On Using Approximate Finite Differences in Matrix-Free Newton-Krylov Methods
On Using Approximate Finite Differences in Matrix-Free Newton-Krylov Methods
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DOI：
10.1137/060652749
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发表时间：
2008-04
期刊：
影响因子：
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通讯作者：
P. Brown;H. Walker;Rebecca Wasyk;C. Woodward
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文献类型：
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作者：
P. Brown;H. Walker;Rebecca Wasyk;C. Woodward
Newton-Krylov 方法是牛顿方法的一种实现，其中 Krylov 子空间方法用于近似求解表征牛顿方法步骤的线性系统。牛顿-克雷洛夫方法通常以“无矩阵”形式实现，其中克雷洛夫求解器所需的雅可比向量乘积通过有限差分来近似。这里我们考虑在这些有限差分中使用近似函数值。我们首先制定一个使用近似函数值的有限差分 Arnoldi 过程。然后，我们概述了一种 Newton-Krylov 方法，该方法使用基于此过程的 GMRES 或 Arnoldi 方法的实现，并为其开发了局部收敛分析，给出了近似函数值的充分条件，以保持所需的局部收敛特性。我们以涉及适用于非线性扩散问题的特定函数值近似的数值实验作为结论。对于这种情况，给出了满足函数评估中的非线性滞后和线性化收敛假设的条件。
A Newton-Krylov method is an implementation of Newton's method in which a Krylov subspace method is used to solve approximately the linear systems that characterize steps of Newton's method. Newton-Krylov methods are often implemented in “matrix-free” form, in which the Jacobian-vector products required by the Krylov solver are approximated by finite differences. Here we consider using approximate function values in these finite differences. We first formulate a finite-difference Arnoldi process that uses approximate function values. We then outline a Newton-Krylov method that uses an implementation of the GMRES or Arnoldi method based on this process, and we develop a local convergence analysis for it, giving sufficient conditions on the approximate function values for desirable local convergence properties to hold. We conclude with numerical experiments involving particular function-value approximations suitable for nonlinear diffusion problems. For this case, conditions are given for meeting the convergence assumptions for both lagging and linearizing the nonlinearity in the function evaluation.