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<p>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-Ampère 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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal{C}^{2,\alpha }\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">C</mi> </mrow> <mrow> <mn>2</mn> <mo>,</mo> <mi>α</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> 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.</p>

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

  • 期刊论文

  • 发表日期：

    2026年1月3日

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  • Liang Chen,
  • Youyicun Lin,
  • Junqi Yang,
  • Wenfan Yi

摘要

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-Ampère 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 \(\mathcal{C}^{2,\alpha }\) 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.