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High fidelity scientific simulations modeling physical phenomena typicallyrequire solving large linear systems of equations which result fromdiscretization of a partial differential equation (PDE) by some numericalmethod. This step often takes a vast amount of computational time to complete,and therefore presents a bottleneck in simulation work. Solving these linearsystems efficiently requires the use of massively parallel hardware with highcomputational throughput, as well as the development of algorithms whichrespect the memory hierarchy of these hardware architectures to achieve highmemory bandwidth.  In this paper, we present an algorithm to accelerate Jacobi iteration forsolving structured problems on graphics processing units (GPUs) using ahierarchical approach in which multiple iterations are performed within on-chipshared memory every cycle. A domain decomposition style procedure is adopted inwhich the problem domain is partitioned into subdomains whose data is copied tothe shared memory of each GPU block. Jacobi iterations are performed internallywithin each block's shared memory, avoiding the need to perform expensiveglobal memory accesses every step. We test our algorithm on the linear systemsarising from discretization of Poisson's equation in 1D and 2D, and observespeedup in convergence using our shared memory approach compared to atraditional Jacobi implementation which only uses global memory on the GPU. Weobserve a x8 speedup in convergence in the 1D problem and a nearly x6 speedupin the 2D case from the use of shared memory compared to a conventional GPUapproach.

Hierarchical Jacobi Iteration for Structured Matrices on GPUs using Shared Memory