43,695 research outputs found
Reach Set Approximation through Decomposition with Low-dimensional Sets and High-dimensional Matrices
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
Reachability analysis of linear hybrid systems via block decomposition
Reachability analysis aims at identifying states reachable by a system within
a given time horizon. This task is known to be computationally expensive for
linear hybrid systems. Reachability analysis works by iteratively applying
continuous and discrete post operators to compute states reachable according to
continuous and discrete dynamics, respectively. In this paper, we enhance both
of these operators and make sure that most of the involved computations are
performed in low-dimensional state space. In particular, we improve the
continuous-post operator by performing computations in high-dimensional state
space only for time intervals relevant for the subsequent application of the
discrete-post operator. Furthermore, the new discrete-post operator performs
low-dimensional computations by leveraging the structure of the guard and
assignment of a considered transition. We illustrate the potential of our
approach on a number of challenging benchmarks.Comment: Accepted at EMSOFT 202
Numerical Verification of Affine Systems with up to a Billion Dimensions
Affine systems reachability is the basis of many verification methods. With
further computation, methods exist to reason about richer models with inputs,
nonlinear differential equations, and hybrid dynamics. As such, the scalability
of affine systems verification is a prerequisite to scalable analysis for more
complex systems. In this paper, we improve the scalability of affine systems
verification, in terms of the number of dimensions (variables) in the system.
The reachable states of affine systems can be written in terms of the matrix
exponential, and safety checking can be performed at specific time steps with
linear programming. Unfortunately, for large systems with many state variables,
this direct approach requires an intractable amount of memory while using an
intractable amount of computation time. We overcome these challenges by
combining several methods that leverage common problem structure. Memory is
reduced by exploiting initial states that are not full-dimensional and safety
properties (outputs) over a few linear projections of the state variables.
Computation time is saved by using numerical simulations to compute only
projections of the matrix exponential relevant for the verification problem.
Since large systems often have sparse dynamics, we use Krylov-subspace
simulation approaches based on the Arnoldi or Lanczos iterations. Our method
produces accurate counter-examples when properties are violated and, in the
extreme case with sufficient problem structure, can analyze a system with one
billion real-valued state variables
Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions
Low-rank matrix approximations, such as the truncated singular value
decomposition and the rank-revealing QR decomposition, play a central role in
data analysis and scientific computing. This work surveys and extends recent
research which demonstrates that randomization offers a powerful tool for
performing low-rank matrix approximation. These techniques exploit modern
computational architectures more fully than classical methods and open the
possibility of dealing with truly massive data sets.
This paper presents a modular framework for constructing randomized
algorithms that compute partial matrix decompositions. These methods use random
sampling to identify a subspace that captures most of the action of a matrix.
The input matrix is then compressed---either explicitly or implicitly---to this
subspace, and the reduced matrix is manipulated deterministically to obtain the
desired low-rank factorization. In many cases, this approach beats its
classical competitors in terms of accuracy, speed, and robustness. These claims
are supported by extensive numerical experiments and a detailed error analysis
Computing covariant vectors, Lyapunov vectors, Oseledets vectors, and dichotomy projectors: a comparative numerical study
Covariant vectors, Lyapunov vectors, or Oseledets vectors are increasingly
being used for a variety of model analyses in areas such as partial
differential equations, nonautonomous differentiable dynamical systems, and
random dynamical systems. These vectors identify spatially varying directions
of specific asymptotic growth rates and obey equivariance principles. In recent
years new computational methods for approximating Oseledets vectors have been
developed, motivated by increasing model complexity and greater demands for
accuracy. In this numerical study we introduce two new approaches based on
singular value decomposition and exponential dichotomies and comparatively
review and improve two recent popular approaches of Ginelli et al. (2007) and
Wolfe and Samelson (2007). We compare the performance of the four approaches
via three case studies with very different dynamics in terms of symmetry,
spectral separation, and dimension. We also investigate which methods perform
well with limited data
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