迭代方法中浮点数据内联ZFP压缩的稳定性分析
Stability Analysis of Inline ZFP Compression for Floating-Point Data in Iterative Methods
AI总结：
本文将ZFP压缩数组的误差分析从单次压缩解压缩推广到时间步进和迭代格式，证明不动点与时间演化迭代的累积误差在标准约束下有界，并给出收敛所需额外迭代次数上界及前后向误差分析，通过1D、2D、3D测试验证了理论界限。
摘要：
目前，在许多高性能计算应用中，主导性约束是数据容量和带宽，既包括节点间通信，更包括节点内数据移动。解决这一限制的一种新方法是利用压缩数据数组形式的数据压缩。将数据存储在压缩数据数组中，并在计算过程中根据需要转换为标准IEEE-754类型，可以减轻带宽和存储压力。然而，反复的转换（有损压缩和解压缩）会引入额外的近似误差，需要证明这些误差不会显著影响模拟结果。我们扩展了最近的工作[J. Diffenderfer等人，浮点数据ZFP压缩的误差分析，SIAM Journal on Scientific Computing，2019]，该工作分析了ZFP压缩数据数组表示[P. Lindstrom，固定速率压缩浮点数组，IEEE Transactions on Visualization and Computer Graphics，2014]的单次压缩和解压缩误差，将其推广到时间步进和迭代格式的情形，其中除转换外还反复应用推进算子。我们证明，在标准约束下，涉及不动点和时间演化迭代的迭代方法的累积误差是有界的。建立了 stationary fixed-point iterations 收敛所需的额外迭代次数的上界。此外，还给出了使用ZFP压缩数组的 stationary iterative methods 的传统前向误差和后向误差分析。提供了若干一维、二维和三维测试问题的结果，以证明理论界限的正确性。
英文摘要：
Currently, the dominating constraint in many high performance computing applications is data capacity and bandwidth, in both inter-node communications and even more-so in on-node data motion. A new approach to address this limitation is to make use of data compression in the form of a compressed data array. Storing data in a compressed data array and converting to standard IEEE-754 types as needed during a computation can reduce the pressure on bandwidth and storage. However, repeated conversions (lossy compression and decompression) introduce additional approximation errors, which need to be shown to not significantly affect the simulation results. We extend recent work [J. Diffenderfer, et al., Error Analysis of ZFP Compression for Floating-Point Data, SIAM Journal on Scientific Computing, 2019] that analyzed the error of a single use of compression and decompression of the ZFP compressed data array representation [P. Lindstrom, Fixed-rate compressed floating-point arrays, IEEE Transactions on Visualization and Computer Graphics, 2014] to the case of time-stepping and iterative schemes, where an advancement operator is repeatedly applied in addition to the conversions. We show that the accumulated error for iterative methods involving fixed-point and time evolving iterations is bounded under standard constraints. An upper bound is established on the number of additional iterations required for the convergence of stationary fixed-point iterations. An additional analysis of traditional forward and backward error of stationary iterative methods using ZFP compressed arrays is also presented. The results of several 1D, 2D, and 3D test problems are provided to demonstrate the correctness of the theoretical bounds.
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