一种基于空间填充曲线的容错区域分解方法
A fault-tolerant domain decomposition method based on space-filling curves
摘要：
我们提出了一个简单的区域分解方法的$d$ -维椭圆偏微分方程，其中涉及到一个重叠分解成局部子域问题和一个整体的粗问题。它依赖于一个空间填充曲线来创建大小相等的子问题，并根据空间填充曲线的一维排序来确定一定的重叠。此外，我们采用凝聚和一个纯粹的代数Galerkin离散在建设的粗糙问题。这样，避免了使用$d$维几何信息。子问题的处理是一个附加的，并行的方式，从而产生一个子空间校正型线性迭代和预条件的共轭梯度法。为了使算法容错，我们存储在每个处理器上，除了相关的子问题的数据，一个副本的粗糙的问题，也是一个固定数量的相邻子问题的数据相对于一维排序的子问题引起的空间填充曲线。如果处理器在计算过程中出现故障，这种冗余可以恢复必要的数据。理论证明，这种线性迭代法的收敛速度与期望值保持一致，只有阶常数因故障而略有恶化。我们观察到这一点，在数值实验中的预条件共轭梯度法在稍微弱的形式。总之，我们得到了一个容错的，并行的和高效的区域分解方法的基础上的空间填充曲线，特别适合于高维椭圆问题。
英文摘要：
We propose a simple domain decomposition method for $d$-dimensional elliptic PDEs which involves an overlapping decomposition into local subdomain problems and a global coarse problem. It relies on a space-filling curve to create equally sized subproblems and to determine a certain overlap based on the one-dimensional ordering of the space-filling curve. Furthermore we employ agglomeration and a purely algebraic Galerkin discretization in the construction of the coarse problem. This way, the use of $d$-dimensional geometric information is avoided. The subproblems are dealt with in an additive, parallel way, which gives rise to a subspace correction type linear iteration and a preconditioner for the conjugate gradient method. To make the algorithm fault-tolerant we store on each processor, besides the data of the associated subproblem, a copy of the coarse problem and also the data of a fixed amount of neighboring subproblems with respect to the one-dimensional ordering of the subproblems induced by the space-filling curve. This redundancy then allows to restore the necessary data if processors fail during the computation. Theory supports that the convergence rate of such a linear iteration method stays the same in expectation, and only its order constant deteriorates slightly due to the faults. We observe this in numerical experiments for the preconditioned conjugate gradient method in slightly weaker form as well. Altogether, we obtain a fault-tolerant, parallel and efficient domain decomposition method based on space-filling curves which is especially suited for higher-dimensional elliptic problems.
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