Optimal order multilevel preconditioners for regularized ill-posed problems
Optimal order multilevel preconditioners for regularized ill-posed problems
复制标题
用于正则化病态问题的最优阶多级预处理器
DOI：
10.1090/s0025-5718-08-02100-5
复制
发表时间：
2008
期刊：
影响因子：
--
通讯作者：
T. Dupont
中科院分区：
文献类型：
--
作者：
Andrei Draganescu;T. Dupont
在这篇文章中，我们设计和分析的线性系统所产生的正则化反问题的多级预条件。使用尺度无关的距离函数，测量谱等价的运营商，它表明，这些预条件近似逆的运营商的最佳顺序相对于空间离散化参数h。因此，当增加层数时，求解系统所需的预处理共轭梯度迭代次数将减少，如果h足够小，则可能仅执行一次精细级残差计算。这些结果是基于先前已知的里德尔（1997）的两级预条件子（也可参见Hanke和Vogel（1999）），以及对算子方程X-1- A = 0应用类牛顿方法。我们要求相关的前向问题具有一定的光滑性质，然而，只有自然的稳定性和逼近性质的离散算子假设。该算法被应用到一个逆时抛物方程，也就是说，找到导致一个给定的最终状态的初始值的问题。我们还提出了一些关于构造具有预先指定的逼近性质的限制算子的结果，这些结果具有独立的意义。
In this article we design and analyze multilevel preconditioners for linear systems arising from regularized inverse problems. Using a scale-independent distance function that measures spectral equivalence of operators, it is shown that these preconditioners approximate the inverse of the operator to optimal order with respect to the spatial discretization parameter h. As a consequence, the number of preconditioned conjugate gradient iterations needed for solving the system will decrease when increasing the number of levels, with the possibility of performing only one fine-level residual computation if h is small enough. The results are based on the previously known two-level preconditioners of Rieder (1997) (see also Hanke and Vogel (1999)), and on applying Newton-like methods to the operator equation X -1 - A = 0. We require that the associated forward problem has certain smoothing properties; however, only natural stability and approximation properties are assumed for the discrete operators. The algorithm is applied to a reverse-time parabolic equation, that is, the problem of finding the initial value leading to a given final state. We also present some results on constructing restriction operators with preassigned approximating properties that are of independent interest.