The Method of Alternating Projections

DMS获奖摘要获奖编号：0204569PI: Deutsch， FrankInstitution: Pennsylvania State University， University park项目：应用数学项目经理：Catherine mavriplis标题：交替投影法交替投影法（MAP）是一种迭代过程，用于从有限个闭凸集的交集中确定最近点。这种方法至少在15个不同的数学领域得到了应用，包括求解线性方程和不等式、信号分析和计算机断层扫描。MAP的主要实际缺点（至少对于某些应用程序）是收敛速度慢。我们将把子空间之间的“角”的概念推广到更一般的凸集，研究MAP的收敛速度，并确定加速MAP收敛的方法。此外，我们希望证明MAP的收敛速度与我们在过去20年的几篇论文中研究的凸集的一个性质（“强圆锥壳相交”性质）直接相关。交替投影法是一种通过重复应用相同的几个简单步骤来计算特定集合中最近点的算法。由于该方法已应用于许多不同的数学领域（最早的应用之一是通过x射线进行医学成像），因此了解该算法需要采取多快（即多少步）才能得到精确解的结果是很重要的。算法的“收敛速度”处理的问题是算法对任何给定问题的速度有多快。对于收敛速度较慢的问题，我们还将研究如何通过对算法中的某些步骤进行适当的修改，使算法更快地工作。日期：2002年5月13日

DMS Award AbstractAward #: 0204569PI: Deutsch, FrankInstitution: Pennsylvania State University, University ParkProgram: Applied MathematicsProgram Manager: Catherine MavriplisTitle: The Method of Alternating ProjectionsThe method of alternating projections (MAP) is an iterative procedure for determining nearest points from a set that is the intersection of a finite number of closed convex sets. This method has found use in at least 15 different areas of mathematics, which includes solving linear equations and inequalities, signal analysis, and computed tomography. The main practical drawback of the MAP, at least for some applications, is its slow convergence. We will study rates of convergence for the MAP by extending the notion of ``angle'' between subspaces to that of more general convex sets, as well as determine means for accelerating the convergence of the MAP. In addition, we expect to show that the speed of convergence of the MAP is directly related to a property (the ``strong conical hull intersection'' property) of the convex sets in question that we have studied in several papers over the last 20 years. The method of alternating projections is an algorithm for computing nearest points in certain sets by a repetitive application of the same few simple steps. Since the method has applications to so many different areas of mathematics (one of the earliest being in medical imaging via X-rays), it is important to know how fast (i.e., how many steps) this algorithm needs to take before the results give good accuracy of the exact solution. The ``rate of convergence'' of the algorithm deals with the question of how fast the algorithm is for any given problem. For those problems when the rate of convergence is slow, we will also study ways to make the algorithm work faster by making appropriate modifications of certain steps in the algorithm. Date: May 13, 2002