Generalized unnormalized optimal transport and its fast algorithms
Generalized unnormalized optimal transport and its fast algorithms
复制标题
DOI：
10.1016/j.jcp.2020.110041
复制
发表时间：
2021-03-26
影响因子：
4.1
通讯作者：
Osher, Stanley
中科院分区：
文献类型：
--
作者：
Lee, Wonjun;Lai, Rongjie;Osher, Stanley
我们介绍了广义非规范化最优传输的快速算法。为了处理不同总质量的密度，我们考虑了一个动力学模型，它混合了Lpoptimal运输与Lpdistance。当p = 1时，我们得到了相应的L1广义非正规化Kantorovich公式.我们进一步表明，该问题成为一个简单的L-1最小化，有效地解决了原始对偶算法。当p = 2时，我们得到了L2广义非正规化Kantorovich公式，一个新的非正规化Monge问题和相应的Monge-Ampere方程.此外，我们引入了一个新的无约束优化公式的问题。相关的梯度流本质上是一个椭圆方程，可以有效地解决。在这里，建议的梯度下降过程连同Nesterov加速涉及HamiltonJacobi方程从KKT条件。数值算例表明了所提算法的有效性。(c)2020爱思唯尔公司All rights reserved.
We introduce fast algorithms for generalized unnormalized optimal transport. To handle densities with different total mass, we consider a dynamic model, which mixes the Lpoptimal transport with Lpdistance. For p = 1, we derive the corresponding L1generalized 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 L2generalized unnormalized Kantorovich formula, a new unnormalized Monge problem and the corresponding Monge-Ampere 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 HamiltonJacobi equation arisingfrom the KKT conditions. Several numerical examples are presented to illustrate the effectiveness of the proposed algorithms. (c) 2020 Elsevier Inc. All rights reserved.