Linear and nonlinear substructured Restricted Additive Schwarz iterations and preconditioning
Linear and nonlinear substructured Restricted Additive Schwarz iterations and preconditioning
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DOI：
10.1007/s11075-022-01255-5
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发表时间：
2022-04
影响因子：
2.1
通讯作者：
F. Chaouqui;M. Gander;P. Kumbhar;T. Vanzan
中科院分区：
文献类型：
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作者：
F. Chaouqui;M. Gander;P. Kumbhar;T. Vanzan
迭代子结构域分解（DD）方法已被广泛研究，它们通常与非重叠分解相关。鲜为人知的是，经典的重叠 DD 方法也可以以子结构形式表示，即作为作用于专门在重叠域分解的接口上定义的变量的迭代方法。我们将这种表述称为子结构化域分解方法。我们在这里引入受限添加剂 Schwarz (RAS) 的子结构版本，我们称之为 SRAS。我们表明，RAS 和 SRAS 在用作迭代求解器时是等效的，因为它们产生相同的迭代，而在用作 GMRES 的预处理器时却有很大不同。我们将体积和子结构 Krylov 空间联系起来，并通过导出每次 GMRES 迭代中解决的最小二乘问题来表明迭代是不同的。当用作迭代求解器时，SRAS 比 RAS 具有计算优势，因为它避免了体积级别的矩阵和向量计算。当用作预处理器时，SRAS 具有进一步的优势，即允许 GMRES 存储较小的向量并在较低维空间中执行正交化。然后我们考虑非线性问题，并引入 SRASPEN（子结构限制加性施瓦茨预条件精确牛顿），其中 SRAS 用作牛顿方法的预条件子。与线性情况相反，我们证明应用于预处理体积和子结构公式的牛顿方法在非线性情况下产生相同的迭代。接下来，我们介绍非线性 SRAS 和 SRASPEN 的两级版本。最后，我们通过数值实验验证了我们的理论结果。
Iterative substructuring Domain Decomposition (DD) methods have been extensively studied, and they are usually associated with nonoverlapping decompositions. It is less known that classical overlapping DD methods can also be formulated in substructured form, i.e., as iterative methods acting on variables defined exclusively on the interfaces of the overlapping domain decomposition. We call such formulations substructured domain decomposition methods. We introduce here a substructured version of Restricted Additive Schwarz (RAS) which we call SRAS. We show that RAS and SRAS are equivalent when used as iterative solvers, as they produce the same iterates, while they are substantially different when used as preconditioners for GMRES. We link the volume and substructured Krylov spaces and show that the iterates are different by deriving the least squares problems solved at each GMRES iteration. When used as iterative solvers, SRAS presents computational advantages over RAS, as it avoids computations with matrices and vectors at the volume level. When used as preconditioners, SRAS has the further advantage of allowing GMRES to store smaller vectors and perform orthogonalization in a lower dimensional space. We then consider nonlinear problems, and we introduce SRASPEN (Substructured Restricted Additive Schwarz Preconditioned Exact Newton), where SRAS is used as a preconditioner for Newton’s method. In contrast to the linear case, we prove that Newton’s method applied to the preconditioned volume and substructured formulation produces the same iterates in the nonlinear case. Next, we introduce two-level versions of nonlinear SRAS and SRASPEN. Finally, we validate our theoretical results with numerical experiments.