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The Monge-Ampère equation

Michael Neilan, Abner J. Salgado, Wujun Zhang

TL;DR

The chapter surveys numerical approaches for the Monge-Ampère equation, emphasizing the prototypical Dirichlet problem $\det D^2u=f$ with convexity constraints and a wide range of geometric applications. It develops monotone, wide-stencil finite difference schemes, variational characterizations of the determinant, and geometric discretizations (Oliker–Prussner) to handle non-smooth solutions through Alexandrov measure. The text also presents two-scale and power-diagram discretizations, lattice basis reduction, and HJB/semilagrangian formulations, with convergence proofs and rates tied to regularity and barrier constructions. Extensions include filtered schemes, transport boundary conditions, and convex-envelope approximations, offering robust tools for degenerate or singular MA problems on unstructured and structured meshes. Collectively, these methods advance reliable convergence and provide practical, implementable frameworks for MA-type problems across differential geometry, optics, and optimal transport.

Abstract

We review recent advances in the numerical analysis of the Monge-Ampère equation. Various computational techniques are discussed including wide-stencil finite difference schemes, two-scaled methods, finite element methods, and methods based on geometric considerations. Particular focus is the development of appropriate stability and consistency estimates which lead to rates of convergence of the discrete approximations. Finally we present numerical experiments which highlight each method for a variety of test problem with different levels of regularity.

The Monge-Ampère equation

TL;DR

The chapter surveys numerical approaches for the Monge-Ampère equation, emphasizing the prototypical Dirichlet problem with convexity constraints and a wide range of geometric applications. It develops monotone, wide-stencil finite difference schemes, variational characterizations of the determinant, and geometric discretizations (Oliker–Prussner) to handle non-smooth solutions through Alexandrov measure. The text also presents two-scale and power-diagram discretizations, lattice basis reduction, and HJB/semilagrangian formulations, with convergence proofs and rates tied to regularity and barrier constructions. Extensions include filtered schemes, transport boundary conditions, and convex-envelope approximations, offering robust tools for degenerate or singular MA problems on unstructured and structured meshes. Collectively, these methods advance reliable convergence and provide practical, implementable frameworks for MA-type problems across differential geometry, optics, and optimal transport.

Abstract

We review recent advances in the numerical analysis of the Monge-Ampère equation. Various computational techniques are discussed including wide-stencil finite difference schemes, two-scaled methods, finite element methods, and methods based on geometric considerations. Particular focus is the development of appropriate stability and consistency estimates which lead to rates of convergence of the discrete approximations. Finally we present numerical experiments which highlight each method for a variety of test problem with different levels of regularity.

Paper Structure

This paper contains 48 sections, 60 theorems, 413 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Geometric applications
  3. Gauss curvature problem
  4. Reflector design problem
  5. Affine plateau problem
  6. Optimal mass transport problem
  7. Solution concepts for the Monge-Ampère equation
  8. Classical solutions
  9. Viscosity solutions
  10. Alexandrov solutions
  11. Wide Stencil Finite Differences
  12. A general framework for approximation schemes
  13. A variational characterization of the determinant
  14. Wide stencil finite difference schemes
  15. Filtered schemes
  16. ...and 33 more sections

Key Result

Theorem 1.8

Let $\alpha \in (0,1)$. Assume that $\Omega$ is a bounded and uniformly convex domain, whose boundary is of class $C^3$, $f \in C^\alpha(\bar{\Omega})$ with $f \geq f_0 >0$, and $g \in C^3({\partial} \Omega)$. Then problem eqn:MA has a unique solution $u \in C^{2,\alpha}(\bar{\Omega})$.

Figures (11)

  • Figure 1: The function $S$ defined in \ref{['eq:exoffilter']} is a filter.
  • Figure 2: The function $\zeta$ used to define the discrete barrier of Proposition \ref{['prop:otherbarrier']}.
  • Figure 3: The construction Proposition \ref{['prop:otherbarrier']} that shows that the function $b$ is convex. The distance between $x_+$ and the supporting hyperplane $P$ equals the sum of the distance from $x_0$ to the boundary ${\partial}\Omega$ and the inner product between $\boldsymbol{n}$ and $\boldsymbol{v}$.
  • Figure 4: Meshes corresponding to convex envelopes $\Gamma(w_1) = |x_1|$ (left), $\Gamma(w_2)= |x_2|$ (middle), and $\Gamma(w_3) = |x_1|+|x_2|$ (right).
  • Figure 5: Mesh induced by the nodal interpolant of $w(x) = (x \cdot e)^2$ where $e = (1,2)^\intercal$. Its convex envelope equals $|x \cdot e|$ in the star of $(0,0)$.
  • ...and 6 more figures

Theorems & Definitions (142)

  • Definition 1.7: classical solution
  • Theorem 1.8: existence of classical solutions
  • Definition 1.9: elliptic operator
  • Definition 1.11: upper and lower semicontinuous envelopes
  • Definition 1.12: viscosity solution
  • Definition 1.13: comparison principle
  • Definition 1.15: viscosity solution
  • Proposition 1.16: continuous dependence
  • Proposition 1.17: smooth approximation
  • Definition 1.18: subdifferential
  • ...and 132 more