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首页> 外文期刊>SIAM Journal on Matrix Analysis and Applications >RANGE-SPACE VARIANTS AND INEXACT MATRIX-VECTOR PRODUCTS IN KRYLOV SOLVERS FOR LINEAR SYSTEMS ARISING FROM INVERSE PROBLEMS
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RANGE-SPACE VARIANTS AND INEXACT MATRIX-VECTOR PRODUCTS IN KRYLOV SOLVERS FOR LINEAR SYSTEMS ARISING FROM INVERSE PROBLEMS

机译：逆问题引起的线性系统Krylov求解中的空间变量和矩阵向量的不精确乘积

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摘要

The objects of this paper are to introduce range-space variants of standard Krylov iterative solvers for unsymmetric and symmetric linear systems and to discuss how inexact matrix-vector products may be used in this context. The new range-space variants are characterized by possibly much lower storage and computational costs than their full-space counterparts, which is crucial in data assimilation applications and other inverse problems. However, this gain is achieved without sacrificing the inherent monotonicity properties of the original algorithms, which are of paramount importance in data assimilation applications. The use of inexact matrix-vector products is shown to further reduce computational cost in a controlled manner. Formal error bounds are derived on the size of the residuals obtained under two different accuracy models, and it is shown why a model controlling forward error on the product result is often preferable to one controlling backward error on the operator. Simple numerical examples finally illustrate the developed concepts and methods.
机译：本文的目的是介绍用于非对称和对称线性系统的标准Krylov迭代求解器的范围空间变体，并讨论在这种情况下如何使用不精确的矩阵矢量乘积。新的距离空间变体的特征是与全空间变体相比，其存储和计算成本可能要低得多，这在数据同化应用和其他反问题中至关重要。但是，在不牺牲原始算法固有的单调性的前提下获得了这种增益，这在数据同化应用中至关重要。示出了使用不精确的矩阵向量乘积以受控的方式进一步降低了计算成本。形式误差边界是根据在两个不同精度模型下获得的残差的大小得出的，并且表明了为什么对乘积结果进行正向误差控制的模型通常优于对算符进行反向误差控制的模型。最后通过简单的数字示例说明了开发的概念和方法。

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