Spline element method for Monge–Ampère equations

Awanou, Gerard

doi:10.1007/s10543-014-0524-y

Cited by 14 publications

(11 citation statements)

References 48 publications

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“…Regardless of the problems, the numerical approximation of fully nonlinear second order elliptic equations, as described in Caffarelli and Cabré [1995], have been the object of considerable recent research, particularly for the case of Monge-Ampère of which Oliker and Prussner [1988], Loeper and Rapetti [2005a], Dean and Glowinski [2006], Feng and Neilan [2009b], Oberman [2008], Awanou [2010], Davydov and Saeed [2012], Brenner et al [2011a], Froese [2011] are selected examples.…”

Section: Introductionmentioning

confidence: 99%

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A Finite Element Method for Nonlinear Elliptic Problems

Lakkis¹,

Pryer²

2013

SIAM J. Sci. Comput.

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We present a continuous finite element method for some examples of fully nonlinear elliptic equation. A key tool is the discretisation proposed in Lakkis & Pryer (2011) allowing us to work directly on the strong form of a linear PDE. An added benefit to making use of this discretisation method is that a recovered (finite element) Hessian is a biproduct of the solution process. We build on the linear basis and ultimately construct two different methodologies for the solution of second order fully nonlinear PDEs. Benchmark numerical results illustrate the convergence properties of the scheme for some test problems as well as the Monge-Ampère equation and the Pucci equation.

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Section: Introductionmentioning

confidence: 99%

“…In Feng and Neilan [2009a,b], Awanou [2010] the authors give a method in which they approximate the general second order fully nonlinear PDE by a sequence of fourth order quasilinear PDEs. These are quasilinear biharmonic equations which are discretised via mixed finite elements, or using high-regularity elements such as splines.…”

Section: Introductionmentioning

confidence: 99%

A Finite Element Method for Nonlinear Elliptic Problems

Lakkis¹,

Pryer²

2013

SIAM J. Sci. Comput.

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“…While this method may be convergent (cf. [19,84,6,3,33,10]), the appearance of global secondorder derivatives in the method necessitates the use of C 1 finite element spaces which can be arduous to implement and are not found in most finite element software packages. In addition, C 1 finite element generally require high-degree polynomial bases, resulting in a relatively large algebraic system.…”

Section: )mentioning

confidence: 99%

The Monge–Ampère equation

Neilan

Salgado

Zhang

2020

Geometric Partial Differential Equations - Part I

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“…In particular, in this case the left-hand side determinant may only depend on the Hessian of the solution, but no perturbations in the determinant like det(D 2 u + A) = f for a matrix A are permitted. Numerical methods for this Monge-Ampère equation of standard type can for example be found in [3,4,9,18,19,27,55].…”

Section: Existence and Uniquenessmentioning

confidence: 99%

Solving the Monge–Ampère equations for the inverse reflector problem

Brix

Hafizogullari

Platen

2015

Math. Models Methods Appl. Sci.

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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-Ampère equation, which consists of a highly nonlinear partial differential equation of second order and constraints. In our approach the Monge-Ampère 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-Ampère equation as for the inverse reflector problem for various images. The designed reflector surfaces are validated by a forward simulation using ray tracing.

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Spline element method for Monge–Ampère equations

Cited by 14 publications

References 48 publications

A Finite Element Method for Nonlinear Elliptic Problems

A Finite Element Method for Nonlinear Elliptic Problems

The Monge–Ampère equation

Solving the Monge–Ampère equations for the inverse reflector problem

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