Partial Differential Equations (PDEs) are used ubiquitously in modelling natural phenomena. It is generally not possible to obtain an analytical solution and hence they are commonly discretized using schemes such as the Finite Difference Method (FDM) and the Finite Element Method (FEM), converting the continuous PDE to a discrete system of sparse algebraic equations. The solution of this system can be approximated using iterative methods, which are better suited to many sparse systems than direct methods.
In this thesis we use the FDM to discretize linear, second order, Elliptic PDEs and consider parallel implementations of standard iterative solvers. The dominant paradigm in this field is distributed memory parallelism which requires the FDM grid to be partitioned across the available computational cores. The orthodox approach to domain partitioning aims to minimize only the communication volume and achieve perfect load-balance on each core. In this work, we re-examine and challenge this traditional method of domain partitioning and show that for well load-balanced problems, minimizing only the communication volume is insufficient for obtaining optimal domain partitions. To this effect we create a high-level, quasi-cache-aware mathematical model that quantifies cache-misses at the sub-domain level and minimizes them to obtain families of high performing domain decompositions. To our knowledge this is the first work that optimizes domain partitioning by analyzing cache misses, establishing a relationship between cache-misses and domain partitioning.
To place our model in its true context, we identify and qualitatively examine multiple other factors such as the Least Recently Used policy, Cache Line Utilization and Vectorization, that influence the choice of optimal sub-domain dimensions. Since the convergence rate of point iterative methods, such as Jacobi, for uniform meshes is not acceptable at a high mesh resolution, we extend the model to Parallel Geometric Multigrid (GMG). GMG is a multilevel, iterative, optimal algorithm for numerically solving Elliptic PDEs. Adaptive Mesh Refinement (AMR) is another multilevel technique that allows local refinement of a global mesh based on parameters such as error estimates or geometric importance. We study a massively parallel, multiphysics, multi-resolution AMR framework called BoxLib, and implement and discuss our model on single level and adaptively refined meshes, respectively.
We conclude that “close to 2-D” partitions are optimal for stencil-based codes on structured 3-D domains and that it is necessary to optimize for both minimizing cache-misses and communication. We advise that in light of the evolving hardware-software ecosystem, there is an imperative need to re-examine conventional domain partitioning strategies
