming series of papers by the authors. Our preconditioner is based on the concept of the auxiliary space method and techniques from spectral element methods such as Legendre-Gauß-Lobatto grids. The presentation for the case of geometrically conforming meshes is complemented by numerical studies that shed some light on constants arising in four basic estimates used in the second stage. Key words: multi stage preconditioner, spectral discontinuous Galerkin method, auxiliary space method, Legendre–Gauß–Lobatto grid