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Formally Verified Computation of Enclosures of Solutions of Ordinary Differential Equations [chapter]

Fabian Immler
2014 Lecture Notes in Computer Science  
Ordinary differential equations (ODEs) are ubiquitous when modeling continuous dynamics.  ...  In this paper, we demonstrate that enclosures of the solution can be verified with a high level of rigor: We implement a functional algorithm that computes enclosures of solutions of ODEs in the interactive  ...  Thanks are due to Johannes Hölzl and the anonymous reviewers for valuable comments on drafts of this paper.  ... 
doi:10.1007/978-3-319-06200-6_9 fatcat:cewze6crlzgcfbzwkdo72skuty

Verified Reachability Analysis of Continuous Systems [chapter]

Fabian Immler
2015 Lecture Notes in Computer Science  
Ordinary differential equations (ODEs) are often used to model the dynamics of (often safety-critical) continuous systems.  ...  It is proved to be sound with respect to the existing formalization of ODEs in Isabelle/HOL.  ...  [6] approximate the solution of one particular partial differential equation with a C-program and verify its correctness in Coq.  ... 
doi:10.1007/978-3-662-46681-0_3 fatcat:c4n2otyxrndu7pz7nwjbldggvq

A Novel Interval Arithmetic Approach for Solving Differential-Algebraic Equations with ValEncIA-IVP

Andreas Rauh, Michael Brill, Clemens Günther
2009 International Journal of Applied Mathematics and Computer Science  
The algorithm is based on ValEncIA-IVP, which has been developed recently for the computation of verified enclosures of the solution sets of initial value problems for ordinary differential equations.  ...  of differential-algebraic equations (DAEs) are presented.  ...  Furthermore, we would like to thank Ekaterina Auer, Faculty of Engineering, INKO, University of Duisburg-Essen, Germany, for her contributions to the development and imple-  ... 
doi:10.2478/v10006-009-0032-4 fatcat:g35xwekisjg4dnn6ges45ht2ym

Uses of Verified Methods for Solving Non-Smooth Initial Value Problems

Ekaterina Auer, Stefan Kiel
2013 Reliable Computing  
After that, we provide an overview of existing methods for verified enclosure of exact solutions to non-smooth IVPs.  ...  However, application of existing interval methods to reallife scenarios is challenging, since they might provide overly conservative enclosures of exact solutions.  ...  The task becomes especially complicated if non-smooth initial value problems (IVPs) for e.g. ordinary differential equations (ODEs) are considered.  ... 
dblp:journals/rc/AuerK13 fatcat:x25l4wfxsnb65l2svjcohnzmr4

A verified method for solving piecewise smooth initial value problems

Ekaterina Auer, Stefan Kiel, Andreas Rauh
2013 International Journal of Applied Mathematics and Computer Science  
Moreover, solution processes from usual numerics are sensitive to roundoff errors so that verified analysis might be more useful if a guarantee of correctness is required or if the system model is influenced  ...  Moreover, we summarize already existing verified methods for solving initial value problems with non-smooth (in fact, even not absolutely continuous) right-hand sides and propose a way of handling a certain  ...  of sliding solutions + − − − − − − − − use of verified IVPS + + + + + o + + + analytic/verified constraint prop. − − − + + − + + − verified equation solvers + − + − + + + − − formal model checkers o −  ... 
doi:10.2478/amcs-2013-0055 fatcat:kuo475r2ovgqlm4qkpxvd4crva

Formal Verification of Robotic Behaviors in Presence of Bounded Uncertainties

Julien Alexandre dit Sandretto, Alexandre Chapoutot, Olivier Mullier
2017 2017 First IEEE International Conference on Robotic Computing (IRC)  
Robotic behaviors are mainly described by differential equations. Those mathematical models are usually not precise enough because of inaccurately known parameters or model simplifications.  ...  It follows an interval constraint programming approach, combined with validated numerical integration methods to deal with differential equations.  ...  Acknowledgment The authors are grateful to François Pessaux for his pertinent comments on the early versions of this work.  ... 
doi:10.1109/irc.2017.17 dblp:conf/irc/SandrettoCM17 fatcat:hnpauptog5hejoa6ytf2kj2lpm

Including Ordinary Differential Equations Based Constraints in the Standard CP Framework [chapter]

Alexandre Goldsztejn, Olivier Mullier, Damien Eveillard, Hiroshi Hosobe
2010 Lecture Notes in Computer Science  
Coupling constraints and ordinary differential equations has numerous applications.  ...  differential equations based constraints.  ...  Acknowledgments This work was supported in part by the International Internship Program of the National Institute of Informatics, Japan.  ... 
doi:10.1007/978-3-642-15396-9_20 fatcat:2wp7iugdsrdita6wi6cnvacos4

Towards a computer-assisted proof for chaos in a forced damped pendulum equation

Tibor Csendes, Balázs Bánhelyi, László Hatvani
2007 Journal of Computational and Applied Mathematics  
In the present paper we provide computational details on a fitting model and on a verified method of solution.  ...  We report on the first steps made towards the computational proof of the chaotic behaviour of the forced damped pendulum.  ...  Our method applies a verified algorithm to solve initial value problems for systems of ordinary differential equations.  ... 
doi:10.1016/ fatcat:davxlnt5nzabdfah4z5ukdwoam

A Rigorous Generic Branch and Bound Solver for Nonlinear Problems

Andrew P. Smith, Cesar A. Munoz, Anthony J. Narkawicz, Mantas Markevicius
2015 2015 17th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC)  
, has been formally verified.  ...  differential equations.  ...  The second instantiation is for the bifurcation analysis of a system of parameterized ordinary differential equations.  ... 
doi:10.1109/synasc.2015.20 dblp:conf/synasc/SmithMNM15 fatcat:yxj3yt7tfnbktlqagedvgww4uq

Interval King method to compute enclosure solutions of nonlinear equations

Taher Lotfi, Parisa Bakhtiari
2013 Journal of Fuzzy Set Valued Analysis  
In this paper, a different variant of the classic King method for computing the enclosure solutions of a given nonlinear equation is introduced.  ...  Some implemented examples with INTLAB are also included to illustrate the validity and applicability of the scheme.  ...  Katayoun Mahdiani for their suggestions and comments during preparation of the manuscript.  ... 
doi:10.5899/2013/jfsva-00124 fatcat:bw63lj6h3fhrdhjtzkw2ocqgoi

A Verified ODE Solver and the Lorenz Attractor

Fabian Immler
2018 Journal of automated reasoning  
The verification is based on a formalization of a diverse variety of mathematics and algorithms. Formalized mathematics include ordinary differential equations and Poincaré maps.  ...  A rigorous numerical algorithm, formally verified with Isabelle/HOL, is used to certify the computations that Tucker used to prove chaos for the Lorenz attractor.  ...  I thankfully acknowledge Christoph Traut's work for his student project on formalizing the variational equation.  ... 
doi:10.1007/s10817-017-9448-y pmid:30069071 pmcid:PMC6044317 fatcat:bfnqyafba5ah3njzjprip2tgz4

Special issue on "uncertainty modeling and analysis with intervals: foundations, tools, applications"

Vladik Kreinovich, Wolfram Luther, Evgenija D. Popova
2013 Soft Computing - A Fusion of Foundations, Methodologies and Applications  
Acknowledgments We thank Professor Vincenzo Loia, Co-Editorin-Chief of Soft Computing Journal, for providing us with the opportunity to edit and publish this special issue.  ...  Neumaier propose an interesting approach for optimization-based computation of verified state enclosures for initial value problems of ordinary differential equations with uncertain initial states in their  ...  The basic idea is to compute the defect estimates of initial value problems by using outer ellipsoidal approximation and then by applying differential inequalities validated state enclosures for IVPs.  ... 
doi:10.1007/s00500-013-1004-z fatcat:qlnuun2eovhfxmenf2rvptah6a

Verified Global Optimization for Estimating the Parameters of Nonlinear Models [chapter]

Michel Kieffer, Mihály Csaba Markót, Hermann Schichl, Eric Walter
2011 Modeling, Design, and Simulation of Systems with Uncertainties  
available or when this output is obtained via the numerical solution of a set of ordinary differential equations.  ...  However, cost functions involved in parameter estimation are usually challenging for interval techniques, if only because of multi-occurrences of the parameters in the formal expression of the cost.  ...  the model output is obtained via the numerical solution of a set of ordinary differential equations.  ... 
doi:10.1007/978-3-642-15956-5_7 fatcat:v3rppdsnjbejzkdcr4pwu6ipyy

Direct Formal Verification of Liveness Properties in Continuous and Hybrid Dynamical Systems [chapter]

Andrew Sogokon, Paul B. Jackson
2015 Lecture Notes in Computer Science  
We present a general method for proving eventuality properties that works with the differential equations directly, without the need to compute their solutions.  ...  This paper is concerned with proof methods for the temporal property of eventuality (a type of liveness) in systems of polynomial ordinary differential equations (ODEs) evolving under constraints.  ...  André Platzer at the same institution for kindly responding to our query concerning the method of differential variants and extend special thanks to the anonymous reviewers for their diligent reading and  ... 
doi:10.1007/978-3-319-19249-9_32 fatcat:tk3lgdgqgngz5l2r6lyrw5xonq

Guaranteed Parameter Set Estimation for Monotone Dynamical Systems Using Hybrid Automata

Nacim Meslem, Nacim Ramdani, Yves Candau
2010 Reliable Computing  
This paper investigates guaranteed methods for estimating feasible parameter sets when the system under study is modeled with ordinary differential equations (ODE).  ...  In order to address high dimensional nonlinear continuous-time systems, a guaranteed but tight enclosure of the solution of the ODE is needed.  ...  The coefficients f [i] are the Taylor coefficients of the solution x(t), and can be computed either numerically by automatic differentiation or analytically via formal methods.  ... 
dblp:journals/rc/MeslemRC10 fatcat:q7cwjl7e75ekbdnyrfrnilrbrq
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