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The Norm Convergence of a Least Squares Approximation Method for Random Maps

Raymond Manna Bangura

Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, P. R. China

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Congming Jin

https://orcid.org/0000-0002-8800-4350

Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, P. R. China

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, and

Jiu Ding

Department of Mathematics, The University of Southern Mississippi, Hattiesburg, MS 39406-5045, USA

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https://doi.org/10.1142/S0218127421500681Cited by:3 (Source: Crossref)

Abstract

We prove the $L^{1}$ $L^{1}$ -norm and bounded variation norm convergence of a piecewise linear least squares method for the computation of an invariant density of the Foias operator associated with a random map with position dependent probabilities. Then we estimate the convergence rate of this least squares method in the $L^{1}$ $L^{1}$ -norm and the bounded variation norm, respectively. The numerical results, which demonstrate a higher order accuracy than the linear spline Markov method, support the theoretical analysis.

Keywords:

References

Bahsoun, W., Bose, C. & Quas, A. [2008] “ Deterministic representation for position dependent random maps,” Discr. Contin. Dyn. Syst. 22, 529–540. Crossref, Web of Science, Google Scholar
Behnia, S., Akhshani, A., Ahadpour, S., Akhavand, A. & Mahmodib, H. [2009] “ Cryptography based on chaotic random maps with position dependent weighting probabilities,” Chaos Solit. Fract. 40, 362–369. Crossref, Web of Science, Google Scholar
Ding, J. & Li, T. Y. [1998] “ A convergence rate analysis for Markov finite approximations to a class of Frobenius–Perron operators,” Nonlin. Anal.: Th. Meth. Appl. 31, 765–777. Crossref, Web of Science, Google Scholar
Ding, J. & Zhou, A. [2009] Statistical Properties of Deterministic Systems (Springer). Crossref, Google Scholar
Ding, J. & Rhee, N. H. [2010] “ Piecewise linear least squares approximations of Frobenius–Perron operators,” Appl. Math. Comput. 217, 3257–3262. Crossref, Web of Science, Google Scholar
Ding, J. & Rhee, N. H. [2012] “ On the norm convergence of a piecewise linear least squares method for Frobenius–Perron operators,” J. Math. Anal. Appl. 386, 91–102. Crossref, Web of Science, Google Scholar
Dunford, N. & Schwartz, L. [1957] Linear Operators, Part I (Interscience). Google Scholar
Gockenbach, M. [2006] Understanding and Implementing the Finite Element Method (Society for Industrial and Applied Mathematics). Crossref, Google Scholar
Góra, P. & Boyarsky, A. [2003] “ Absolutely continuous invariant measures for random maps with position dependent probabilities,” J. Math. Anal. Appl. 278, 225–242. Crossref, Web of Science, Google Scholar
Horbacz, K., Myjak, J. & Szarek, T. [2005] “ On stability of some general random dynamical system,” J. Stat. Phys. 119, 35–60. Crossref, Web of Science, Google Scholar
Islam, M. S., Gora, P. & Boyarsky, A. [2005] “ Approximation of absolutely continuous invariant measures for Markov switching position dependent random maps,” Int. J. Pure Appl. Math. 1, 51–78. Google Scholar
Islam, M. S. [2011] “ Maximum entropy method for position dependent random maps,” Int. J. Bifurcation and Chaos 6, 1805–1811. Link, Web of Science, Google Scholar
Islam, M. S. [2015] “ Piecewise linear least squares approximations of invariant measures for random maps,” Neu. Paral. Sci. Comput. 23, 129–136. Google Scholar
Islam, M. S. [2017] “ A piecewise quadratic maximum entropy method for invariant measures of position dependent random maps,” Dyn. Contin. Discr. Impuls. Syst. Ser. A Math. Anal. 24, 431–445. Google Scholar
Jaeger, H. & Haas, H. [2004] “ Harnessing nonlinearity: Predicting chaotic systems and saving energy in wireless communication,” Science 304, 78–80. Crossref, Web of Science, Google Scholar
Jin, C., Upadhyay, T. & Ding, J. [2018] “ A picewise linear maximum entropy method for invariant measures of random maps with position dependent probabilities,” Int. J. Bifurcation and Chaos 28, 1850154-1–9. Link, Web of Science, Google Scholar
Jin, C. & Ding, J. [2020] “ A linear spline Markov approximation method for random maps with position dependent probabilities,” Int. J. Bifurcation and Chaos 30, 2050046-1–10. Link, Web of Science, Google Scholar
Lasota, A. & Yorke, J. A. [1973] “ On the existence of invariant measures for piecewise monotonic transformations,” Trans. Amer. Math. Soc. 186, 481–488. Crossref, Web of Science, Google Scholar
Lasota, A. & Mackey, M. C. [1994] Chaos, Fractals and Noise, 2nd edition (Springer). Crossref, Google Scholar
Lasota, A. & Mackey, M. C. [1999] “ Cell division and the stability of cellular population,” J. Math. Biol. 38, 241–261. Crossref, Web of Science, Google Scholar
Lasota, A. & Szarek, T. [2004] “ Dimension of measures invariant with respect to Ważewska partial differential equations,” J. Diff. Eqs. 196, 448–465. Crossref, Web of Science, Google Scholar
Lipniacki, T., Paszek, P., Marciniak-Czochra, A., Brasier, A. R. & Kimel, M. [2006] “ Transcriptional stochasticity in gene expression,” J. Theoret. Biol. 238, 348–367. Crossref, Web of Science, Google Scholar
Lorenz, E. [1963] Deterministic Nonperiodic Flow (University of Massachusetts Press). Crossref, Google Scholar
Morita, T. [1985] “ Random iteration of one-dimensional transformations,” Osaka J. Math. 22, 489–518. Web of Science, Google Scholar
Natanson, I. P. [1955] Theory of Functions of a Real Variable (Frederick Ungar Publishing Co., NY). Google Scholar
Pelikan, S. [1984] “ Invariant densities for random maps of the interval,” Proc. Amer. Math. Soc. 281, 813–825. Crossref, Web of Science, Google Scholar
Ulam, S. & von Neumann, J. [1945] “ Random ergodic theorems,” Bull. Amer. Math. Soc. 51, 660. Web of Science, Google Scholar