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Acceleration of randomized Kaczmarz method via the Johnson–Lindenstrauss Lemma | ||
| Computational Sciences and Engineering | ||
| دوره 4، شماره 2، آذر 2024، صفحه 331-349 اصل مقاله (778.57 K) | ||
| نوع مقاله: Original Article | ||
| شناسه دیجیتال (DOI): 10.22124/cse.2025.31590.1119 | ||
| نویسنده | ||
| Somayeh Aghaei Khomami* | ||
| Rasht Municipality | ||
| چکیده | ||
| 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. | ||
| کلیدواژهها | ||
| Randomized Kaczmarz Method؛ Dimensionality Reduction؛ Johnson–Lindenstrauss Lemma؛ Monte Carlo Technique؛ Iterative Algorithms | ||
| مراجع | ||
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[1] Yousefpanah, K., (2017), Resolved Monte Carlo Algorithms for solving linear systems, MSc dissertations, University of Guilan. [2] Eldar, Yonina C., and Deanna Needell. "Acceleration of randomized Kaczmarz method via the Johnson–Lindenstrauss lemma." Numerical Algorithms 58.2 (2011): 163-177. [3] Sabelfeld, Karl, and Nadja Loshchina. "Stochastic iterative projection methods for large linear systems." Monte Carlo Methods and Applications16.3-4 (2010): 343-359. [4] Dasgupta, Sanjoy, and Anupam Gupta. "An elementary proof of the Johnson-Lindenstrauss lemma." International Computer Science Institute, Technical Report (1999): 99-006. [5] Chan, T.F., Wan, W.L.: Analysis of projection methods for solving linear systems with multiple right-hand sides. SIAM J. Sci. Comput. 18, 1698–1721 (1997). [6] Dasgupta, S., Gupta, A.: An elementary proof of a theorem of Johnson and Lindenstrauss. Random Struct. Algorithms 22(1), 60–65 (2003). [7] B. Fathi Vajargah, M. Moradi, Diagonal scaling of ill-conditioned matrixes by genetic algorithm, Journal of Applied Mathematics, Statistics and Informatics (JAMSI) 8 (1), (2012). [8] B. Fathi Vajargah , A way to obtain Monte Carlo matrix inversion with minimal error, Applied mathematics and computation 191 (1), 225-233, (2007). | ||
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