Rescheduling for Locality in Sparse

This talk corresponds to the paper 8] with the same name. 1 Motivation Multigrid solves a set of simultaneous linear equations optimally. That is the time and space used by the multigrid computation are proportional to N, the number of unknowns. In reality the performance of any computation is not just dictated by how many operations performed. The order of the operations and the layout of data in memory determine how well levels of cache in the target architecture are used to avoid memory latency. When a piece of data is still in cache upon reuse the computation is said to exhibit data locality. We have developed a technique for reordering computation and data in order to improve data locality. We apply this technique to Gauss-Seidel operating on an irregular mesh, because iterative algorithms like Gauss-Seidel are used as the smoother in Multigrid. In many scientiic computation the most time-consuming operation is solving a set of simultaneous linear equations, Au = f where u is a vector of unknowns. When using the Finite Diierence or Finite Element methods the equations arise from unknowns on a mesh, for example pressure or temperature. Gauss-Seidel is an example of an iterative computation used to solve Au = f. On the left we show an Iteration Space Graph of the Gauss-Seidel computation with 3 iterations shown. Each point represents an update of the u value for the corresponding mesh node. The arrows represent data dependences for once such mesh node. Gauss-Seidel uses the most recent version of the neighbors possible, so some data dependences come from points in the same i iteration and some data dependences come from points in the previous i iteration. On the right we show the overhead view of this 3D iteration space. Typically, this pattern of computation on a regular mesh is referred to as a 5 pt stencil. Since solving Au = f occurs frequently in scientiic computations, it is important for the solvers to exhibit good performance. On machines with caches improved data locality typically improves performance. Data locality occurs in a computation when the reuse is close enough in time that the data being reused is still in the faster levels of the memory hierarchy such as cache. Good data locality in Gauss-Seidel means that the data dependences for the current iteration point we are calculating are already in cache. For the purposes of this talk, …