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Convergence Analysis of the Random Bisection Method

Ludovick Bouthat, Philippe-André Luneau, Philippe Petitclerc

Abstract

We propose a generalized version of the bisection method where the cutting point between the two subintervals is chosen at random following an arbitrary distribution. We compute expected convergence rates with respect to any arbitrary a priori distribution for the position of the root in the initial interval and proved that it depends only on the the expectation $\mathbb{E}[c(1-c)]$ of the cut $c$. We also provide a generalization of the method for $K$ random cuts and study its convergence properties. Most probabilistic derivations are kept fairly simple for the ease of understanding of a larger audience. Our theoretical results are then validated numerically using statistical simulation.

Convergence Analysis of the Random Bisection Method

Abstract

We propose a generalized version of the bisection method where the cutting point between the two subintervals is chosen at random following an arbitrary distribution. We compute expected convergence rates with respect to any arbitrary a priori distribution for the position of the root in the initial interval and proved that it depends only on the the expectation of the cut . We also provide a generalization of the method for random cuts and study its convergence properties. Most probabilistic derivations are kept fairly simple for the ease of understanding of a larger audience. Our theoretical results are then validated numerically using statistical simulation.
Paper Structure (17 sections, 14 theorems, 84 equations, 27 figures, 4 tables, 2 algorithms)

This paper contains 17 sections, 14 theorems, 84 equations, 27 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. The Stochastic Bisection Algorithm and Related Definitions
  4. Stochastic Dynamical Systems
  5. Outline of the paper
  6. The Case When r is Uniform
  7. The interval length after a single iteration
  8. Stationarity of the Uniform Distribution
  9. The probability density function of ℓₙ
  10. Properties of Lₙ
  11. The Case When r is Not Uniform
  12. The case with multiple cuts
  13. Numerical Experiments
  14. Expected Convergence Rate
  15. Convergence to the Uniform Distribution
  16. ...and 2 more sections

Key Result

Proposition 3

If $r_0\sim\mathcal{U}(0,1)$, then $r_n\sim \mathcal{U}(0,1)$ for all $n\geq1$. That is, $\mathcal{U}(0,1)$ is a stationary distribution for the random process described in alg:RRbisection.

Figures (27)

  • Figure 1: Skewed Dyadic map with parameter $1/c$.
  • Figure 2: Expected length after one iteration if $c_0\sim \mathcal{U}(0,1)$.
  • Figure 3: Expected length after one iteration if $c_0\sim \mathrm{Beta}(2,2)$.
  • Figure 4: The region $L^-_t(T)$. The blue section corresponds to $r_0\geq c_0$, and the red region corresponds to $r_0<c_0$.
  • Figure 5: The region $\{(c,r) : \ell \leq t \}$. The blue section corresponds to $r>c$, and the red region corresponds to $r<c$.
  • ...and 22 more figures

Theorems & Definitions (24)

  • Proposition 3
  • proof
  • Corollary 4
  • Theorem 5
  • proof
  • Corollary 6
  • proof
  • Corollary 7
  • Theorem 8
  • proof
  • ...and 14 more