VALENCIA-IVP: A Comparison with Other Initial Value Problem Solvers. Validated integration of ordinary differential equations with uncertain initial conditions and uncertain parameters is important for many practical applications. If guaranteed bounds for the uncertainties are known, interval methods can be applied to obtain validated enclosures of all states. However, validated computations are often affected by overestimation, which, in naive implementations, might even lead to meaningless results. Parallelepiped and QR preconditioning of the state equations, Taylor model arithmetic, as well as simulation techniques employing splitting and merging routines are a few existing approaches for reduction of overestimation. In this paper, the recently developed validated solver ValEncIA-IVP and several methods implemented there for reduction of overestimation are described. Furthermore, a detailed comparison of this solver with COSY VI and VNODE, two of the most well- known validated ODE solvers, is presented. Simulation results for simplified system models in mechanical and bio- process engineering show specific properties, advantages, and limitations of each tool.

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  1. Bünger, Florian: A Taylor model toolbox for solving ODEs implemented in Matlab/INTLAB (2020)
  2. Rauh, Andreas; Kersten, Julia; Aschemann, Harald: Interval and linear matrix inequality techniques for reliable control of linear continuous-time cooperative systems with applications to heat transfer (2020)
  3. Jauberthie, Carine; Travé-Massuyès, Louise; Verdière, Nathalie: Set-membership identifiability of nonlinear models and related parameter estimation properties (2016)
  4. Konečný, Michal; Taha, Walid; Bartha, Ferenc A.; Duracz, Jan; Duracz, Adam; Ames, Aaron D.: Enclosing the behavior of a hybrid automaton up to and beyond a Zeno point (2016)
  5. Pérez-Galván, Carlos; Bogle, I. David L.: Dynamic global optimization methods for determining guaranteed solutions in chemical engineering (2016)
  6. Dzetkulič, Tomáš: Rigorous integration of non-linear ordinary differential equations in Chebyshev basis (2015)
  7. Villanueva, Mario E.; Houska, Boris; Chachuat, Benoît: Unified framework for the propagation of continuous-time enclosures for parametric nonlinear ODEs (2015)
  8. Auer, Ekaterina; Kiel, Stefan; Rauh, Andreas: A verified method for solving piecewise smooth initial value problems (2013)
  9. Fazal, Qaisra; Neumaier, Arnold: Error bounds for initial value problems by optimization (2013)
  10. Auer, Ekaterina; Rauh, Andreas: VERICOMP: A system to compare and assess verified IVP solvers (2012)
  11. Rauh, Andreas; Aschemann, Harald: Parameter identification and observer-based control for distributed heating systems - the basis for temperature control of solid oxide fuel cell stacks (2012)
  12. Aschemann, H.; Minisini, J.; Rauh, A.: Interval arithmetic techniques for the design of controllers for nonlinear dynamical systems with applications in mechatronics (2010)
  13. Krasnochtanova, Irina; Rauh, Andreas; Kletting, Marco; Aschemann, Harald; Hofer, Eberhard P.; Schoop, Karl-Michael: Interval methods as a simulation tool for the dynamics of biological wastewater treatment processes with parameter uncertainties (2010)
  14. Rump, Siegfried M.: Verification methods: rigorous results using floating-point arithmetic (2010)
  15. Auer, Ekaterina; Luther, Wolfram: Uses of new sensitivity and DAE solving methods in SmartMobile for verified analysis of mechanical systems (2009)
  16. Freihold, Mareile; Hofer, Eberhard P.: Derivation of physically motivated constraints for efficient interval simulations applied to the analysis of uncertain dynamical systems (2009)
  17. Rauh, Andreas; Brill, Michael; Günther, Clemens: A novel interval arithmetic approach for solving differential-algebraic equations with \textscValEncIA-IVP (2009)
  18. Rauh, Andreas; Minisini, Johanna; Hofer, Eberhard: Verification techniques for sensitivity analysis and design of controllers for nonlinear dynamic systems with uncertainties (2009)
  19. Auer, Ekaterina; Rauh, Andreas; Hofer, Eberhard P.; Luther, Wolfram: Validated modeling of mechanical systems with SmartMOBILE: Improvement of performance by ValEncIA-IVP (2008)