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

Efficient reachability analysis of parametric linear hybrid systems with time-triggered transitions

Efficiently handling time-triggered and possibly nondeterministic switches for hybrid systems reachability is a challenging task. In this paper we present an approach based on conservative set-based enclosure of the dynamics that can handle systems with uncertain parameters and inputs, where the uncertainties are bound to given intervals. The method is evaluated on the plant model of an experimental electro-mechanical braking system with periodic controller. In this model, the fast-switching controller dynamics requires simulation time scales of the order of nanoseconds. Accurate set-based computations for relatively large time horizons are known to be expensive. However, by appropriately decoupling the time variable with respect to the spatial variables, and enclosing the uncertain parameters using interval matrix maps acting on zonotopes, we show that the computation time can be lowered to 5000 times faster with respect to previous works. This is a step forward in formal verification of hybrid systems because reduced run-times allow engineers to introduce more expressiveness in their models with a relatively inexpensive computational cost.Comment: Submitte

Algoritmos de optimización para secuenciación adaptativa de rutas reales en entregas de última milla

This article explores the design and application of machine learning techniques to enhance traditional approaches for solving NP-hard optimization problems. Specifically, it focuses on the Last-Mile Routing Research Challenge (LMRRC), supported by Amazon and MIT, which sought innovative solutions for cargo routing optimization. While the challenge provided travel times and zone identifiers, the dependency on these factors raises concerns about the algorithms’ generalizability to different contexts and regions with standard delivery services registries. To address these concerns, this study proposes personalized cost matrices that incorporate both distance and time models, along with the relationships between delivery stops. Additionally, it presents an improved approach to sequencing stops by combining exact and approximate algorithms, utilizing a customized regression technique alongside fine-tuned metaheuristics and heuristics refinements. The resulting methodology achieves competitive scores on the LMRRC validation dataset, which comprises routes from the USA. By carefully delineating route characteristics, the study enables the selection of specific technique combinations for each route, considering its geometrical and geographical attributes. Furthermore, the proposed methodologies are successfully applied to real-case scenarios of last-mile deliveries in Montevideo (Uruguay), demonstrating similar average scores and accuracy on new testing routes. This research contributes to the advancement of last-mile delivery optimization by leveraging personalized cost matrices and algorithmic refinements. The findings highlight the potential for improving existing approaches and their adaptability to diverse geographic contexts, paving the way for more efficient and effective delivery services in the future.Este artículo explora el diseño y aplicación de técnicas de aprendizaje automático para mejorar los enfoques tradicionales y así resolver problemas de optimización NP-hard. En particular, se enfoca en el Last-Mile Routing Research Challenge (LMRRC), apoyado por Amazon y MIT, que buscaba soluciones innovadoras para la optimización de rutas de carga. Si bien el desafío proporcionó tiempos de viaje e identificadores de zona, la dependencia de estos factores plantea preocupaciones sobre la generalización de los algoritmos a diferentes contextos y regiones con registros de servicios de entrega estándar. Para abordar estas interrogantes, este estudio propone matrices de costos personalizadas que incorporan modelos de distancia y tiempo, junto con las relaciones entre las paradas de entrega. Además, presenta enfoques mejorados para la secuenciación de paradas mediante la combinación de algoritmos exactos y aproximados, utilizando técnicas de regresión personalizada junto con metaheurísticas y refinamientos heurísticos ajustados. La metodología resultante logra puntajes competitivos en el conjunto de datos de validación LMRRC, que usa rutas de EE. UU. Al delinear cuidadosamente las características de la ruta, el estudio permite la selección de combinaciones de técnicas específicas para cada ruta, considerando sus atributos geométricos y geográficos. Además, las metodologías propuestas se aplican con éxito a escenarios de casos reales de entregas en Montevideo (Uruguay), demostrando puntajes promedio y precisión similares en nuevas rutas de prueba. Esta investigación contribuye al avance de la optimización de la entrega de última milla al aprovechar matrices de costos personalizadas y refinamientos algorítmicos. Los hallazgos resaltan el potencial para mejorar los enfoques existentes y su adaptabilidad a diversos contextos geográficos, allanando el camino para servicios de entrega más eficientes y efectivos en el futuro

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

Marches quantiques et mécanique quantique relativiste

This thesis is devoted to the development of two well-known models of computation for their application in quantum computer simulations. These models are the quantum walk (QW) and quantum cellular automata (QCA) models, and they constitute doubly strategic topics in this respect. First, they are privileged mathematical settings in which to encode the description of the actual physical system to be simulated. Second, they offer an experimentally viable architecture for actual physical devices performing the simulation.For QWs, we prove precise error bounds and convergence rates of the discrete scheme towards the Dirac equation, thus validating the QW as a quantum simulation scheme. Furthermore, for both models we formulate a notion of discrete Lorentz covariance, which admits a diagrammatic representation in terms of local, circuit equivalence rules. We also study the continuum limit of a wide class of QWs, and show that it leads to a class of PDEs which includes the Hamiltonian form of the massive Dirac equation in (1+1)-dimensional curved spacetime.Finally, we study the two particle sector of a QCA. We find the conditions for the existence of discrete spectrum (interpretable as molecular binding) for short-range and for long-range interactions. This is achieved using perturbation techniques of trace class operators and spectral analysis of unitary operators.Cette thèse étudie deux modèles de calcul: les marches quantiques (QW) et les automates cellulaires quantiques (QCA), en vue de les appliquer en simulation quantique. Ces modèles ont deux avantages stratégiques pour aborder ce problème: d'une part, ils constituent un cadre mathématique privilégié pour coder la description du système physique à simuler; d'autre part, ils correspondent à des architectures expérimentalement réalisables.Nous effectuons d'abord une analyse des QWs en tant que schéma numérique pour l'équation de Dirac, en établissant leur borne d'erreur globale et leur taux de convergence. Puis nous proposons une notion de transformée de Lorentz discrète pour les deux modèles, QW et QCA, qui admet une représentation diagrammatique s'exprimant par des règles locales et d'équivalence de circuits. Par ailleurs, nous avons caractérisé la limite continue d'une grande classe de QWs, et démontré qu'elle correspond à une classe d'équations aux dérivées partielles incluant l'équation de Dirac massive en espace-temps courbe de (1+1)-dimensions.Finalement, nous étudions le secteur à deux particules des automates cellulaires quantiques. Nous avons trouvé les conditions d'existence du spectre discret (interprétable comme une liaison moléculaire) pour des interactions à courte et longue portée, à travers des techniques perturbatives et d'analyse spectrale des opérateurs unitaires

Marches quantiques et mécanique quantique relativiste

This thesis is devoted to the development of two well-known models of computation for their application in quantum computer simulations. These models are the quantum walk (QW) and quantum cellular automata (QCA) models, and they constitute doubly strategic topics in this respect. First, they are privileged mathematical settings in which to encode the description of the actual physical system to be simulated. Second, they offer an experimentally viable architecture for actual physical devices performing the simulation.For QWs, we prove precise error bounds and convergence rates of the discrete scheme towards the Dirac equation, thus validating the QW as a quantum simulation scheme. Furthermore, for both models we formulate a notion of discrete Lorentz covariance, which admits a diagrammatic representation in terms of local, circuit equivalence rules. We also study the continuum limit of a wide class of QWs, and show that it leads to a class of PDEs which includes the Hamiltonian form of the massive Dirac equation in (1+1)-dimensional curved spacetime.Finally, we study the two particle sector of a QCA. We find the conditions for the existence of discrete spectrum (interpretable as molecular binding) for short-range and for long-range interactions. This is achieved using perturbation techniques of trace class operators and spectral analysis of unitary operators.Cette thèse étudie deux modèles de calcul: les marches quantiques (QW) et les automates cellulaires quantiques (QCA), en vue de les appliquer en simulation quantique. Ces modèles ont deux avantages stratégiques pour aborder ce problème: d'une part, ils constituent un cadre mathématique privilégié pour coder la description du système physique à simuler; d'autre part, ils correspondent à des architectures expérimentalement réalisables.Nous effectuons d'abord une analyse des QWs en tant que schéma numérique pour l'équation de Dirac, en établissant leur borne d'erreur globale et leur taux de convergence. Puis nous proposons une notion de transformée de Lorentz discrète pour les deux modèles, QW et QCA, qui admet une représentation diagrammatique s'exprimant par des règles locales et d'équivalence de circuits. Par ailleurs, nous avons caractérisé la limite continue d'une grande classe de QWs, et démontré qu'elle correspond à une classe d'équations aux dérivées partielles incluant l'équation de Dirac massive en espace-temps courbe de (1+1)-dimensions.Finalement, nous étudions le secteur à deux particules des automates cellulaires quantiques. Nous avons trouvé les conditions d'existence du spectre discret (interprétable comme une liaison moléculaire) pour des interactions à courte et longue portée, à travers des techniques perturbatives et d'analyse spectrale des opérateurs unitaires

JuliaReach: a Toolbox for Set-Based Reachability

We present JuliaReach, a toolbox for set-based reachability analysis of dynamical systems. JuliaReach consists of two main packages: Reachability, containing implementations of reachability algorithms for continuous and hybrid systems, and LazySets, a standalone library that implements state-of-the-art algorithms for calculus with convex sets. The library offers both concrete and lazy set representations, where the latter stands for the ability to delay set computations until they are needed. The choice of the programming language Julia and the accompanying documentation of our toolbox allow researchers to easily translate set-based algorithms from mathematics to software in a platform-independent way, while achieving runtime performance that is comparable to statically compiled languages. Combining lazy operations in high dimensions and explicit computations in low dimensions, JuliaReach can be applied to solve complex, large-scale problems.Comment: Accepted in Proceedings of HSCC'19: 22nd ACM International Conference on Hybrid Systems: Computation and Control (HSCC'19