SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 749-774, April 2024.
Abstract. We consider a Beckmann formulation of an unbalanced optimal transport (UOT) problem. The [math]-convergence of this formulation of UOT to the corresponding optimal transport (OT) problem is established as the balancing parameter [math] goes to infinity. The discretization of the problem is further shown to be asymptotic preserving regarding the same limit, which ensures that a numerical method can be applied uniformly and the solutions converge to the one of the OT problem automatically. Particularly, there exists a critical value, which is independent of the mesh size, such that the discrete problem reduces to the discrete OT problem for [math] being larger than this critical value. The discrete problem is solved by a convergent primal-dual hybrid algorithm and the iterates for UOT are also shown to converge to that for OT. Finally, numerical experiments on shape deformation and partial color transfer are implemented to validate the theoretical convergence and the proposed numerical algorithm.

中文翻译：

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1-Wasserstein距离连续与离散不平衡最优输运模型的收敛性

SIAM 数值分析杂志，第 62 卷，第 2 期，第 749-774 页，2024 年 4 月。
摘要。我们考虑不平衡最优传输 (UOT) 问题的贝克曼公式。当平衡参数[数学]趋于无穷大时，UOT 公式与相应的最优传输 (OT) 问题的[数学]收敛性得以建立。问题的离散化进一步证明对于相同的极限是渐近保持的，这确保了数值方法可以统一应用并且解自动收敛到OT问题的解。特别地，存在一个与网格尺寸无关的临界值，使得当[math]大于该临界值时，离散问题简化为离散OT问题。离散问题通过收敛的原对偶混合算法解决，并且 UOT 的迭代也收敛于 OT 的迭代。最后，进行了形状变形和部分颜色转移的数值实验，以验证理论收敛性和所提出的数值算法。