Parallelization Strategies for the Randomized Kaczmarz Algorithm on Large-Scale Dense Systems
CoRR(2024)
摘要
The Kaczmarz algorithm is an iterative technique designed to solve consistent
linear systems of equations. It falls within the category of row-action
methods, focusing on handling one equation per iteration. This characteristic
makes it especially useful in solving very large systems. The recent
introduction of a randomized version, the Randomized Kaczmarz method, renewed
interest in the algorithm, leading to the development of numerous variations.
Subsequently, parallel implementations for both the original and Randomized
Kaczmarz method have since then been proposed. However, previous work has
addressed sparse linear systems, whereas we focus on solving dense systems. In
this paper, we explore in detail approaches to parallelizing the Kaczmarz
method for both shared and distributed memory for large dense systems. In
particular, we implemented the Randomized Kaczmarz with Averaging (RKA) method
that, for inconsistent systems, unlike the standard Randomized Kaczmarz
algorithm, reduces the final error of the solution. While efficient
parallelization of this algorithm is not achievable, we introduce a block
version of the averaging method that can outperform the RKA method.
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