Mathematical Sciences: Solutions-Adaptive Grid Partitioning and Variable Ordering for PDEs

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Dear Al, Alex has kept me in the loop as you and he have converged on a revised scope of our proposed investigation into partitioning and ordering methods that take advantage of knowledge of coefficient magnitude, instead of coefficient structure only. Thanks for your support of this work, whose outcomes could feed directly into several other projects. To complete what I believe you need before processing this grant further, he and I have abstracted the project into the following two paragraphs for peers and nontechnical readers. I hope it is okay. Best Regards, David ========================================================================== Solution-adaptive Grid Partitioning and Variable Ordering for PDEs Alex Pothen and David E. Keyes The proposers consider grid partitioning and variable ordering problems that come to the fore in the parallel computation of problems modeled by partial differential equations discretized on unstructured grids. Partitioning algorithms that maintain load balance among the processors while incurring low communication costs for distributed memory parallelism have been developed in earlier work. Here solution-adaptive partitioning methods are proposed that strive, in addition, to attain good convergence rates in such data parallel contexts. Communication volume per iteration may rise, but only modestly for moderate-granularity parallelism, while the more important number of message-startups per iteration should not be strongly affected. Solution-adaptive ordering methods are also proposed for the fast solution of the problems on the local partitions. Two recently developed classes of partitioning and ordering algorithms that have proved to be successful in non-adaptive contexts will feature prominently in both the partitioning and ordering phases of this proposal. One class consists of spectral algorithms, (i.e., algorithms that make use of an approximation to a certain eigenvector of a Laplacian matrix derived from the grid and the discretization sche me), while the second class includes multilevel algorithms. Both algorithms will be generalized to the solution-adaptive context. Many computational problems in science and engineering can be modeled by partial differential equations, and subsequently solved as a system of algebraic equations defined on an unstructured grid. The most natural form of parallelism for such problems requires the original domain to be decomposed into subdomains, which are mapped whole on to individual processors. Iterative domain decomposition algorithms fit naturally into this framework, wherein a solution is assembled from subproblems whose boundary conditions are set by transmitting a limited amount of information between nearest neighbors. The partitioning and ordering methods proposed herein will be useful for solution methods of such domain decomposition type, which are designed with respect for the high communication-to-computation cost ratio in contemporary distributed-memory computers. The key opportunity to be exploited in this work is that of synergistically joining two ideas that have matured separately, and are usually used as ``black boxes''-- partitioning and ordering based on sparsity structure only, and domain-blocked iteration based on understanding the physical dependencies reflected in the coefficients. The software tools developed in this work will be demonstrated for ``grand challenge'' problems in application areas such as aerodynamics and geophysics.