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Extensions of ValEncIA‐IVP for reduction of overestimation, for simulation of differential algebraic systems, and for dynamical optimization
Author(s) -
Rauh Andreas,
Auer Ekaterina,
Minisini Johanna,
Hofer Eberhard P.
Publication year - 2007
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200700022
Subject(s) - interval arithmetic , correctness , context (archaeology) , interval (graph theory) , ordinary differential equation , rounding , algebraic number , reduction (mathematics) , mathematics , dynamical systems theory , computation , computer science , state (computer science) , mathematical optimization , differential equation , algorithm , mathematical analysis , paleontology , physics , geometry , combinatorics , quantum mechanics , bounded function , biology , operating system
Abstract Simulation techniques are commonly used to analyze the influence of uncertainties of initial conditions and systemparameters on the trajectories of the state variables of dynamical systems. In this context, interval arithmetic approaches are of interest. They are capable of determining guaranteed bounds of all reachable states if worst‐case bounds of the above‐mentioned uncertainties are known. Furthermore, interval algorithms ensure the correctness of numerical results in spite of rounding errors which inevitably arise if floating point operations are carried out on a computer. However, naive implementations of interval algorithms often lead to overestimation, i.e., too conservative enclosures which can make the results meaningless. In this contribution, we summarize the basic routines of ValEncIA‐IVP which computes interval enclosures of all reachable states of dynamical systems described by ordinary differential equations ODEs. ValEncIA‐IVP , VAL idation of state ENC losures using I nterval A rithmetic for I nitial V alue P roblems, can be applied to the simulation of systems with both uncertain parameters and uncertain initial conditions. Advanced techniques for reduction of overestimation are demonstrated for a simplified catalytic reactor. Afirst approach to using VanEncIA‐IVP for the simulation of sets of differential algebraic equations is outlined. Finally, an outlook on the integration of ValEncIA‐IVP in an interval arithmetic framework for computation of optimal and robust control strategies for continuous‐time processes is given. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)