Reach Set Approximation through Decomposition with Low-dimensional Sets and High-dimensional Matrices
This work addresses the scalability bottleneck of reachability analysis for safety verification of dynamical systems, enabling application to high-dimensional industrial problems.
The paper proposes a decomposition method for reach set approximation that performs set operations in low dimensions and matrix operations in high dimensions, achieving up to 100x speedup on standard benchmarks with modest accuracy loss, and scaling to over 10,000 variables in dense-time settings.
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.