We introduce fast algorithms for generalized unnormalized optimal transport. To handle densities with different total mass, we consider a dynamic model, which mixes the $L^p$ optimal transport with $L^p$ distance. For $p=1$, we derive the corresponding $L^1$ generalized unnormalized Kantorovich formula. We further show that the problem becomes a simple $L^1$ minimization which is solved efficiently by a primal-dual algorithm. For $p=2$, we derive the $L^2$ generalized unnormalized Kantorovich formula, a new unnormalized Monge problem and the corresponding Monge-Ampère equation. Furthermore, we introduce a new unconstrained optimization formulation of the problem. The associated gradient flow is essentially related to an elliptic equation which can be solved efficiently. Here the proposed gradient descent procedure together with the Nesterov acceleration involves the Hamilton-Jacobi equation arising from the KKT conditions. Several numerical examples are presented to illustrate the effectiveness of the proposed algorithms.

我们介绍了用于广义非归一化最优运输的快速算法。为了处理具有不同总质量的密度，我们考虑一个动态模型，该模型将${\mathrm{大号}}^p$ 最佳运输 ${\mathrm{大号}}^p$距离。为了$p=\mathrm{1个}$，我们得出相应的 ${\mathrm{大号}}^{\mathrm{1个}}$广义非规范化的Kantorovich公式。我们进一步证明问题变得简单了${\mathrm{大号}}^{\mathrm{1个}}$通过原始对偶算法有效解决的最小化问题。为了$p=\mathrm{2个}$，我们得出 ${\mathrm{大号}}^{\mathrm{2个}}$广义非规范化Kantorovich公式，新的非规范化Monge问题和相应的Monge-Ampère方程。此外，我们介绍了该问题的新的无约束优化公式。相关的梯度流基本上与可以有效求解的椭圆方程有关。此处提出的梯度下降过程以及Nesterov加速度涉及KKT条件引起的汉密尔顿-雅各比方程。给出了几个数值示例，以说明所提出算法的有效性。