Postdoctoral Research Associateship in Computer Science: Parallel Conjugate Gradient and Multilevel Algorithms for PDEs

在新兴的并行计算机上，研究偏微分方程组数值解的一种新的对称化方法。重点介绍了大规模并行化背景下该方法的算法发展。对称化方法对于椭圆偏微分算子A=S+BT很有吸引力，因为它将A中的问题转化为一个自伴且与S的自伴部分非常相似的算子。因此，通过调用对称性，即使对于非自伴问题，它也允许使用高效和健壮的多水平和共轭梯度算法来求解所产生的线性方程组。在并行计算机上实现该方法时使用多层算法和共轭梯度算法将提供在多层算法本身用作由内迭代和外迭代组成的算法的内求解器的情况下研究并行化的机会。算法将在Connection Machine提供的大规模并行环境中进行测试。

A new symmetrization method for the numerical solution of partial differential equations on emerging parallel computers will be studied. The algorithmic development of the method in the context of massive parallelization will be emphasized. The method of symmetrization is attractive for an elliptic partial differential operator, A=S+BT in that it transforms the problem in A to one in an operator that is self-adjoint and that closely resembles S, the self-adjoint part. Thus, by invoking symmetry even for nonself-adjoint problems, it allows the use of efficient and robust versions of multilevel and conjugate gradient algorithms for the solution of the arising linear systems of equations. The use of multilevel and conjugate gradient algorithms in the implementation of the method on a parallel computer will provide the opportunity to study parallelization in a context where a multilevel algorithm itself is used as in inner solver for an algorithm composed of inner and outer iterations. The algorithms will be tested in a massively parallel environment offered by the Connection Machine.