Reach Set Approximation through Decomposition with Low-dimensional Sets and High-dimensional Matrices

TL;DR

This paper introduces a decomposition method for reach set approximation that combines low-dimensional set operations with high-dimensional matrix computations, significantly improving scalability and efficiency in analyzing large dynamical systems.

Contribution

The authors propose a novel decomposition approach that enables scalable reach set computations by performing set operations in low dimensions and matrix operations in full dimension, applicable to dense and discrete-time systems.

Findings

Achieves up to 100x speed-up over state-of-the-art tools.

Handles systems with over 10,000 variables, two orders of magnitude larger than previous methods.

Maintains modest accuracy loss despite significant efficiency gains.

Abstract

Approximating the set of reachable states of a dynamical system is an algorithmic yet mathematically rigorous way to reason about its safety. Although progress has been made in the development of efficient algorithms for affine dynamical systems, available algorithms still lack scalability to ensure their wide adoption in the industrial setting. While modern linear algebra packages are efficient for matrices with tens of thousands of dimensions, set-based image computations are limited to a few hundred. We propose to decompose reach set computations such that set operations are performed in low dimensions, while matrix operations like exponentiation are carried out in the full dimension. Our method is applicable both in dense- and discrete-time settings. For a set of standard benchmarks, it shows a speed-up of up to two orders of magnitude compared to the respective state-of-the art tools, with only modest losses in accuracy. For the dense-time case, we show an experiment with more than 10.000 variables, roughly two orders of magnitude higher than possible with previous approaches.