适用于GPU加速的快速迭代矩阵求逆高阶方法的推广
Generalization of Higher Order Methods for Fast Iterative Matrix Inversion Suitable for GPU Acceleration
AI总结：
本文提出一种基于Neumann Series的Nested Neumann迭代矩阵求逆算法，通过预处理矩阵的无代价迭代更新推广高阶方法，在保证收敛性的同时适用于GPU加速和各类线性系统。
摘要：
最近的技术发展带来了大数据处理，这在求解大规模线性系统或矩阵求逆时导致了显著的计算困难。因此，通过图形处理单元（GPU）加速的快速近似迭代矩阵求逆方法成为了广泛研究的课题，旨在寻找当经典和直接求逆代价过高时的解决方案。目前常用的一些方法包括 Neumann Series (NS)、Newton iteration (NI)、Chebyshev Iteration (CI) 和 Successive Over-Relaxation 等。在这项工作中，我们开发了一种基于 NS 的新迭代算法，我们将其命名为 'Nested Neumann' (NN)。这种新方法通过利用预处理矩阵作为给定 'inception depth' 的函数进行计算上无代价的迭代更新，推广了 NI（或 CI）的高阶形式。数学上已经证明 NN 具有以下性质： 在预处理满足 NS 的谱范数条件时收敛， 其收敛速率的阶数被证明等同于该阶数（inception depth 加一）， 最优的 inception depth 是一或最好是二，具体取决于 RAM 限制。此外，我们推导了 NN 的一个显式公式，该公式适用于大型稀疏矩阵，但计算成本会有所增加。重要的是，NN 发现了 NS 与 NN（NI、CI 及更高阶）之间的解析等价性陈述，这对 mMIMO 系统具有重要意义。最后，NN 方法适用于正半定矩阵的求逆，并且适用于任何线性系统（稀疏、非稀疏、复数等）。
英文摘要：
Recent technological developments have led to big data processing, which resulted in significant computational difficulties when solving large-scale linear systems or inverting matrices. As a result, fast approximate iterative matrix inversion methodologies via Graphical Processing Unit (GPU) acceleration has been a subject of extensive research, to find solutions where classic and direct inversion are too expensive to conduct. Some currently used methods are Neumann Series (NS), Newton iteration (NI), Chebyshev Iteration (CI), and Successive Over-Relaxation, to cite a few. In this work, we develop a new iterative algorithm based off the NS, which we named 'Nested Neumann' (NN). This new methodology generalizes higher orders of the NI (or CI), by taking advantage of a computationally free iterative update of the preconditioning matrix as a function of a given 'inception depth'. It has been mathematically demonstrated that the NN: (i) convergences given the preconditioning satisfies the spectral norm condition of the NS, (ii) has an order of rate of convergence has been shown to be equivalent to the order (inception depth plus one), and (iii) has an optimal inception depth is an inception depth of one or preferably two, depending on RAM constraints. Furthermore, we derive an explicit formula for the NN, which is applicable to massive sparse matrices, given an increase in computational cost. Importantly, the NN finds an analytic equivalancy statement between the NS and the the NN (NI, CI, and higher orders), which is of importance for mMIMO systems. Finally, the NN method is applicable positive semi-definite matrices for matrix inversion, and applicable to any linear system (sparse, non-sparse, complex, etc.).
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