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Computational Complexity Analysis of Interval Methods in Solving Uncertain Nonlinear Systems

Rudra Prakash, S. Janardhanan, Shaunak Sen

Abstract

This paper analyses the computational complexity of validated interval methods for uncertain nonlinear systems. Interval analysis produces guaranteed enclosures that account for uncertainty and round-off, but its adoption is often limited by computational cost in high dimensions. We develop an algorithm-level worst-case framework that makes the dependence on the initial search volume $\mathrm{Vol}(X_0)$, the target tolerance $\varepsilon$, and the costs of validated primitives explicit (inclusion-function evaluation, Jacobian evaluation, and interval linear algebra). Within this framework, we derive worst-case time and space bounds for interval bisection, subdivision$+$filter, interval constraint propagation, interval Newton, and interval Krawczyk. The bounds quantify the scaling with $\mathrm{Vol}(X_0)$ and $\varepsilon$ for validated steady-state enclosure and highlight dominant cost drivers. We also show that determinant and inverse computation for interval matrices via naive Laplace expansion is factorial in the matrix dimension, motivating specialised interval linear algebra. Finally, interval Newton and interval Krawczyk have comparable leading-order costs; Krawczyk is typically cheaper in practice because it inverts a real midpoint matrix rather than an interval matrix. These results support the practical design of solvers for validated steady-state analysis in applications such as biochemical reaction network modelling, robust parameter estimation, and other uncertainty-aware computations in systems and synthetic biology.

Computational Complexity Analysis of Interval Methods in Solving Uncertain Nonlinear Systems

Abstract

This paper analyses the computational complexity of validated interval methods for uncertain nonlinear systems. Interval analysis produces guaranteed enclosures that account for uncertainty and round-off, but its adoption is often limited by computational cost in high dimensions. We develop an algorithm-level worst-case framework that makes the dependence on the initial search volume , the target tolerance , and the costs of validated primitives explicit (inclusion-function evaluation, Jacobian evaluation, and interval linear algebra). Within this framework, we derive worst-case time and space bounds for interval bisection, subdivisionfilter, interval constraint propagation, interval Newton, and interval Krawczyk. The bounds quantify the scaling with and for validated steady-state enclosure and highlight dominant cost drivers. We also show that determinant and inverse computation for interval matrices via naive Laplace expansion is factorial in the matrix dimension, motivating specialised interval linear algebra. Finally, interval Newton and interval Krawczyk have comparable leading-order costs; Krawczyk is typically cheaper in practice because it inverts a real midpoint matrix rather than an interval matrix. These results support the practical design of solvers for validated steady-state analysis in applications such as biochemical reaction network modelling, robust parameter estimation, and other uncertainty-aware computations in systems and synthetic biology.
Paper Structure (31 sections, 18 theorems, 34 equations, 2 figures, 4 tables, 5 algorithms)

This paper contains 31 sections, 18 theorems, 34 equations, 2 figures, 4 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Interval Analysis
  4. Computational Complexity
  5. Results & Discussion
  6. Cost assumptions and symbols
  7. Interval Bisection Method
  8. Assumptions and enclosure goal for the complexity analysis
  9. Notation
  10. Time Complexity
  11. Space Complexity
  12. Subdivision $+$ Filter Method
  13. Time Complexity
  14. Space Complexity
  15. Interval Constraint Propagation
  16. ...and 16 more sections

Key Result

Theorem 2.1

Given a real-valued rational function $f$ and a natural interval extension $F$ such that $F(X)$ is well defined for some $X \in \mathbb{IR}^n$, we have:

Figures (2)

  • Figure 1: Interval-bisection recursion tree: the roots corresponds to the initial box $X_0$; each internal node is a box that passes the inclusion test $0\in F(X,U)$ and is split into two child boxes; recursion stops at leaves when the box diameter is $\leq\varepsilon$ (or when the inclusion test fails and the branch is discarded).
  • Figure 2: Uniform subdivision of the initial search box $X_0$ into $k$ sub-boxes (shown schematically as equal-volume partitions).

Theorems & Definitions (34)

  • Theorem 2.1: moore_introduction_2009
  • Theorem 2.2: moore_introduction_2009
  • Theorem 3.1
  • Proof 1
  • Corollary 3.2
  • Proof 2
  • Theorem 3.3
  • Proof 3
  • Theorem 3.4
  • Proof 4
  • ...and 24 more