An Efficient Operator-Splitting Method for the Eigenvalue Problem of the Monge-Ampere Equation

Hao Liu, Shingyu Leung*, Jianliang Qian

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review


We develop an efficient operator-splitting method for the eigenvalue problem of the Monge-Ampère operator in the Aleksandrov sense. The backbone of our method relies on a convergent Rayleigh inverse iterative formulation proposed by Abedin and Kitagawa (Inverse iteration for the Monge-Ampère eigenvalue problem, Proceedings of the American Mathematical Society, 148 (2020) 4975-4886). Modifying the theoretical formulation, we develop an efficient algorithm for computing the eigenvalue and eigenfunction of the Monge-Ampère operator by solving a constrained Monge-Ampère equation during each iteration. Our method consists of four essential steps: (i) Formulate the Monge-Ampère eigenvalue problem as an optimization problem with a constraint; (ii) Adopt an indicator function to treat the constraint; (iii) Introduce an auxiliary variable to decouple the original constrained optimization problem into simpler optimization subproblems and associate the resulting new optimization problem with an initial value problem; and (iv) Discretize the resulting initial-value problem by an operator-splitting method in time and a mixed finite element method in space. The performance of our method is demonstrated by several experiments. Compared to existing methods, the new method is more efficient in terms of computational cost and has a comparable rate of convergence in terms of accuracy.
Original languageEnglish
Article number7
Number of pages22
JournalCommunications in Optimization Theory
Issue number2022
Publication statusPublished - 5 Aug 2022

User-Defined Keywords

  • Eigenvalue problem
  • Monge–Ampére equation
  • Inverse iteration
  • Operator–splitting method


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