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Application codes reliably achieve performance far less than the advertised capabilities of existing architectures, and this problem is worsening with increasingly-parallel machines. For large-scale numerical applications, stencil operations often impose the greater part of the computational cost, and the primary sources of inefficiency are the costs of message passing and poor cache utilization. This paper proposes and demonstrates optimizations for stencil and stencil-like computations for both serial and parallel environments that ameliorate these sources of inefficiency. Additionally, they argue that when stencil-like computations are encoded at a high level using object-oriented parallel array class libraries, these optimizations, which are beyond the capability of compilers, may be automated

Improving Cache Utilization of Linear Relaxation Methods: Theory and Practice