We consider a Beckmann formulation of an unbalanced optimal transport (UOT) problem. The $\Gamma$-convergence of this formulation of UOT to the corresponding optimal transport (OT) problem is established as the balancing parameter $\alpha$ 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 $\alpha$ 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.

翻译：我们考虑非平衡最优输运（UOT）问题的Beckmann形式。当平衡参数α趋于无穷时，证明了该UOT形式到经典最优输运（OT）问题的Γ-收敛性。进一步表明，该问题的离散化在相同极限下具有渐进保持性，从而确保数值方法可以统一应用，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α大于该临界值时，离散问题退化为离散OT问题。采用收敛的原-对偶混合算法求解离散问题，并证明UOT的迭代解也收敛到OT的迭代解。最后，通过形状变形和部分颜色转移的数值实验验证了理论收敛性及所提数值算法的有效性。