Parallel Multilevel Elliptic Preconditioners

在过去的几年里，人们对结合共轭梯度法求解椭圆型问题的多层预条件算子的研究越来越感兴趣。部分动机是这些方法是高度可并行化的，除了通常具有良好的收敛特性外，这是大多数经典预条件(例如，SSOR、不完全因式分解和多项式预条件)所缺乏的组合。作者在这方面的工作使他们相信，这是一种在大规模并行计算机上构造求解大型椭圆型差分方程组的高效且稳健的迭代方法的非常有前途的方法。在这一领域的继续研究将包括对诸如CM.2的大规模并行计算机的实现问题的仔细研究，不同的多级预处理器和一类区域分解方法之间的关系的理论研究，关于非光滑系数保留不完全因式分解方法的良好性质的混合多级方法，以及对自伴二阶椭圆算子(如不定算子、对流扩散算子和双调和算子)的推广。

In the last few years, there has been an increased interest in the study of multilevel preconditioners for solving elliptic problems in conjunction with the conjugate gradient method. Part of the motivation is that these methods are highly parallelizable, in addition to usually possessing good convergence properties, a combination that most of the classical preconditioners (e.g. SSOR, incomplete factorizations and polynomial preconditioners) lack. The authors' previous work in this area has convinced them that this is a very promising approach for constructing efficient and robust iterative methods for solving large elliptic difference equations on massively parallel computers. Continued research in this area, will include a careful study of implementation issues for massively parallel computers such as the CM.2, a theoretical study of the relationships among different multilevel preconditioners and the class of domain decomposition methods, hybrid multilevel methods which retain the nice properties of incomplete factorization methods with respect to non.smooth coefficients, and extensions beyond self-adjoint second order elliptic operators (e.g.indefinite, convection-diffusion and biharmonic operators.)