数学建模及其应用

本研究针对2024年“高教社杯”全国大学生数学建模竞赛D题“反潜航空深弹命中概率问题”,提出了一种基于互相关函数与二值化运算的高效概率计算模型，通过构建潜艇与深弹杀伤范围的动态相遇模型，给出了命中概率的解析表达式，并采用空间离散化与三维卷积实现快速数值求解.本方法在普适性、高效性、可解释性和可拓展性方面均优于传统方法：适用于任意的深弹排布、潜艇形状及定位误差分布，突破了传统方法对场景简化的依赖；能够快速求解三维空间中的概率分布；计算过程透明，中间结果具有明确的物理含义；理论框架可推广至其他动态相遇概率问题.本文方法不仅为反潜作战提供了精准的理论依据，也为军事运筹学中的概率建模提供了新思路.

This study addresses the problem of calculating the hit probability of anti-submarine aviation depth charges(2024 CUMCM Problem D) by proposing an efficient model based on cross-correlation and binarization operations. A dynamic encounter model between the submarine and the lethal zone of the depth charges is constructed to derive an analytical expression for the hit probability, with a rapid numerical solution achieved through spatial discretization and three-dimensional convolution. The proposed method demonstrates significant advantages over traditional approaches in terms of generality(it is applicable to arbitrary depth charge patterns, submarine shapes and positioning error distributions, overcoming the reliance on simplified scenarios), efficiency(enabling fast computation of the three-dimensional probability distribution P(x, y, z), interpretability(featuring a transparent process where intermediate results have clear physical meanings), and extensibility(its theoretical framework is generalizable to other complex dynamic encounter probability problems). The results provide not only a precise theoretical foundation for anti-submarine warfare but also a novel methodology for probability modeling in military operations research.