论文标题
最佳运输正则化的动态逆问题的通用条件梯度方法
A generalized conditional gradient method for dynamic inverse problems with optimal transport regularization
论文作者
论文摘要
我们开发了一种动态通用条件梯度法（DGCG），用于最佳运输正则化的动态反问题。我们考虑（Bredies and Fanzon，Esaim：M2an，54：2351-2382，2020）中介绍的框架，该框架由忠诚度组成，在该框架中构成了一个忠诚术语，惩罚了观察到的时间差异，并通过时间变化的Homgers nightiber of the Hombert nightiber of the Homgers nodimentiber the Homgers nodimifies，并保持定期的能力，并保持正常的跟踪，并延续了定期的范围 - 方程式。采用贝纳莫 - 布雷尼啤酒能量的极端点的表征（Bredies等，Arxiv：1907.11589，2019），我们将问题的原子定义为衡量标准的原子，因为措施集中在域中的绝对连续曲线上。我们提出了一种有条件梯度方法的动态概括，该方法包括迭代地添加适当选择的原子到当前的稀疏迭代中，并随后优化了所得线性组合中的系数。我们证明该方法以均方根速率收敛到目标功能的最小化器。此外，我们提出了启发式策略和加速步骤，允许有效地实施该算法。最后，我们提供了数值示例，这些示例证明了我们的算法和模型在重建大量不足的动态数据以及噪声的存在方面的有效性。
We develop a dynamic generalized conditional gradient method (DGCG) for dynamic inverse problems with optimal transport regularization. We consider the framework introduced in (Bredies and Fanzon, ESAIM: M2AN, 54:2351-2382, 2020), where the objective functional is comprised of a fidelity term, penalizing the pointwise in time discrepancy between the observation and the unknown in time-varying Hilbert spaces, and a regularizer keeping track of the dynamics, given by the Benamou-Brenier energy constrained via the homogeneous continuity equation. Employing the characterization of the extremal points of the Benamou-Brenier energy (Bredies et al., arXiv:1907.11589, 2019) we define the atoms of the problem as measures concentrated on absolutely continuous curves in the domain. We propose a dynamic generalization of a conditional gradient method that consists in iteratively adding suitably chosen atoms to the current sparse iterate, and subsequently optimize the coefficients in the resulting linear combination. We prove that the method converges with a sublinear rate to a minimizer of the objective functional. Additionally, we propose heuristic strategies and acceleration steps that allow to implement the algorithm efficiently. Finally, we provide numerical examples that demonstrate the effectiveness of our algorithm and model at reconstructing heavily undersampled dynamic data, together with the presence of noise.
