Block Floating Point (BFP) arithmetic is currently seeing a resurgence in interest because it requires less power, less chip area, and is less complicated to implement in hardware than standard floating point arithmetic. This paper explores the application of BFP to mixed- and progressive-precision multigrid methods, enabling the solution of linear elliptic partial differential equations (PDEs) in energy- and hardware-efficient integer arithmetic. While most existing applications of BFP arithmetic tend to use small block sizes, the block size here is chosen to be maximal such that matrices and vectors share a single exponent for all entries. This is sometimes also referred to as a scaled fixed-point format. We provide algorithms for BLAS-like routines for BFP arithmetic that ensure exact vector-vector and matrix-vector computations up to a specified precision. Using these algorithms, we study the asymptotic precision requirements to achieve discretization-error-accuracy. We demonstrate that some computations can be performed using as little as 4-bit integers, while the number of bits required to attain a certain target accuracy is similar to that of standard floating point arithmetic. Finally, we present a heuristic for full multigrid in BFP arithmetic based on saturation and truncation that still achieves discretization-error-accuracy without the need for expensive normalization steps of intermediate results.

翻译：块浮点（BFP）算术当前重新引起关注，因其相比标准浮点算术所需功耗更低、芯片面积更小，且硬件实现复杂度更低。本文探索将BFP应用于混合精度与渐进精度多重网格方法，从而在节能且硬件高效的整数算术中求解线性椭圆型偏微分方程（PDE）。现有BFP算术大多采用小规模块结构，而本文选择最大化块尺寸，使矩阵和向量的所有条目共享单一指数。这种格式有时也被称为缩放定点格式。我们提供了BFP算术中类BLAS（基础线性代数子程序）算法的实现，确保向量-向量及矩阵-向量运算在指定精度内精确执行。利用这些算法，我们研究了达到离散化误差精度所需的渐进精度条件。结果表明，部分计算仅需4位整数即可完成，而达到特定目标精度所需的位数与标准浮点算术相当。最后，我们提出一种基于饱和与截断的全多重网格BFP启发式算法，该算法无需对中间结果进行昂贵的归一化步骤即可实现离散化误差精度。