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Plantinga-Vegter algorithm takes average polynomial time

Felipe Cucker, Alperen A. Ergür, Josue Tonelli-Cueto

TL;DR

This paper analyzes the Plantinga–Vegter algorithm for approximating real plane curves via adaptive subdivision with interval arithmetic. By marrying condition-number analysis with continuous amortization and geometric functional-analysis techniques, it derives polynomial-time bounds on the average and smoothed complexities, in contrast to the prior exponential worst-case results. The key idea is to bound the final subdivision size by the expectation of a local condition number $\kappa_{\sf aff}$ and to show tight probabilistic controls under broad randomness models (dobro polynomials) and perturbations. The results explain why the PV algorithm performs well in practice and provide a rigorous framework connecting numerical geometry, condition numbers, and probabilistic analysis for continuous solvers.

Abstract

We exhibit a condition-based analysis of the adaptive subdivision algorithm due to Plantinga and Vegter. The first complexity analysis of the PV Algorithm is due to Burr, Gao and Tsigaridas who proved a $O\big(2^{τd^{4}\log d}\big)$ worst-case cost bound for degree $d$ plane curves with maximum coefficient bit-size $τ$. This exponential bound, it was observed, is in stark contrast with the good performance of the algorithm in practice. More in line with this performance, we show that, with respect to a broad family of measures, the expected time complexity of the PV Algorithm is bounded by $O(d^7)$ for real, degree $d$, plane curves. We also exhibit a smoothed analysis of the PV Algorithm that yields similar complexity estimates. To obtain these results we combine robust probabilistic techniques coming from geometric functional analysis with condition numbers and the continuous amortization paradigm introduced by Burr, Krahmer and Yap. We hope this will motivate a fruitful exchange of ideas between the different approaches to numerical computation.

Plantinga-Vegter algorithm takes average polynomial time

TL;DR

This paper analyzes the Plantinga–Vegter algorithm for approximating real plane curves via adaptive subdivision with interval arithmetic. By marrying condition-number analysis with continuous amortization and geometric functional-analysis techniques, it derives polynomial-time bounds on the average and smoothed complexities, in contrast to the prior exponential worst-case results. The key idea is to bound the final subdivision size by the expectation of a local condition number and to show tight probabilistic controls under broad randomness models (dobro polynomials) and perturbations. The results explain why the PV algorithm performs well in practice and provide a rigorous framework connecting numerical geometry, condition numbers, and probabilistic analysis for continuous solvers.

Abstract

We exhibit a condition-based analysis of the adaptive subdivision algorithm due to Plantinga and Vegter. The first complexity analysis of the PV Algorithm is due to Burr, Gao and Tsigaridas who proved a worst-case cost bound for degree plane curves with maximum coefficient bit-size . This exponential bound, it was observed, is in stark contrast with the good performance of the algorithm in practice. More in line with this performance, we show that, with respect to a broad family of measures, the expected time complexity of the PV Algorithm is bounded by for real, degree , plane curves. We also exhibit a smoothed analysis of the PV Algorithm that yields similar complexity estimates. To obtain these results we combine robust probabilistic techniques coming from geometric functional analysis with condition numbers and the continuous amortization paradigm introduced by Burr, Krahmer and Yap. We hope this will motivate a fruitful exchange of ideas between the different approaches to numerical computation.

Paper Structure

This paper contains 20 sections, 23 theorems, 66 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation
  3. Outline
  4. The $\mathsf{PV}$ Algorithm
  5. Main result
  6. Randomness Model
  7. Average and Smoothed Complexity
  8. Geometric framework
  9. Lipschitz properties
  10. Interval approximations
  11. Condition number
  12. What does $\kappa_{\sf aff}$ measure?
  13. A fundamental proposition
  14. Adaptive Complexity Analysis
  15. Local size bound framework
  16. ...and 5 more sections

Key Result

theorem 1

Let $f\in\mathcal{P}_{n,d}$, $\sigma>0$, and $g\in\mathcal{P}_{n,d}$ a dobro random polynomial with parameters $K$ and $\rho$. The expected number of $n$-cubes in the final subdivision of Algorithm alg:PVAlgorithm on input $(f,a)$ is at most if the interval approximations satisfy eq:intcond1 and eq:intcond2 and if they satisfy the hypothesis of burr2017.

Theorems & Definitions (47)

  • Definition 3.1
  • Remark 3.2
  • theorem 1
  • theorem 2
  • Proposition 4.1
  • proof
  • Corollary 4.2
  • proof
  • Theorem 4.3
  • Lemma 4.4
  • ...and 37 more