本文针对稀疏反卷积问题引入了一种新的非线性演化偏微分方程。所提出的PDE具有连续性方程的形式，该形式出现在流体动力学和最优输运等多个研究领域，因此具有一些有趣的物理解释和几何解释。我们考虑的基础优化模型是带有线性等式约束的标准ℓ1最小化，即$\min_u\{\|u\|_1 : Au=f\}$，其中$A$是欠采样卷积算子。我们证明，我们的PDE在降低残差$\|Au-f\|_2$的同时保持ℓ1范数。更重要的是，PDE的解渐近地变得更稀疏，这在数值上得到了验证。因此，它可以作为某些ℓ1最小化问题算法（例如，在[W. Yin, S. Osher, D. Goldfarb, and J. Darbon, SIAM J. Imaging Sci., 1 (2008), pp. 143-168]中为稀疏重建问题引入的Bregman迭代法）的一种自然且有益的插件。数值实验表明，在收敛速度和重建质量方面都有显著提升。
英文摘要：
In this paper, we introduce a new nonlinear evolution partial differential equation for sparse deconvolution problems. The proposed PDE has the form of continuity equation that arises in various research areas, e.g. fluid dynamics and optimal transportation, and thus has some interesting physical and geometric interpretations. The underlying optimization model that we consider is the standard $\ell_1$ minimization with linear equality constraints, i.e. $\min_u\{\|u\|_1 : Au=f\}$ with $A$ being an under-sampled convolution operator. We show that our PDE preserves the $\ell_1$ norm while lowering the residual $\|Au-f\|_2$. More importantly the solution of the PDE becomes sparser asymptotically, which is illustrated numerically. Therefore, it can be treated as a natural and helpful plug-in to some algorithms for $\ell_1$ minimization problems, e.g. Bregman iterative methods introduced for sparse reconstruction problems in [W. Yin, S. Osher, D. Goldfarb, and J. Darbon,SIAM J. Imaging Sci., 1 (2008), pp. 143-168]. Numerical experiments show great improvements in terms of both convergence speed and reconstruction quality.
Multichannel Sparse Blind Deconvolution on the Sphere
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Coordinated Science Laboratory（协同科学实验室）；Department of Electrical and Computer Engineering（电气与计算机工程系）
Jointly Sparse Blind Deconvolution via Riemannian Optimization
School of Mathematics and Statistics, Xi’an Jiaotong University（西安交通大学数学与统计学院）；College of Electronic Engineering, National University of Defense Technology（国防科技大学电子工程学院）；School of Mathematical Sciences,University of Science and Technology of China（中国科学技术大学数学科学学院）
Blind Deconvolution of Binary and Pattern Images with Pixel Intensity Constraints and Sparse Gradient Prior
Beijing Normal University（北京师范大学）；Anhui University（安徽大学）；University of Electronic Science and Technology of China（电子科技大学）；Northeast Normal University（东北师范大学）
