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Explicit Error Bounds for Carleman Linearization [article]

Marcelo Forets, Amaury Pouly
2017 arXiv   pre-print
We revisit the method of Carleman linearization for systems of ordinary differential equations with polynomial right-hand sides. This transformation provides an approximate linearization in a higher-dimensional space through the exact embedding of polynomial nonlinearities into an infinite-dimensional linear system, which is then truncated to obtain a finite-dimensional representation with an additive error. To the best of our knowledge, no explicit calculation of the error bound has been
more » ... d. In this paper, we propose two strategies to obtain a time-dependent function that locally bounds the truncation error. In the first approach, we proceed by iterative backwards-integration of the truncated system. However, the resulting error bound requires an a priori estimate of the norm of the exact solution for the given time horizon. To overcome this difficulty, we construct a combinatorial approach and solve it using generating functions, obtaining a local error bound that can be computed effectively.
arXiv:1711.02552v1 fatcat:fckewconlrhjnoqtanvywletni

Le potentiel de la production forestière paysanne en Amazonie brésilienne

Philippe Sablayrolles, Hildemberg Cruz, Marcelo Santos Melo, Isabel Garcia Drigo, Plinio Sist
Dans l'État du Pará, 60 % des forêts destinées à l'aménagement sont des forêts paysannes.  ...  Il est proposé des pistes de politiques publiques qui doivent aider à la diffusion de l'aménagement paysan dans différentes contextes fonciers : unités de conservation et forêts de production, périmètres  ...  le bois de forêts plantées.  ... 
doi:10.19182/bft2013.315.a20539 fatcat:oesi2prkcjchjb76wgws2vwu7e

Quantum walking in curved spacetime [article]

Pablo Arrighi, Stefano Facchini, Marcelo Forets
2016 arXiv   pre-print
A discrete-time Quantum Walk (QW) is essentially a unitary operator driving the evolution of a single particle on the lattice. Some QWs admit a continuum limit, leading to familiar PDEs (e.g. the Dirac equation). In this paper, we study the continuum limit of a wide class of QWs, and show that it leads to an entire class of PDEs, encompassing the Hamiltonian form of the massive Dirac equation in (1+1) curved spacetime. Therefore a certain QW, which we make explicit, provides us with a unitary
more » ... screte toy model of a test particle in curved spacetime, in spite of the fixed background lattice. Mathematically we have introduced two novel ingredients for taking the continuum limit of a QW, but which apply to any quantum cellular automata: encoding and grouping.
arXiv:1505.07023v2 fatcat:blqodcrntrh4zl6quhmko2wdxe

Conservative Time Discretization: A Comparative Study [article]

Marcelo Forets, Christian Schilling
2022 arXiv   pre-print
We present the first review of methods to overapproximate the set of reachable states of linear time-invariant systems subject to uncertain initial states and input signals for short time horizons. These methods are fundamental to state-of-the-art reachability algorithms for long time horizons, which proceed in two steps: First they use such a method to discretize the system for a short time horizon, and then they efficiently obtain a solution of the new discrete system for the long time
more » ... . Traditionally, both qualitative and quantitative comparison between different reachability algorithms has only considered the combination of both steps. In this paper we study the first step in isolation. We perform a variety of numerical experiments for six fundamental discretization methods from the literature. As we show, these methods have different trade-offs regarding accuracy and computational cost and, depending on the characteristics of the system, some methods may be preferred over others. We also discuss preprocessing steps to improve the results and efficient implementation strategies.
arXiv:2111.01454v2 fatcat:ekwe5643ovg5lmh4ssdz6mbsqa

LazySets.jl: Scalable Symbolic-Numeric Set Computations [article]

Marcelo Forets, Christian Schilling
2021 arXiv   pre-print
LazySets.jl is a Julia library that provides ways to symbolically represent sets of points as geometric shapes, with a special focus on convex sets and polyhedral approximations. LazySets provides methods to apply common set operations, convert between different set representations, and efficiently compute with sets in high dimensions using specialized algorithms based on the set types. LazySets is the core library of JuliaReach, a cutting-edge software addressing the fundamental problem of
more » ... hability analysis: computing the set of states that are reachable by a dynamical system from all initial states and for all admissible inputs and parameters. While the library was originally designed for reachability and formal verification, its scope goes beyond such topics. LazySets is an easy-to-use, general-purpose and scalable library for computations that mix symbolics and numerics. In this article we showcase the basic functionality, highlighting some of the key design choices.
arXiv:2110.01711v1 fatcat:g6grvdiayrg3hefym5cfl3wuaa

Semidefinite Approximations of Invariant Measures for Polynomial Systems [article]

Victor Magron and Marcelo Forets and Didier Henrion
2018 arXiv   pre-print
We consider the problem of approximating numerically the moments and the supports of measures which are invariant with respect to the dynamics of continuous- and discrete-time polynomial systems, under semialgebraic set constraints. First, we address the problem of approximating the density and hence the support of an invariant measure which is absolutely continuous with respect to the Lebesgue measure. Then, we focus on the approximation of the support of an invariant measure which is singular
more » ... with respect to the Lebesgue measure. Each problem is handled through an appropriate reformulation into a linear optimization problem over measures, solved in practice with two hierarchies of finite-dimensional semidefinite moment-sum-of-square relaxations, also called Lasserre hierarchies. Under specific assumptions, the first Lasserre hierarchy allows to approximate the moments of an absolutely continuous invariant measure as close as desired and to extract a sequence of polynomials converging weakly to the density of this measure. The second Lasserre hierarchy allows to approximate as close as desired in the Hausdorff metric the support of a singular invariant measure with the level sets of the Christoffel polynomials associated to the moment matrices of this measure. We also present some application examples together with numerical results for several dynamical systems admitting either absolutely continuous or singular invariant measures.
arXiv:1807.00754v1 fatcat:uhr3keoe6ve4zbxwq4pjnqw6ou

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

Marcelo Forets, Daniel Freire, Christian Schilling
2020 arXiv   pre-print
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
more » ... troller 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.
arXiv:2006.12325v1 fatcat:kngtoye4ajc4ngd3i2mva3d2h4

Reachability of weakly nonlinear systems using Carleman linearization [article]

Marcelo Forets, Christian Schilling
2021 arXiv   pre-print
In this article we introduce a solution method for a special class of nonlinear initial-value problems using set-based propagation techniques. The novelty of the approach is that we employ a particular embedding (Carleman linearization) to leverage recent advances of high-dimensional reachability solvers for linear ordinary differential equations based on the support function. Using a global error bound for the Carleman linearization abstraction, we are able to describe the full set of
more » ... of the system for sets of initial conditions and in dense time.
arXiv:2108.10390v1 fatcat:liamuxx74zdd5l7s4biqhj2eh4

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

Sergiy Bogomolov, Marcelo Forets, Goran Frehse, Andreas Podelski, Christian Schilling, Frédéric Viry
2018 arXiv   pre-print
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
more » ... 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.
arXiv:1801.09526v1 fatcat:3vf22wugefcbpojbanzqacljui

Verification of Neural-Network Control Systems by Integrating Taylor Models and Zonotopes [article]

Christian Schilling, Marcelo Forets, Sebastian Guadalupe
2022 arXiv   pre-print
Set representation and set conversion is implemented in the library LazySets (Forets and Schilling 2021) . For the Taylor-model analysis we use the implementation by ; .  ... 
arXiv:2112.09197v2 fatcat:n5syezrt2zfw7pagquarvz7hj4

Occupation measure methods for modelling and analysis of biological hybrid automata [article]

Alexandre Rocca, Marcelo Forets, Victor Magron, Eric Fanchon, Thao Dang
2018 arXiv   pre-print
Mechanistic models in biology often involve numerous parameters about which we do not have direct experimental information. The traditional approach is to fit these parameters using extensive numerical simulations (e.g. by the Monte-Carlo method), and eventually revising the model if the predictions do not correspond to the actual measurements. In this work we propose a methodology for hybrid automaton model revision, when new type of functions are needed to capture time varying parameters. To
more » ... his end, we formulate a hybrid optimal control problem with intermediate points as successive infinite-dimensional linear programs (LP) on occupation measures. Then, these infinite-dimensional LPs are solved using a hierarchy of semidefinite relaxations. The whole procedure is exposed on a recent model for haemoglobin production in erythrocytes.
arXiv:1710.03158v2 fatcat:z6zrtma3xfdzneavje5m2ilyzy

Constructing verification models of nonlinear Simulink systems via syntactic hybridization

Nikolaos Kekatos, Marcelo Forets, Goran Frehse
2017 2017 IEEE 56th Annual Conference on Decision and Control (CDC)  
Constructing Verification Models of Nonlinear Simulink Systems via Syntactic Hybridization* Nikolaos Kekatos 1 , Marcelo Forets 1 and Goran Frehse 1 Abstract-In this paper, we present a methodology that  ...  Nikolaos Kekatos, Marcelo Forets and Goran Frehse are with VERIMAG Laboratory, University Grenoble Alpes, Bâtiment IMAG, 700 Avenue Centrale, 38400 Saint-Martin-d'Hères, France.  ... 
doi:10.1109/cdc.2017.8263907 dblp:conf/cdc/KekatosFF17 fatcat:3mjjeu6y5vdftifdm2zuw6c6mm

The "Naked Coral" Hypothesis Revisited – Evidence for and Against Scleractinian Monophyly

Marcelo V. Kitahara, Mei-Fang Lin, Sylvain Forêt, Gavin Huttley, David J. Miller, Chaolun Allen Chen, John Murray Roberts
2014 PLoS ONE  
The relationship between Scleractinia and Corallimorpharia, Orders within Anthozoa distinguished by the presence of an aragonite skeleton in the former, is controversial. Although classically considered distinct groups, some phylogenetic analyses have placed the Corallimorpharia within a larger Scleractinia/Corallimorpharia clade, leading to the suggestion that the Corallimorpharia are "naked corals" that arose via skeleton loss during the Cretaceous from a Scleractinian ancestor. Scleractinian
more » ... paraphyly is, however, contradicted by a number of recent phylogenetic studies based on mt nucleotide (nt) sequence data. Whereas the "naked coral" hypothesis was based on analysis of the sequences of proteins encoded by a relatively small number of mt genomes, here a much-expanded dataset was used to reinvestigate hexacorallian phylogeny. The initial observation was that, whereas analyses based on nt data support scleractinian monophyly, those based on amino acid (aa) data support the "naked coral" hypothesis, irrespective of the method and with very strong support. To better understand the bases of these contrasting results, the effects of systematic errors were examined. Compared to other hexacorallians, the mt genomes of "Robust" corals have a higher (A+T) content, codon usage is far more constrained, and the proteins that they encode have a markedly higher phenylalanine content, leading us to suggest that mt DNA repair may be impaired in this lineage. Thus the "naked coral" topology could be caused by high levels of saturation in these mitochondrial sequences, long-branch effects or model violations. The equivocal results of these extensive analyses highlight the fundamental problems of basing coral phylogeny on mitochondrial sequence data.
doi:10.1371/journal.pone.0094774 pmid:24740380 pmcid:PMC3989238 fatcat:rf666xn46zgq3li7jef6q3uk4e

Reachability analysis of linear hybrid systems via block decomposition [article]

Sergiy Bogomolov, Marcelo Forets, Goran Frehse, Kostiantyn Potomkin, Christian Schilling
2020 arXiv   pre-print
Forets is with Universidad de la República, CURE, Maldonado, Uruguay, (e-mail: G.  ... 
arXiv:1905.02458v3 fatcat:yvcr72wkabg5vbhmkjqvsfq2mm

Discrete Lorentz covariance for quantum walks and quantum cellular automata

Pablo Arrighi, Stefano Facchini, Marcelo Forets
2014 New Journal of Physics  
We formalize a notion of discrete Lorentz transforms for Quantum Walks (QW) and Quantum Cellular Automata (QCA), in (1 + 1)-dimensional discrete spacetime. The theory admits a diagrammatic representation in terms of a few local, circuit equivalence rules. Within this framework, we show the first-order-only covariance of the Dirac QW. We then introduce the Clock QW and the Clock QCA, and prove that they are exactly discrete Lorentz covariant. The theory also allows for non-homogeneous Lorentz transforms, between non-inertial frames.
doi:10.1088/1367-2630/16/9/093007 fatcat:rrcapy2byzhdxfalkdqgav6hri
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