预解式集的不动点：几何与算法
Fixed Points of Averages of Resolvents: Geometry and Algorithms
摘要：
在数学和自然科学中，如果一个给定的问题没有实际的解，那么给出它的广义解是一项重要的任务。它有丰富的历史可以追溯到19世纪初，当卡尔弗里德里希高斯开发的最小二乘法系统的线性方程组-其解决方案可以被视为不动点的平均投影到超平面。这个问题的一个强有力的推广是找到平均预解式的不动点（即，坚决非扩张映射）。本文研究了平均预解式的不动点集与预解式的某些复合不动点集之间的关系。它部分地推广了最近的工作，两个映射的问题C。伯恩分析表明，在一个积空间的重构。此外，提出了两种新的算法。一个全收敛的证明是基于平均映射的第一个算法。第二个算法，目前还没有收敛性证明，迭代映射，甚至不是非扩张的。数值实验表明，这些算法的潜力相比，迭代的平均解。
英文摘要：
To provide generalized solutions if a given problem admits no actual solution is an important task in mathematics and the natural sciences. It has a rich history dating back to the early 19th century when Carl Friedrich Gauss developed the method of least squares of a system of linear equations - its solutions can be viewed as fixed points of averaged projections onto hyperplanes. A powerful generalization of this problem is to find fixed points of averaged resolvents (i.e., firmly nonexpansive mappings). This paper concerns the relationship between the set of fixed points of averaged resolvents and certain fixed point sets of compositions of resolvents. It partially extends recent work for two mappings on a question of C. Byrne. The analysis suggests a reformulation in a product space. Furthermore, two new algorithms are presented. A complete convergence proof that is based on averaged mappings is provided for the first algorithm. The second algorithm, which currently has no convergence proof, iterates a mapping that is not even nonexpansive. Numerical experiments indicate the potential of these algorithms when compared to iterating the average of the resolvents.
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