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A Convergent Inexact Abedin-Kitagawa Iteration Method for Monge-Ampère Eigenvalue Problems

Liang Chen, Youyicun Lin, Junqi Yang, Wenfan Yi

TL;DR

The paper tackles computing Monge-Ampère eigenpairs by adapting the Abedin–Kitagawa iteration to an inexact setting with a summable error sequence in subproblem solves, preserving convergence to the MAE eigenpair. It introduces fixed-point reformulations in 2D and 3D to efficiently solve the MA subproblems, and provides rigorous convergence analysis for both general and nonnegative-error cases under $\mathcal{C}^{2,\alpha}$ boundary conditions. Theoretical results show that the Rayleigh quotient updates converge to $\lambda_{\rm MA}$ and the iterates converge to the corresponding eigenfunction $u_{\infty}$, with subsequences coinciding and the full sequence converging. Numerical experiments in 2D and 3D demonstrate that the inexact fixed-point AKI method achieves substantial speedups (often by factors of 5–40) while maintaining or improving accuracy relative to existing approaches, validating the method's efficiency and robustness for MAE problems.

Abstract

This paper proposes an inexact Aleksandrov-solution-based iteration method, formulated by adapting the convergent Rayleigh inverse iterative scheme introduced by Abedin and Kitagawa, to solve real Monge-Amp{è}re eigenvalue (MAE) problems. The central feature of the proposed approach is the introduction of a flexible error tolerance criterion for computing inexact Aleksandrov solutions to the required subproblems. This allows the inner iteration to be solved approximately without compromising the global convergence properties of the overall scheme, as we established under a ${\cal C}^{2,α}$ boundary condition, and has the potential of achieving reduced computational cost compared to the original algorithm. In practice, for both two- and three-dimensional problems, by leveraging the flexibility of the inexact iterative formulation in conjunction with a fixed-point approach for solving the subproblems, the proposed method performs several times faster than its original version of Abedin and Kitagawa, across all tested problem instances in the numerical experiments.

A Convergent Inexact Abedin-Kitagawa Iteration Method for Monge-Ampère Eigenvalue Problems

TL;DR

The paper tackles computing Monge-Ampère eigenpairs by adapting the Abedin–Kitagawa iteration to an inexact setting with a summable error sequence in subproblem solves, preserving convergence to the MAE eigenpair. It introduces fixed-point reformulations in 2D and 3D to efficiently solve the MA subproblems, and provides rigorous convergence analysis for both general and nonnegative-error cases under boundary conditions. Theoretical results show that the Rayleigh quotient updates converge to and the iterates converge to the corresponding eigenfunction , with subsequences coinciding and the full sequence converging. Numerical experiments in 2D and 3D demonstrate that the inexact fixed-point AKI method achieves substantial speedups (often by factors of 5–40) while maintaining or improving accuracy relative to existing approaches, validating the method's efficiency and robustness for MAE problems.

Abstract

This paper proposes an inexact Aleksandrov-solution-based iteration method, formulated by adapting the convergent Rayleigh inverse iterative scheme introduced by Abedin and Kitagawa, to solve real Monge-Amp{è}re eigenvalue (MAE) problems. The central feature of the proposed approach is the introduction of a flexible error tolerance criterion for computing inexact Aleksandrov solutions to the required subproblems. This allows the inner iteration to be solved approximately without compromising the global convergence properties of the overall scheme, as we established under a boundary condition, and has the potential of achieving reduced computational cost compared to the original algorithm. In practice, for both two- and three-dimensional problems, by leveraging the flexibility of the inexact iterative formulation in conjunction with a fixed-point approach for solving the subproblems, the proposed method performs several times faster than its original version of Abedin and Kitagawa, across all tested problem instances in the numerical experiments.
Paper Structure (33 sections, 16 theorems, 149 equations, 9 figures, 8 tables, 2 algorithms)

This paper contains 33 sections, 16 theorems, 149 equations, 9 figures, 8 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. An inexact AKI method for MAE problems
  4. Convergence analysis for the general case
  5. Convergence analysis for nonnegative error functions
  6. Step 1: The whole sequence $\{R(u_{k})\}_{k\geq 0}$ converges to $\lambda_{\rm MA}$.
  7. Step 2: A subsequence $\{u_{k(j)}\}_{j\geq 1}$ and the corresponding shifted subsequence $\{u_{k(j)+1}\}_{j\geq 1}$ converge to the same nontrivial eigenfunction $u_{\infty}$.
  8. Step 3: Convergence of the full sequence.
  9. Fixed-point approaches for solving subproblems
  10. A fixed-point approach for Omega subset R2
  11. Step 1: Contraction property of the operator ${Q_2}$.
  12. Step 2: Verifying that the operator ${Q_2}$ maps $B_{\rho}$ into itself.
  13. Step 3: The sequence $\{u_{k}\}_{k\ge0}$ converges to the solution to the MA equation \ref{['3.6']}.
  14. A fixed-point approach for Omega subset R3
  15. Step 1: Contraction property of the operator $Q_3$.
  16. ...and 18 more sections

Key Result

Lemma 2.1

Let $\Omega$ be a bounded convex open domain in $\mathbb{R}^{d}$ and $\nu$ be a nonnegative Borel measure in $\Omega$, then there exists a unique convex function $u\in \mathcal{C}(\overline{\Omega})$ constituting an Aleksandrov solution of the Dirichlet problem

Figures (9)

  • Figure 1: The domains of the problems and the meshes in $\mathbb{R}^2$ (edge length $h=1/20$)
  • Figure 2: The domains of the problems and the meshes in $\mathbb{R}^3$ (edge length $h=1/16$)
  • Figure 3: Performance of the inexact AKI-FP method on the unit disk domain \ref{['eq.disk']}
  • Figure 4: Performance of the inexact AKI-FP method on the ellipse domain \ref{['eq.ellipse']}
  • Figure 5: Performance of the inexact AKI-FP method on the smoothed square \ref{['eq.smoothsq']}
  • ...and 4 more figures

Theorems & Definitions (26)

  • Definition 1: Monge-Ampère measure; figalli2017monge
  • Definition 2: Aleksandrov solutions; figalli2017monge
  • Lemma 2.1: Solvability of Dirichlet problem; hartenstine2006dirichlet
  • Lemma 2.2: Aleksandrov maximum principle; gutierrez2016monge
  • Lemma 2.3: Comparison principle; gutierrez2016monge
  • Lemma 2.4: Continuity property of the Monge-Ampère energy; abedin2020inverse
  • Lemma 2.5: Comparability of $L^p$ norms; abedin2020inverse
  • Lemma 2.6: Existence and uniqueness of solution; le2017eigenvalue
  • Proposition 3.1
  • proof
  • ...and 16 more