Nonsymmetric Preconditioner Updates in Newton-Krylov Methods for Nonlinear Systems
Nonsymmetric Preconditioner Updates in Newton-Krylov Methods for Nonlinear Systems
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DOI：
10.1137/100789786
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发表时间：
2011-09
期刊：
影响因子：
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通讯作者：
S. Bellavia;D. Bertaccini;B. Morini
中科院分区：
文献类型：
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作者：
S. Bellavia;D. Bertaccini;B. Morini
牛顿-Krylov方法是牛顿类方法和Krylov子空间方法的结合，通常需要足够的预条件才能成功。雅可比矩阵的近似需要形成预条件子，而这一步往往是牛顿-克雷洛夫方法的主要代价。因此，使用预条件函数可能会破坏“无雅可比”(或无矩阵)设置，在该设置中可以提供单个雅可比向量积，而无需形成和存储真雅可比的元素。本文提出并分析了一种针对非对称雅可比矩阵序列的预条件技术，该技术是基于一种更新的预条件方法。所提出的策略可以以无矩阵的方式实施。通过对常见测试问题的数值实验，验证了该方法的有效性，并与标准的ILU-预条件牛顿-克里洛夫方法进行了比较。
Newton-Krylov methods, a combination of Newton-like methods and Krylov subspace methods for solving the Newton equations, often need adequate preconditioning in order to be successful. Approximations of the Jacobian matrices are required to form preconditioners, and this step is very often the dominant cost of Newton-Krylov methods. Therefore, working with preconditioners may destroy the “Jacobian-free” (or matrix-free) setting where the single Jacobian-vector product can be provided without forming and storing the element of the true Jacobian. In this paper, we propose and analyze a preconditioning technique for sequences of nonsymmetric Jacobian matrices based on the update of an earlier preconditioner. The proposed strategy can be implemented in a matrix-free manner. Numerical experiments on popular test problems confirm the effectiveness of the approach in comparison with the standard ILU-preconditioned Newton-Krylov approaches.