AI 中文总结
受基于图的非线性插值和半监督学习启发，研究人员提出有限图上的Monge–Ampère方程，推导其非齐次Dirichlet问题的Bellman型公式，建立相关原理，研究齐次方程并提出两类问题的数值格式。
AI 中文摘要
受基于图的非线性插值和半监督学习的启发，我们引入了有限图上的Monge–Ampère方程版本。该算子定义为Hessian特征值离散模拟的乘积，这些特征值通过相邻顶点函数值的局部顺序统计量获得。我们推导了非齐次Dirichlet问题的等价Bellman型公式，建立了严格图凸类中的比较原理和唯一性，并通过Perron方法研究存在性，确定了某些图论障碍。我们还研究了齐次方程，其问题简化为涉及最小离散特征值的非线性插值规则。最后，我们为齐次和非齐次问题提出了数值格式。
英文摘要
We introduce a version of the Monge--Ampère equation on finite graphs, motivated by nonlinear graph-based interpolation and semi-supervised learning. The operator is defined as the product of discrete analogs of the Hessian eigenvalues, obtained via local order statistics of function values at neighboring vertices. We derive an equivalent Bellman-type formulation of the inhomogeneous Dirichlet problem, establish a comparison principle and uniqueness in the strictly graph-convex class, and investigate existence via Perron's method, identifying certain graph-theoretic obstructions. We also study the homogeneous equation, for which the problem reduces to a nonlinear interpolation rule involving the smallest discrete eigenvalue. Finally, we propose numerical schemes for both the homogeneous and inhomogeneous problems.
