Derivation of Physically Motivated Constraints for Efficient Interval Simulations Applied to the Analysis of Uncertain Dynamical Systems
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 algori...
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| Vydáno v: | International Journal of Applied Mathematics and Computer Science Ročník 19; číslo 3; s. 485 - 499 |
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| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
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Zielona Góra
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01.09.2009
De Gruyter Brill Sp. z o.o., Paradigm Publishing Services |
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| ISSN: | 1641-876X, 2083-8492 |
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| Abstract | 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. |
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| AbstractList | Derivation of Physically Motivated Constraints for Efficient Interval Simulations Applied to the Analysis of Uncertain Dynamical Systems 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. 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. |
| Author | Hofer, Eberhard Freihold, Mareile |
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| Cites_doi | 10.2478/v10006-009-0032-4 10.1007/978-3-642-77020-3_4 10.1137/040604388 |
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| References | B. Maschke (9) 2000 R. Moore (10) 1964 F. Pfeiffer (12) 1987 13 R. Kearfott (7) 1992; 3 17 A. Rauh (14) 2007 18 19 E. Hofer (5) 1991a N. Nedialkov (11) 2007 1 2 3 4 E. Hofer (6) 1991b; 275 A. Rauh (15) 2009; 3 8 A. Singer (16) 2006; 6 |
| References_xml | – ident: 2 – volume: 3 start-page: 381 year: 2009 ident: 15 article-title: A novel interval arithmetic approach for solving differential-algebraic equations with ValEncIA-IVP publication-title: International Journal of Applied Mathematics and Computer Science doi: 10.2478/v10006-009-0032-4 – ident: 17 – ident: 3 – ident: 18 – volume: 275 start-page: 33 year: 1991b ident: 6 publication-title: Analyse dynamischer Systeme in Medizin, Biologie, Ökologie doi: 10.1007/978-3-642-77020-3_4 – ident: 4 – year: 2007 ident: 14 article-title: ValEncIA-IVP: A comparison with other initial value problem solvers – ident: 1 – year: 2007 ident: 11 article-title: Interval tools for ODEs and DAEs – volume: 6 start-page: 2167 year: 2006 ident: 16 article-title: Bounding the solutions of parameter dependent nonlinear ordinary differential equations publication-title: SIAM Journal on Scientific Computing doi: 10.1137/040604388 – volume: 3 start-page: 259 year: 1992 ident: 7 article-title: An interval branch and bound algorithm for bound constrained optimization problems publication-title: Journal of Global Optimization – ident: 19 – start-page: 28 year: 2000 ident: 9 article-title: Portcontrolled Hamiltonian representation of distributed parameter systems – ident: 13 – year: 1987 ident: 12 publication-title: Roboterdynamik – start-page: 58 year: 1991a ident: 5 article-title: Extraction of rules for model based estimation of granulocytopoiesis – ident: 8 – year: 1964 ident: 10 publication-title: Error in Digital Computation, the Automatic Analysis and Control of Error |
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| Snippet | Interval arithmetic techniques such as ValEncIA-IVP allow calculating guaranteed enclosures of all reachable states of continuous-time dynamical systems with... Derivation of Physically Motivated Constraints for Efficient Interval Simulations Applied to the Analysis of Uncertain Dynamical Systems Interval arithmetic... |
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| SubjectTerms | branch and prune algorithms consistency tests for the reduction of overestimation Hamiltonian systems identification of dynamical constraints ValEncIA-IVP |
| Title | Derivation of Physically Motivated Constraints for Efficient Interval Simulations Applied to the Analysis of Uncertain Dynamical Systems |
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