An implicit algorithm for validated enclosures of the solutions to variational equations for ODEs

We propose a new algorithm for computing validated bounds for the solutions to the first order variational equations associated to ODEs. These validated solutions are the kernel of numerics computer-assisted proofs in dynamical systems literature. The method uses a high-order Taylor method as a predictor step and an implicit method based on the Hermite-Obreshkov interpolation as a corrector step. The proposed algorithm is an improvement of the $C^1$-Lohner algorithm proposed by Zgliczy\'nski and it provides sharper bounds. As an application of the algorithm, we give a computer-assisted proof of the existence of an attractor set in the R\"ossler system, and we show that the attractor contains an invariant and uniformly hyperbolic subset on which the dynamics is chaotic, that is, conjugated to subshift of finite type with positive topological entropy.Comment: 33 pages, 11 figure

Optimization techniques for error bounds of ODEs

Fehlerschranken von Anfangswertproblemen mit unbestimmten Anfangsbedingungen werden herkömmlicherweise mit Hilfe von Intervallanalysis berechnet, allerdings mit mäßigem Erfolg. Die traditionelle Herangehensweise führt zu asymptotischen Fehlerabschätzungen, die nur gültig sind, wenn die maximale Schrittweite gegen Null geht. Jedoch benötigt eine effiziente Approximation größtmögliche Schrittweiten, ohne die Genauigkeit zu mindern. Neue Entwicklungen in der globalen Optimierung ermöglichen es, das Finden von Fehlerschranken als globales Optimierungsproblem aufzufassen. Das ist insbesondere wichtig im Fall, dass die Differentialgleichungen oder die Anfangsbedingungen bedeutende Unschärfen enthalten. Es wurde ein neuer Solver - DIVIS (Differential Inequality based Validated IVP Solver) - entwickelt, um die Fehlerschranken für Anfangswertprobleme mit Hilfe von Fehlerabschätzungen und Optimierungstechniken zu berechnen. Die Idee dabei ist, die Fehlerabschätzung von Anfangswertproblemen durch elliptische Approximation zu berechnen. Die validierten Zustandseinschliessungen werden mit Hilfe von Differentialungleichungen berechnet. Die Konvergenz dieser Methode hängt von der Wahl geeigneter Vorkonditionierer ab. Das beschriebene Schema wurde in MATLAB und AMPL implementiert. Die Ergebnisse wurden mit VALENCIA-IVP, VNODE-LP und VSPODE verglichen.Error bounds of initial value problems with uncertain initial conditions are traditionally computed by using interval analysis but with limited success. Traditional analysis only leads to asymptotic error estimates valid when the maximal step size tends to zero, while efficiency in the approximation requires that step sizes are as large as possible without compromising accuracy. Recent progress in global optimization makes it feasible to treat the error bounding problem as a global optimization problem. This is particularly important in the case where the differential equations or the initial conditions contain significant uncertainties. A new solver DIVIS (Differential Inequality based Validated IVP Solver) has been developed to compute the error bounds of initial value problems by using defect estimates and optimization techniques. The basic idea is to compute the defect estimates of initial value problems by using outer ellipsoidal approximation. The validated state enclosures are computed by applying differential inequalities. Convergence of the method depends upon a suitable choice of preconditioner. The scheme is implemented in MATLAB and AMPL and the resulting enclosures are compared with VALENCIA-IVP, VNODE-LP and VSPODE

Recursive Solution of Initial Value Problems with Temporal Discretization

We construct a continuous domain for temporal discretization of differential equations. By using this domain, and the domain of Lipschitz maps, we formulate a generalization of the Euler operator, which exhibits second-order convergence. We prove computability of the operator within the framework of effectively given domains. The operator only requires the vector field of the differential equation to be Lipschitz continuous, in contrast to the related operators in the literature which require the vector field to be at least continuously differentiable. Within the same framework, we also analyze temporal discretization and computability of another variant of the Euler operator formulated according to Runge-Kutta theory. We prove that, compared with this variant, the second-order operator that we formulate directly, not only imposes weaker assumptions on the vector field, but also exhibits superior convergence rate. We implement the first-order, second-order, and Runge-Kutta Euler operators using arbitrary-precision interval arithmetic, and report on some experiments. The experiments confirm our theoretical results. In particular, we observe the superior convergence rate of our second-order operator compared with the Runge-Kutta Euler and the common (first-order) Euler operators.Comment: 50 pages, 6 figure

Enclosing the behavior of a hybrid automaton up to and beyond a Zeno point

Even simple hybrid automata like the classic bouncing ball can exhibit Zeno behavior. The existence of this type of behavior has so far forced a large class of simulators to either ignore some events or risk looping indefinitely. This in turn forces modelers to either insert ad-hoc restrictions to circumvent Zeno behavior or to abandon hybrid automata. To address this problem, we take a fresh look at event detection and localization. A key insight that emerges from this investigation is that an enclosure for a given time interval can be valid independent of the occurrence of a given event. Such an event can then even occur an unbounded number of times. This insight makes it possible to handle some types of Zeno behavior. If the post-Zeno state is defined explicitly in the given model of the hybrid automaton, the computed enclosure covers the corresponding trajectory that starts from the Zeno point through a restarted evolution

Global optimisation for dynamic systems using novel overestimation reduction techniques

The optimisation of dynamic systems is of high relevance in chemical engineering as many practical systems can be described by ordinary differential equations (ODEs) or differential algebraic equations (DAEs). The current techniques for solving these problems rigorously to global optimality rely mainly on sequential approaches in which a branch and bound framework is used for solving the global optimisation part of the problem and a verified simulator (in which rounding errors are accounted for in the computations) is used for solving the dynamic constraints. The verified simulation part is the main bottleneck since tight bounds are difficult to obtain for high dimensional dynamic systems. Additionally, uncertainty in the form of, for example, intervals is introduced in the parameters of the dynamic constraints which are also the decision variables of the optimisation problem. Nevertheless, in the verified simulation the accumulation of trajectories that do not belong to the exact solution (overestimation) makes the state bounds overconservative and in the worst case they blow up and tend towards ±∞. In this thesis, methods for verified simulation in global optimisation for dynamic systems were investigated. A novel algorithm that uses an interval Taylor series (ITS) method with enhanced overestimation reduction capabilities was developed. These enhancements for the reduction of the overestimation rely on interval contractors (Krawczyk, Newton, ForwardBackward) and model reformulation based on pattern substitution and input scaling. The method with interval contractors was also extended to Taylor Models (TM) for comparison purposes. The two algorithms were tested on several case studies to demonstrate the effectiveness of the methods. The case studies have a different number of state variables and system parameters and they use uncertain amounts in some of the system parameters and initial conditions. Both of the methods were also used in a sequential approach to address the global optimisation for dynamic systems problem subject to uncertainty. The simulation results demonstrated that the ITS method with overestimation reduction techniques provided tighter state bounds with less computational expense than the traditional method. In the case of the forward-backward contractor additional constraints can be introduced that can potentially contribute significantly to the reduction of the overestimation. Similarly, the novel TM method with enhanced overestimation reduction capabilities provided tighter bounds than the TM method alone. On the other hand, the optimisation results showed that the global optimisation algorithm with the novel ITS method with overestimation reduction techniques converged faster to a rigorous solution due to the improved state bounds

Global Optimisation for Dynamic Systems using Interval Analysis

Engineers seek optimal solutions when designing dynamic systems but a crucial element is to ensure bounded performance over time. Finding a globally optimal bounded trajectory requires the solution of the ordinary differential equation (ODE) systems in a verified way. To date these methods are only able to address low dimensional problems and for larger systems are unable to prevent gross overestimation of the bounds. In this paper we show how interval contractors can be used to obtain tightly bounded optima. A verified solver constructs tight upper and lower bounds on the dynamic variables using contractors for initial value problems (IVP) for ODEs within a global optimisation method. The solver provides guaranteed bound on the objective function and on the first order sensitivity equations in a branch and bound framework. The method is compared with three previously published methods on three examples from process engineering

CAPD::DynSys: a flexible C++ toolbox for rigorous numerical analysis of dynamical systems

We present the CAPD::DynSys library for rigorous numerical analysis of dynamical systems. The basic interface is described together with several interesting case studies illustrating how it can be used for computer-assisted proofs in dynamics of ODEs.Comment: 25 pages, 4 figures, 11 full C++ example