Document Type : Original Article

Author

• Somayeh Aghaei Khomami

Rasht Municipality

Abstract

In this paper, we propose an accelerated variant of the randomized Kaczmarz method for solving large-scale linear systems, including both standard and inequality-constrained systems. The key innovation lies in integrating the Johnson–Lindenstrauss (JL) lemma into the row-selection process, which allows high-dimensional rows to be projected onto lower-dimensional spaces while approximately preserving pairwise distances. This enables near-optimal row selection with reduced computational cost, improving both convergence rate and stability, particularly for ill-conditioned systems. Furthermore, Monte Carlo techniques are employed to efficiently construct the projection matrices, enhancing the overall computational performance. Numerical experiments demonstrate that the proposed method achieves faster convergence and higher accuracy compared to traditional randomized Kaczmarz and other conventional techniques, making it highly suitable for large-scale problems in applied mathematics and engineering.

Keywords

• Randomized Kaczmarz Method
• Dimensionality Reduction
• Johnson–Lindenstrauss Lemma
• Monte Carlo Technique
• Iterative Algorithms