Total Variation Minimization with Finite Elements: Convergence and Iterative Solution
Total Variation Minimization with Finite Elements: Convergence and Iterative Solution
复制标题
有限元总变差最小化：收敛和迭代解决方案
DOI：
10.1137/11083277x
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
["S. Bartels
中科院分区：
文献类型：
--
作者：
["S. Bartels
分析了一类含非光滑全变分范数的凸极小化问题的数值解。一致的有限元离散，避免正则化导致简单的收敛性证明的情况下，分段仿射，全局连续的有限元。对于分段常数有限元的逼近，证明了一般不能期望收敛到精确解。迭代解基于能量泛函的正则化L^2 $流，并证明了在适当的时间步长约束下迭代收敛到一个稳定点。扩展的技术，涉及一个负序项的能量泛函进行了讨论。数值实验表明，理论结果。
The numerical solution of a convex minimization problem involving the nonsmooth total variation norm is analyzed. Consistent finite element discretizations that avoid regularizations lead to simple convergence proofs in the case of piecewise affine, globally continuous finite elements. For the approximation with piecewise constant finite elements it is proved that convergence to the exact solution cannot be expected in general. The iterative solution is based on a regularized $L^2$ flow of the energy functional, and convergence of the iteration to a stationary point is proved under a moderate constraint on the time-step size. The extension of the techniques to an energy functional that involves a negative order term is discussed. Numerical experiments that illustrate the theoretical results are presented.