Performance of the extended precision VRP processor on various Krylov subspace solvers
Résumé
Numerical stability is a central aspect of high performance scientific computation, since the
growing scale of modern problems has led researchers to complex numerical techniques which
are vulnerable to round-off and quantization errors. This is particularly true for iterative Krylov
subspace projective solvers, which are the workhorse the modern scientific applications. Here,
higher precision, e.g. larger significand size, can mitigate these numerical instabilities and allow
for simpler and more memory efficient computations.
The VRP Processor [1] is designed to accelerate extended precision arithmetics in hardware.
It supports 1/ a fast arithmetic unit for up to 512 bits of mantissa and 2/ support in memory for
unaligned floating-point (FP) arrays.
The objective of this study is to assess the performance impact of using the VRP with extended
precision on common solver algorithms, e.g. conjugate gradient (CG), its preconditionned
variant (PCG) and biconjugate gradient (BiCG).
We consider two metrics: 1/ convergence speed, which refers to the number of iterations
necessary for reaching that objective and 2/ execution time, including memory access time,
measured in clock cycles. Execution time depends from the actual implementation of both
hardware and low-level software, and from the input matrix structure and values.
Our sample matrices principally come from the Florida sparse Matrix Collection [2]. We
restrict ourselves to real matrices, which may be symmetric (for CG and PCG) or asymmetric
(for BiCG).We compare execution time between different precisions (including standard double
format) on the same VRP platform
Our results confirm the benefits of extended precision for Krylov subspace solvers. For the
CG solver, extending precision around 128 mantissa bits appears optimal in terms of iteration
count and mostly beneficial for cycle count. The benefit for the BiCG solver is even greater.
When using precisions above 256 bits, BiCG convergence becomes predictable.
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