1. What are the numerical results for two model problems that suggest precision bounds in BFP-arithmetic?
Using the ideas described above, Subsection 6.1 presents numerical results for two model problems that suggest that the precision bounds in (5.1) also apply in BFP-arithmetic. Subsection 6.2 covers the a priori estimation of the individual BFP precisions wj, w j, wj, and compares multigrid convergence rates of fixed-precision floating point, fixed-precision BFP, and progressive-precision BFP setups. Additionally, it reports on the estimation of g and w tmp to avoid recomputations in Algorithm_1 (Subsection 6.3), and studies the effect of skipping BFP-vector normalization altogether (Subsection 6.4).
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2. How to confirm BFP-precision bounds?
To confirm BFP-precision bounds, iteratively increase mantissa widths wj, wj, and wj until the BFP-approximation is close to a reference solution. Accept uh if u - uh L /u - u h L <= 1.5. Aim for discretization accuracy with O(h k-m ) error. Compare results with reference solution using 400-bit mantissa. Plot results for different test cases to observe asymptotic behavior. Use minimal mantissa widths for convergence. Compare 1D and 2D cases for consistency. Trade-off between precision and computational performance should be considered.
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3. How to estimate precisions for BFP-precision?
Precisions for BFP-precision can be estimated using discretization-dependent constants computed on coarse grids. The estimated precisions are extrapolated for finer grids based on their asymptotic behavior. In this paper, the precisions are determined using Algorithm 2 with specific parameters. The minima are determined using binary search over O(h^3), O(h^4), and O(h^5) float64 ('double') values. The results show that progressive precision BFP-F MG algorithm with estimated precisions provides discretization-error accurate solutions regardless of the refinement level.
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4. How can BFP arithmetic be efficient in FMG?
BFP arithmetic is particularly efficient when applied to FMG, as our results suggest that normalization of the BFP vectors is not necessary with proper estimates of upper bounds of the infinity norm of the intermediate results. This efficiency is achieved by ensuring exact computation of common BLAS-like vector-vector and matrix-vector operations up to a specified precision. The number of bits required to attain a certain level of error accuracy is similar to that of standard floating point arithmetic. This efficiency in BFP arithmetic can lead to energy-efficient implementations on specialized hardware, making it a promising approach for iterative linear solvers in BFP arithmetic.
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