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Certified surface approximations using the interval Krawczyk test

Michael Burr, Jonathan D. Hauenstein, Kisun Lee

Abstract

We propose an algorithm to construct a certified approximation of a surface by generalizing the Krawczyk test. The Krawczyk test is based on interval arithmetic, and confirms the existence and uniqueness of a solution to a square system of analytic equations in a region. By generalizing this test, we extend the reach of this technique to non-square systems and higher-dimensional varieties. We provide a prototype implementation and illustrate its use on several examples.

Certified surface approximations using the interval Krawczyk test

Abstract

We propose an algorithm to construct a certified approximation of a surface by generalizing the Krawczyk test. The Krawczyk test is based on interval arithmetic, and confirms the existence and uniqueness of a solution to a square system of analytic equations in a region. By generalizing this test, we extend the reach of this technique to non-square systems and higher-dimensional varieties. We provide a prototype implementation and illustrate its use on several examples.
Paper Structure (16 sections, 5 theorems, 9 equations, 5 figures, 7 algorithms)

This paper contains 16 sections, 5 theorems, 9 equations, 5 figures, 7 algorithms.

Table of Contents

  1. Introduction
  2. Interval Krawczyk test
  3. Interval arithmetic
  4. Interval Krawczyk test
  5. An initial case: graphs
  6. Graph approximations
  7. Extensions
  8. Discussion of limitations
  9. Changing coordinate systems
  10. Unitary transformation
  11. Topological correctness
  12. Surface approximation
  13. Constructing boxes
  14. Covering the surface
  15. Correctness
  16. ...and 1 more sections

Key Result

Theorem 2.1

Suppose $F$, $\hat{z}$, $r=(r_1,r_2)$, $A$, $I$, and $J$ are defined as above. If there exists $\rho\in(0,1)$ with then, for every $\hat{x}\in I$, there exists a unique $y^*\in J$ such that $F(\hat{x},y^*)=0$ with $\|y^*-\pi_{-d}(\hat{z})\|\le r_2\rho$. In this case, we say that $\hat{z}$ is a $\rho$-approximate solution to $F$ with certification radius$r$.

Figures (5)

  • Figure 1: An approximation of a unit sphere $\bm{x^2+y^2+z^2-1=0}$ with an initial point $\bm{\hat{z}=(0,0,1)}$ and $\bm{\rho=\frac{1}{8}}$.
  • Figure 2: Approximations of a sphere $\bm{x^2+y^2+z^2-10=0}$ with different $\bm{\rho=\frac{1}{8}}$ (left) and $\bm{\frac{1}{80}}$ (right) at an initial point $\bm{\hat{z}=(0,0,3.16)}$. Each approximation shows $\bm{2000}$ interval boxes respectively.
  • Figure 3: Approximations of a sphere $\bm{x^2+y^2+z^2-10=0}$ with different $\bm{\rho=\frac{1}{8}}$ at $\bm{\hat{z}=(0,0,3.16)}$ and with $\bm{\frac{1}{80}}$ at an initial point $\bm{\hat{z}=(0,0,-3.16)}$. Each approximation shows $\bm{2000}$ interval boxes respectively.
  • Figure 4: An approximation of a torus $\bm{(\sqrt{x^2 + y^2} - 2)^2 + z^2 - 0.64 = 0}$ with an initial point $\bm{\hat{z}=(2.8,0,0)}$ and $\bm{\rho=\frac{7}{8}}$.
  • Figure 5: An approximation of a saddle surface $\bm{-0.125 x y^2 + 0.25 x^2 - z = 0}$ with an initial point $\bm{\hat{z}=(2,2,0)}$ and $\bm{\rho=\frac{7}{8}}$.

Theorems & Definitions (17)

  • Theorem 2.1: Interval Krawczyk test
  • proof
  • Remark 2.2
  • Remark 2.3
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • Remark 4.1
  • Theorem 4.2
  • proof
  • ...and 7 more