A Convergent Inexact Abedin-Kitagawa Iteration Method for Monge-Amp\`ere Eigenvalue Problems

TL;DR

This paper introduces a flexible inexact iterative method for solving Monge-Ampère eigenvalue problems, improving computational efficiency while maintaining convergence, demonstrated through faster performance in numerical experiments.

Contribution

It adapts the Abedin-Kitagawa iterative scheme with an inexact Aleksandrov solution approach, reducing computational cost without losing convergence guarantees.

Findings

Method performs several times faster than original in tests.

Maintains convergence under ${\cal C}^{2,\alpha}$ boundary conditions.

Effective for 2D and 3D Monge-Ampère eigenvalue 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{\`e}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 ${\cal C}^{2,\alpha}$ 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.