Communications in Mathematical Sciences

Volume 22 (2024)

Number 2

A quadratic spline projection method for computing stationary densities of random maps

Pages: 519 – 531

DOI: https://dx.doi.org/10.4310/CMS.2024.v22.n2.a9

Authors

Azzah Alshekhi (Department of Mathematics, Faculty of Science and Art, Al-Baha University, Al-Baha, Saudi Arabia)

Jiu Ding (School of Mathematics and Natural Sciences, University of Southern Mississippi, Hattiesburg, Miss., U.S.A.)

Noah Rhee (Department of Mathematics and Statistics, University of Missouri, Kansas City, Mo., U.S.A.)

Abstract

We propose a quadratic spline projection method that computes stationary densities of random maps with position-dependent probabilities. Using a key variation inequality for the corresponding Markov operator, we prove the norm convergence of the numerical scheme for a family of random maps consisting of the Lasota–Yorke class of interval maps. The numerical experimental results show that the new method improves the $L^1$-norm errors and increases the convergence rate greatly, compared with the previous operator-approximation-based numerical methods for random maps.

Keywords

projection method, Frobenius–Perron operator, Foias operator, Markov operator, absolutely continuous probability measure, invariant measure, stationary density, random map

2010 Mathematics Subject Classification

41A35, 65D07, 65J10

Full Text (PDF format)

Received 29 November 2022

Received revised 13 July 2023

Accepted 14 July 2023

Published 1 February 2024