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2024 | OriginalPaper | Chapter

Novel Monte Carlo Algorithm for Linear Algebraic Systems

Authors : Venelin Todorov, Slavi Georgiev, Ivan Dimov

Published in: New Trends in the Applications of Differential Equations in Sciences

Publisher: Springer Nature Switzerland

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Abstract

A novel Monte Carlo algorithm has been introduced and analyzed for solving linear algebraic equations. This hybrid algorithm is comparable to the ”Walk on Equations”; Monte Carlo approach developed by Ivan Dimov, Sylvain Maire, and Jean Michel Sellier. A comparison with the Gauss-Siedel method has been made for matrices up to a size of 10000. The algorithms’ performance is enhanced by choosing appropriate values for the relaxation parameters, resulting in a significant reduction in computation time and lower relative errors for a given number of iterations. A theorem proving the algorithms convergence has been presented. By balancing the iteration matrix, the original algorithm can be optimized. In addition, a sequential Monte Carlo method developed by John Halton based on an iterative application of the control variate method has been employed. The most notable numerical experiment involves a large system derived from a finite element approximation of a problem that describes a beam structure in constructive mechanics.

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previous chapter Reconstruction of Boundary Conditions of a Parabolic-Hyperbolic Transmission Problem
next chapter A Note on the Hwang-Kim’s Universal Activation Function
Literature
1.
Curtiss, J.H.: Monte Carlo methods for the iteration of linear operators. J. Math Phys., vol 32(4). (1954), pp. 209–232.
2.
Curtiss, J.H.: A Theoretical Comparison of the Efficiencies of two classical methods and a Monte Carlo method for Computing one component of the solution of a set of Linear Algebraic Equations. Proc. Symp. MC Meth., John Wiley and Sons, 1956, pp. 191–233.
3.
Dimov, I.T.: Monte Carlo Methods for Applied Scientists, New Jersey, London, Singapore, World Scientific, 291p (2008).
4.
Dimov, I.T., Maire, S., Sellier, J.M.: A New Walk on Equations Monte Carlo Method for Linear Algebraic Problems, Applied Mathematical Modelling 39 (15), 4494–4510 (2015).
5.
Halton, J.: Sequential Monte Carlo, Proceedings of the Cambridge Philosophical Society, Vol. 58 (1962) pp. 57–78.
6.
Halton, J.: Sequential Monte Carlo. University of Wisconsin, Madison, Mathematics Research Center Technical Summary Report No. 816 (1967) 38 pp.
7.
Halton, J. and Zeidman, E. A.: Monte Carlo integration with sequential stratification. University of Wisconsin, Madison, Comp. Scien. Departm. Tech. Report No. 61 (1969) 31 pp.
8.
Halton, J.: Sequential Monte Carlo for linear systems - a practical summary, Monte Carlo Methods & Applications, 14 (2008) pp. 1–27.
9.
Maire, S.: Reducing variance using iterated control variates, Journal of Statistical Computation and Simulation, Vol. 73(1), pp. 1–29, 2003.
10.
I.M. Sobol, Monte Carlo Numerical Methods, Nauka, Moscow (1973).
Metadata
Title
Novel Monte Carlo Algorithm for Linear Algebraic Systems
Authors
Venelin Todorov
Slavi Georgiev
Ivan Dimov
Copyright Year
2024
Book
New Trends in the Applications of Differential Equations in Sciences

Print ISBN: 978-3-031-53211-5

Electronic ISBN: 978-3-031-53212-2

Copyright Year: 2024

DOI
https://doi.org/10.1007/978-3-031-53212-2_39

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