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

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

C^{r}-Lohner algorithm

We present a Lohner type algorithm for the computation of rigorous bounds for the solutions of ordinary differential equations and its derivatives with respect to the initial conditions up to an arbitrary order

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

Lagrangian Reachtubes: The Next Generation

We introduce LRT-NG, a set of techniques and an associated toolset that computes a reachtube (an over-approximation of the set of reachable states over a given time horizon) of a nonlinear dynamical system. LRT-NG significantly advances the state-of-the-art Langrangian Reachability and its associated tool LRT. From a theoretical perspective, LRT-NG is superior to LRT in three ways. First, it uses for the first time an analytically computed metric for the propagated ball which is proven to minimize the ball's volume. We emphasize that the metric computation is the centerpiece of all bloating-based techniques. Secondly, it computes the next reachset as the intersection of two balls: one based on the Cartesian metric and the other on the new metric. While the two metrics were previously considered opposing approaches, their joint use considerably tightens the reachtubes. Thirdly, it avoids the "wrapping effect" associated with the validated integration of the center of the reachset, by optimally absorbing the interval approximation in the radius of the next ball. From a tool-development perspective, LRT-NG is superior to LRT in two ways. First, it is a standalone tool that no longer relies on CAPD. This required the implementation of the Lohner method and a Runge-Kutta time-propagation method. Secondly, it has an improved interface, allowing the input model and initial conditions to be provided as external input files. Our experiments on a comprehensive set of benchmarks, including two Neural ODEs, demonstrates LRT-NG's superior performance compared to LRT, CAPD, and Flow*.Comment: 12 pages, 14 figure

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

Cr-Lohner algorithm

We present a Lohner type algorithm for the computation of rigorous bounds for the solutions of ordinary differential equations and its derivatives with respect to the initial conditions up to an arbitrary order