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Convergence of a least-squares splitting method for the Monge-Ampère equation

Anna Peruso, Massimo Sorella

TL;DR

This work analyzes a nonlinear least-squares splitting method for the Monge-Ampère equation by recasting the iteration as an alternating-projection process in Sobolev spaces $H^m$, enabling a contraction-based convergence framework. By studying tangent-space transversality of the constraint sets, the authors prove local linear convergence on the two-dimensional torus $\mathbb{T}^2$ under standard regularity ($u\in H^4$) and ellipticity, and establish $H^m$-convergence on bounded convex domains for $m\ge 2$. The analysis combines an $H^m$-projection viewpoint with an $L^2$-based treatment on $\mathbb{T}^2$, yielding a first continuous-level convergence result for this splitting method and explaining observed numerical robustness. The results suggest a flexible framework that could extend to other fully nonlinear PDEs and to nonperiodic settings, with potential adaptations to higher dimensions or gradient-dependent right-hand sides.

Abstract

We study the theoretical convergence of the nonlinear least-squares splitting method for the Monge-Ampère equation in which each iteration decouples the pointwise nonlinearity from the differential operator and consists of a local nonlinear update followed by the solution of two sequential Poisson-type elliptic problems. While the method performs well in computations, a rigorous convergence theory has remained unavailable. We observe that the iteration admits a reformulation as an alternating-projection scheme on Sobolev spaces $H^m$, $m\ge 0$. At a solution, the Gâteaux differentials of the projection maps are the linear projections onto the corresponding tangent spaces. We prove that these tangent spaces are transverse, and hence the linearization of the alternating-projection map is a contraction by classical Hilbert-space theory for alternating projections. Building on this geometric characterization, we prove linear convergence in $H^2$ of the splitting method on the two-dimensional torus $\mathbb{T}^2$ for initial data sufficiently close to a solution $u\in H^4$. To the best of our knowledge, this yields the first rigorous convergence result for this splitting method in the periodic setting and provides a functional-analytic explanation for its observed numerical robustness.

Convergence of a least-squares splitting method for the Monge-Ampère equation

TL;DR

This work analyzes a nonlinear least-squares splitting method for the Monge-Ampère equation by recasting the iteration as an alternating-projection process in Sobolev spaces , enabling a contraction-based convergence framework. By studying tangent-space transversality of the constraint sets, the authors prove local linear convergence on the two-dimensional torus under standard regularity () and ellipticity, and establish -convergence on bounded convex domains for . The analysis combines an -projection viewpoint with an -based treatment on , yielding a first continuous-level convergence result for this splitting method and explaining observed numerical robustness. The results suggest a flexible framework that could extend to other fully nonlinear PDEs and to nonperiodic settings, with potential adaptations to higher dimensions or gradient-dependent right-hand sides.

Abstract

We study the theoretical convergence of the nonlinear least-squares splitting method for the Monge-Ampère equation in which each iteration decouples the pointwise nonlinearity from the differential operator and consists of a local nonlinear update followed by the solution of two sequential Poisson-type elliptic problems. While the method performs well in computations, a rigorous convergence theory has remained unavailable. We observe that the iteration admits a reformulation as an alternating-projection scheme on Sobolev spaces , . At a solution, the Gâteaux differentials of the projection maps are the linear projections onto the corresponding tangent spaces. We prove that these tangent spaces are transverse, and hence the linearization of the alternating-projection map is a contraction by classical Hilbert-space theory for alternating projections. Building on this geometric characterization, we prove linear convergence in of the splitting method on the two-dimensional torus for initial data sufficiently close to a solution . To the best of our knowledge, this yields the first rigorous convergence result for this splitting method in the periodic setting and provides a functional-analytic explanation for its observed numerical robustness.
Paper Structure (13 sections, 7 theorems, 85 equations, 2 figures)

This paper contains 13 sections, 7 theorems, 85 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Contributions
  3. Outline
  4. Problem setting and algorithm
  5. Notation.
  6. Model problem and algorithm.
  7. Convergence in $H^m(\Omega,\mathbb{S}^d)$
  8. Convergence in $L^2(\mathbb{T}^2,\mathbb{S}^2)$
  9. Differentiability of $\Pi_{\mathcal{B}}$ and $T$
  10. Local convergence of \ref{['eq:algorithm-torus']} on $\mathbb{T}^2$
  11. Further discussion
  12. Numerical examples
  13. Conclusions

Key Result

Lemma 3.1

Let $f\in H^m(\Omega)$. Let $\mathbf{P}\in \mathcal{B}^m\cap \mathcal{V}_g^m$ such that A1 is satisfied. Then $\Pi^{(m)}_{\mathcal{B}^m}:H^m\to H^m$ is a well defined map in a neighborhood of $\mathbf{P}$ and is Fréchet differentiable in $\mathbf{P}$, with where $\ker (\mathop{\mathrm{cof}}\limits\mathbf{P}):=\{\mathbf{X}\in H^m(\Omega,\mathbb{S}^2):\, \mathop{\mathrm{cof}}\limits\mathbf{P}(x):\m

Figures (2)

  • Figure 1: Results for $\varepsilon = 0.002$. Left: $\|u-u^n_h\|_{L^2}$ vs. the splitting iteration $n$ for different mesh sizes. Center: $\|D^2u-D^2_hu^n_h\|_{L^2}$ vs. the splitting iteration $n$ for different mesh sizes. Right: $\|u-u_h\|$ in different norms vs. mesh size, where $u_h$ is the approximated solution when the splitting algorithm reaches convergence.
  • Figure 2: Results for $\varepsilon = 0.02$. Left: $\|u-u^n_h\|_{L^2}$ vs. the splitting iteration $n$ for different mesh sizes. Center: $\|D^2u-D^2_hu^n_h\|_{L^2}$ vs. the splitting iteration $n$ for different mesh sizes. Right: $\|u-u_h\|$ in different norms vs. mesh size, where $u_h$ is the approximated solution when the splitting algorithm reaches convergence.

Theorems & Definitions (18)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Remark 3.4
  • Remark 3.5
  • Lemma 4.1
  • proof
  • ...and 8 more