Graph Partitioning for High Performance Scientiic Simulations 0.2 Modeling Mesh-based Computations as Graphs 0.3 Static Graph Partitioning Techniques 0.3.2 Combinatorial Techniques

1 CONTENTS 2 Figure 1: A partitioned 2D irregular mesh of an airfoil. The shading of a mesh element indicates the processor to which it is mapped. 0.1 Introduction Algorithms that nd good partitionings of unstructured and irregular graphs are critical for the eecient execution of scientiic simulations on high performance parallel computers. In these simulations, computation is performed iteratively on each element (and/or node) of a physical two-or three-dimensional mesh and then information is exchanged between adjacent mesh elements. For example, computation is performed on each triangle of the two-dimensional irregular mesh shown in Figure 1. Then information is exchanged for every face between adjacent triangles. The eecient execution of such simulations on parallel machines requires a mapping of the computational mesh onto the processors such that each processor gets roughly an equal number of mesh elements and that the amount of inter-processor communication required to perform the information exchange between adjacent elements is minimized. Such a mapping is commonly found by solving a graph partitioning problem. For example, a graph partitioning algorithm was used to decompose the mesh in Figure 1. Here, the mesh elements have been shaded to indicate the processor to which they have been mapped. In many scientiic simulations, the structure of the computation evolves from time-step to time-step. These require an initial decomposition of the mesh prior to the start of the simulation (as described above), and also periodic load balancing to be performed during the course of the simulation. Other classes of simulations (i. e., multi-phase simulations) consist of a number of computational phases separated by synchronization steps. These require that each of the phases be individually load balanced. Still other scientiic simulations model multiple physical phenomenon (i. e., multi-physics simulations) or employ multiple meshes simultaneously (i. e., multi-mesh simulations). These impose additional requirements that the partitioning algorithm must take into account. Traditional graph partitioning algorithms are not adequate to ensure the eecient execution of these classes of simulations on high performance parallel computers. Instead, generalized graph partitioning algorithms have been developed for such simulations. This chapter presents an overview of graph partitioning algorithms used for scientiic simulations on high performance parallel computers. Recent developments in graph partitioning for adaptive and dynamic simulations , as well as partitioning algorithms for sophisticated simulations such as multi-phase, multi-physics, and multi-mesh computations are also discussed. Speciically, Section 0.2 presents the graph partitioning formulation to model the …