Journal
SIAM JOURNAL ON SCIENTIFIC COMPUTING
Volume 43, Issue 5, Pages S420-S447Publisher
SIAM PUBLICATIONS
DOI: 10.1137/20M1349230
Keywords
Key words; mixed precision; progressive precision; multigrid; rounding-error analysis
Categories
Ask authors/readers for more resources
This paper presents discretization-error-accurate solutions for linear elliptic partial differential equations using mixed-precision multigrid solvers. It reveals that quantization errors quickly dominate the total error as discretization is refined. The progressive-precision scheme proposed in the paper balances quantization, algebraic, and discretization errors, leading to memory savings and reliable solutions with relatively few V-cycles.
This paper builds on the algebraic theory in the companion paper [S. F. McCormick, J. Benzaken, and R. Tamstorf, SIAM J. Sci. Comput., 43 (2021), pp. S392--S419] to obtain discretization-error-accurate solutions for linear elliptic partial differential equations (PDEs) by mixed-precision multigrid solvers. It is often assumed that the achievable accuracy is limited by discretization or algebraic errors. On the contrary, we show that the quantization error incurred by simply storing the matrix in any fixed precision quickly begins to dominate the total error as the discretization is refined. We extend the existing theory to account for these quantization errors and use the resulting bounds to guide the choice of four different precision levels in order to balance quantization, algebraic, and discretization errors in the progressive-precision scheme proposed in the companion paper. A remarkable result is that while iterative refinement is susceptible to quantization errors during the residual and update computations, the V-cycle used to compute the correction in each iteration is much more resilient and continues to work if the system matrices in the hierarchy become indefinite due to quantization. As a result, the V-cycle only requires relatively few bits of precision per level. Based on our findings, we outline a simple way to implement a progressive-precision full multigrid (FMG) solver with minimal overhead and demonstrate as an example that the one-dimensional biharmonic equation can be solved reliably to any desired accuracy using just a few V-cycles when the underlying smoother works well. Additionally, we show that the progressive-precision scheme leads to memory savings of up to 50\% compared to fixed precision.
Authors
I am an author on this paper
Click your name to claim this paper and add it to your profile.
Reviews
Recommended
No Data Available