🤖 AI Summary
This study addresses the computational challenges in solving high-dimensional uncertain nonlinear systems via interval methods, which are often hindered by prohibitive costs. The authors establish an algorithmic worst-case analysis framework to systematically quantify how initial search volume, target tolerance, and fundamental verification operations influence computational complexity. They derive upper bounds on time and space complexity for several interval-based approaches, integrating techniques such as interval bisection, subdivision-filtering, constraint propagation, interval Newton, and Krawczyk methods, alongside interval linear algebra and inclusion functions. Their analysis reveals that naive interval computations of matrix determinants and inverses suffer from factorial-time bottlenecks. Notably, while the Krawczyk and interval Newton methods exhibit comparable dominant-order complexity, the former requires only real-matrix inversion at the midpoint, yielding significantly higher efficiency—providing crucial theoretical support for practical applications such as biochemical reaction networks.
📝 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.