VALENCIA-IVP: A Comparison with Other Initial Value Problem Solvers

Rauh, Andreas; Hofer, Eberhard P.; Auer, Ekaterina

doi:10.1109/scan.2006.47

Cited by 46 publications

(43 citation statements)

References 4 publications

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“…, N when x is only known to belong to some box [x]. Guaranteed numerical integration techniques could be employed, such as AWA [15], VNODE [20], COSY IV [6], VSPODE [13], or ValEncIA-IVP [25,26]. The main difficulty comes from the fact that obtaining accurate enclosures for the solutions when there are uncertain parameters, as here, and uncertain initial conditions is quite difficult.…”

Section: Guaranteed Numerical Integrationmentioning

confidence: 99%

Verified Global Optimization for Estimating the Parameters of Nonlinear Models

Kieffer

Markót

Schichl

et al. 2011

Modeling, Design, and Simulation of Systems With Uncertainties

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Nonlinear parameter estimation is usually achieved via the minimization of some possibly non-convex cost function. Interval analysis allows to derive algorithms for the guaranteed characterization of the set of all global minimizers of such a cost function when an explicit expression for the output of the model is 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. This paper addresses parameter estimation via the verified global optimization of quadratic cost functions. It introduces tools instrumental for the minimization of generic cost functions. When an explicit expression of the output of the parametric model is available, significant improvements may be obtained by a new box exclusion test and by careful manipulations of the quadratic cost function. When the model is described by ODEs, some of the techniques available in the previous case may still be employed, provided that sensitivity functions of the model output with respect to the parameters are available.

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Section: Guaranteed Numerical Integrationmentioning

confidence: 99%

Verified Global Optimization for Estimating the Parameters of Nonlinear Models

Kieffer

Markót

Schichl

et al. 2011

Modeling, Design, and Simulation of Systems With Uncertainties

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“…VALENCIA-IVP is a verified solver for initial value problems (IVPs) for ODEs (Rauh, 2008;Rauh et al, 2007). It has the ability to work with interval variables which result from the propagation of uncertain initial conditions and uncertain parameters.…”

Section: Valencia-ivpmentioning

confidence: 99%

“…Then, verified error bounds [R (t)] are determined using an iteration procedure which can be derived using Banach's fixed-point theorem. A detailed explanation of this two-stage approach together with its proof can be found in (Rauh, 2008;Rauh et al, 2007). For further information, see also (Rauh et al, 2009).…”

Section: Calculation Of Guaranteed State Enclosuresmentioning

confidence: 99%

“…VALENCIA-IVP is a verified solver for sets of ordinary differential equations (ODEs) that allows for bounded uncertainties in the initial states and system parameters (Rauh et al, 2007). The uncertainties are considered through the use of interval arithmetic, the advantage hereby being that calculations always result in guaranteed enclosures (Moore, 1964).…”

Section: Introductionmentioning

confidence: 99%

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Derivation of Physically Motivated Constraints for Efficient Interval Simulations Applied to the Analysis of Uncertain Dynamical Systems

Freihold

Hofer

2009

International Journal of Applied Mathematics and Computer Science

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Interval arithmetic techniques such as VALENCIA-IVP allow calculating guaranteed enclosures of all reachable states of continuous-time dynamical systems with bounded uncertainties of both initial conditions and system parameters. Considering the fact that, in naive implementations of interval algorithms, overestimation might lead to unnecessarily conservative results, suitable consistency tests are essential to obtain the tightest possible enclosures. In this contribution, a general framework for the use of constraints based on physically motivated conservation properties is presented. The use of these constraints in verified simulations of dynamical systems provides a computationally efficient procedure which restricts the state enclosures to regions that are physically meaningful. A branch and prune algorithm is modified to a consistency test, which is based on these constraints. Two application scenarios are studied in detail. First, the total energy is employed as a conservation property for the analysis of mechanical systems. It is shown that conservation properties, such as the energy, are applicable to any Hamiltonian system. The second scenario is based on constraints that are derived from decoupling properties, which are considered for a high-dimensional compartment model of granulopoiesis in human blood cell dynamics.Keywords: VALENCIA-IVP, consistency tests for the reduction of overestimation, identification of dynamical constraints, Hamiltonian systems, branch and prune algorithms.

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“…The excess sludge Q EX is removed from the process. According to [6,8], this process is described by the nonlinear ODEṡ…”

Section: Sensitivity Analysis Of a Wastewater Treatment Processmentioning

confidence: 99%

Towards the Development of an Interval Arithmetic Environment for Validated Computer-Aided Design and Verification of Systems in Control Engineering

Rauh

Minisini

Hofer

2009

Numerical Validation in Current Hardware Architectures

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Abstract. In this paper, an overview of the potential use of validated techniques for the analysis and design of controllers for linear and nonlinear dynamical systems with uncertainties is given. In addition to robust pole assignment for linear dynamical systems with parameter uncertainties, mathematical system models and computational techniques are considered in which constraints for both state and control variables are taken into account. For that purpose, the use of interval arithmetic routines for calculation of guaranteed enclosures of the solutions of sets of ordinary differential equations and for the calculation of validated sensitivity measures of state variables with respect to parameter variations are discussed. Simulation results as well as further steps towards the development of a general-purpose interval arithmetic framework for the design and verification of systems in control engineering are summarized.

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VALENCIA-IVP: A Comparison with Other Initial Value Problem Solvers

Cited by 46 publications

References 4 publications

Verified Global Optimization for Estimating the Parameters of Nonlinear Models

Verified Global Optimization for Estimating the Parameters of Nonlinear Models

Derivation of Physically Motivated Constraints for Efficient Interval Simulations Applied to the Analysis of Uncertain Dynamical Systems

Towards the Development of an Interval Arithmetic Environment for Validated Computer-Aided Design and Verification of Systems in Control Engineering

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