Monotone and consistent discretization of the Monge-Ampère operator

DOI10.1090/mcom/3080zbMath1416.65400arXiv1409.6694OpenAlexW2335594173MaRDI QIDQ5741491

Jean-Marie Mirebeau, Francis Collino, Jean-David Benamou

Publication date: 25 July 2016

Published in: Mathematics of Computation (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1409.6694

zbMATH Keywords

Stern-Brocot treelattice basis reductionMonge-Ampère PDEmonotone finite differences scheme

Mathematics Subject Classification ID

Finite difference methods for boundary value problems involving PDEs (65N06) Monge-Ampère equations (35J96)

Related Items (30)

Adaptive, anisotropic and hierarchical cones of discrete convex functions ⋮ Convergence of a regularized finite element discretization of the two-dimensional Monge–Ampère equation ⋮ Optimal transportation, modelling and numerical simulation ⋮ A convergence framework for optimal transport on the sphere ⋮ Rates of Convergence in $W^2_p$-Norm for the Monge--Ampère Equation ⋮ Two-scale method for the Monge-Ampère equation: Convergence to the viscosity solution ⋮ Monotone discretization of the Monge–Ampère equation of optimal transport ⋮ Discrete Aleksandrov solutions of the Monge-Ampere equation ⋮ The Entropic Regularization of the Monge Problem on the Real Line ⋮ Optimal Pointwise Error Estimates for Two-Scale Methods for the Monge--Ampère Equation ⋮ High-order filtered schemes for time-dependent second order HJB equations ⋮ Numerical analysis of strongly nonlinear PDEs ⋮ Convergent approximation of non-continuous surfaces of prescribed Gaussian curvature ⋮ A Convergent Quadrature-Based Method for the Monge–Ampère Equation ⋮ Multigrid methods for convergent mixed finite difference scheme for Monge-Ampère equation ⋮ A Reconstructed Discontinuous Approximation to Monge-Ampère Equation in Least Square Formulation ⋮ A note on the Monge–Ampère type equations with general source terms ⋮ Pointwise rates of convergence for the Oliker-Prussner method for the Monge-Ampère equation ⋮ Convergent Two-Scale Filtered Scheme for the Monge--Ampère Equation ⋮ Convergence Framework for the Second Boundary Value Problem for the Monge--Ampère Equation ⋮ Monotone and Second Order Consistent Scheme for the Two Dimensional Pucci Equation ⋮ Monotone mixed finite difference scheme for Monge-Ampère equation ⋮ Second order monotone finite differences discretization of linear anisotropic differential operators ⋮ Fast-marching methods for curvature penalized shortest paths ⋮ Minimal Stencils for Discretizations of Anisotropic PDEs Preserving Causality or the Maximum Principle ⋮ A convexity enforcing \(C^0\) interior penalty method for the Monge-Ampère equation on convex polygonal domains ⋮ Improved Accuracy of Monotone Finite Difference Schemes on Point Clouds and Regular Grids ⋮ Minimal convex extensions and finite difference discretisation of the quadratic Monge–Kantorovich problem ⋮ Riemannian Fast-Marching on Cartesian Grids, Using Voronoi's First Reduction of Quadratic Forms ⋮ Minimal Geodesics Along Volume-Preserving Maps, Through Semidiscrete Optimal Transport

Cites Work

• Numerical solution of the optimal transportation problem using the Monge-Ampère equation
• On the numerical solution of the equation \(\frac{\partial ^ 2z\partial ^ 2z}{\partial x^ 2\partial y^ 2}-(\frac{\partial ^ 2z}{\partial x\partial y})^ 2=f\) and its discretizations. I
• An optimal convex hull algorithm in any fixed dimension
• Numerical solution of the Monge--Ampère equation by a Newton's algorithm
• Approximation of viscosity solutions of elliptic partial differential equations on minimal grids
• Sparse non-negative stencils for anisotropic diffusion
• Nonlinear elliptic partial differential equations and \(p\)-harmonic functions on graphs.
• Efficient fast marching with Finsler metrics
• Finite element approximations of the three dimensional Monge-Ampère equation
• Recent Developments in Numerical Methods for Fully Nonlinear Second Order Partial Differential Equations
• Anisotropic Fast-Marching on Cartesian Grids Using Lattice Basis Reduction
• Convergent Finite Difference Solvers for Viscosity Solutions of the Elliptic Monge–Ampère Equation in Dimensions Two and Higher
• Discrete Methods for Fully Nonlinear Elliptic Equations
• Low-dimensional lattices. VI. Voronoi reduction of three-dimensional lattices
• User’s guide to viscosity solutions of second order partial differential equations
• On the second boundary value problem for equations of Monge-Ampère type.
• A Numerical Method for Variational Problems with Convexity Constraints
• Convergent Filtered Schemes for the Monge--Ampère Partial Differential Equation
• A fast algorithm for the two dimensional HJB equation of stochastic control
• Convergent Difference Schemes for Degenerate Elliptic and Parabolic Equations: Hamilton--Jacobi Equations and Free Boundary Problems
• The Monge-Ampère equation

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