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Mixed-precision Orthogonalization Scheme and Adaptive Step Size for CA-GMRES on GPUs

Ichitaro Yamazaki, Stanimire Tomov, Tingxing Dong, Jack Dongarra
University of Tennessee, Knoxville, U.S.A.
University of Tennessee, Technical report ut-eecs-14-730, 2014

@article{yamazaki2014mixed,

   title={Mixed-Precision Orthogonalization Scheme and Adaptive Step Size for CA-GMRES on GPUs},

   author={Yamazaki, Ichitaro and Tomov, Stanimire and Dong, Tingxing and Dongarra, Jack},

   year={2014}

}

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We propose a mixed-precision orthogonalization scheme that takes the input matrix in a standard 32 or 64-bit floating-point precision, but accumulates its intermediate results in the doubled-precision. For a 64-bit input matrix, we use software emulation for the higher-precision arithmetics. Compared with the standard orthogonalization scheme, we require about 8:5 more computation but a much smaller increase in communication. Since the computation is becoming less expensive compared to the communication on new and emerging architectures, the relative cost of our mixed-precision scheme is decreasing. Our case studies with CA-GMRES on a GPU demonstrate that using mixed-precision for this small but critical segment of CA-GMRES can improve not only its overall numerical stability but also, in some cases, its performance. We also study an adaptive scheme to dynamically adjust the step size of the matrix powers kernel. Our experiments on multiple GPUs show that a near optimal step size can be chosen based on the performance measurements from the first restart loop of CA-GMRES.
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