Nonlinear Preconditioning: How to Use a Nonlinear Schwarz Method to Precondition Newton's Method
Nonlinear Preconditioning: How to Use a Nonlinear Schwarz Method to Precondition Newton's Method
复制标题
DOI：
10.1137/15m102887x
复制
发表时间：
2016-05
期刊：
影响因子：
--
通讯作者：
V. Dolean;M. Gander;W. Kheriji;Felix Kwok;R. Masson
中科院分区：
文献类型：
--
作者：
V. Dolean;M. Gander;W. Kheriji;Felix Kwok;R. Masson
对于线性问题，区域分解方法可以直接用作迭代求解器，也可以用作Krylov方法的预处理器。在实践中，Krylov加速几乎总是被使用，因为Krylov方法找到比静态迭代更好的残差多项式，因此收敛得更快。在本文中，我们表明，也为非线性问题，区域分解方法可以直接用作迭代求解器，或者可以使用它们作为预条件的牛顿法。对于具体情况下的并行施瓦茨方法，我们表明，我们得到一个预条件，我们称之为RASPEN（限制添加剂施瓦茨预处理精确牛顿），这是类似于ASPIN（添加剂施瓦茨预处理不精确牛顿），但所有组件直接定义的迭代方法。这具有RASPEN作为迭代求解器时已经收敛的优点，与ASPIN相比，因此我们得到了牛顿方法的更好的预处理器。迭代构造还允许我们自然地使用多重网格全近似方案来定义粗校正，这导致收敛的两级非线性迭代区域分解方法和两级RASPEN非线性预处理器。我们用Forchheimer方程和非线性扩散问题的数值结果来说明我们的发现。
For linear problems, domain decomposition methods can be used directly as iterative solvers, but also as preconditioners for Krylov methods. In practice, Krylov acceleration is almost always used, since the Krylov method finds a much better residual polynomial than the stationary iteration, and thus converges much faster. We show in this paper that also for non-linear problems, domain decomposition methods can either be used directly as iterative solvers, or one can use them as preconditioners for Newton's method. For the concrete case of the parallel Schwarz method, we show that we obtain a preconditioner we call RASPEN (Restricted Additive Schwarz Preconditioned Exact Newton) which is similar to ASPIN (Additive Schwarz Preconditioned Inexact Newton), but with all components directly defined by the iterative method. This has the advantage that RASPEN already converges when used as an iterative solver, in contrast to ASPIN, and we thus get a substantially better preconditioner for Newton's method. The iterative construction also allows us to naturally define a coarse correction using the multigrid full approximation scheme, which leads to a convergent two level non-linear iterative domain decomposition method and a two level RASPEN non-linear preconditioner. We illustrate our findings with numerical results on the Forchheimer equation and a non-linear diffusion problem.