Derivation of physically motivated constraints for efficient interval simulations applied to the analysis of uncertain dynamical systems

Mareile Freihold; Eberhard P. Hofer

International Journal of Applied Mathematics and Computer Science (2009)

  • Volume: 19, Issue: 3, page 485-499
  • ISSN: 1641-876X

Abstract

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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.

How to cite

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Mareile Freihold, and Eberhard P. Hofer. "Derivation of physically motivated constraints for efficient interval simulations applied to the analysis of uncertain dynamical systems." International Journal of Applied Mathematics and Computer Science 19.3 (2009): 485-499. <http://eudml.org/doc/207950>.

@article{MareileFreihold2009,
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.},
author = {Mareile Freihold, Eberhard P. Hofer},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {VALENCIA-IVP; consistency tests for the reduction of overestimation; identification of dynamical constraints; Hamiltonian systems; branch and prune algorithms; ValEncIA-IVP},
language = {eng},
number = {3},
pages = {485-499},
title = {Derivation of physically motivated constraints for efficient interval simulations applied to the analysis of uncertain dynamical systems},
url = {http://eudml.org/doc/207950},
volume = {19},
year = {2009},
}

TY - JOUR
AU - Mareile Freihold
AU - Eberhard P. Hofer
TI - Derivation of physically motivated constraints for efficient interval simulations applied to the analysis of uncertain dynamical systems
JO - International Journal of Applied Mathematics and Computer Science
PY - 2009
VL - 19
IS - 3
SP - 485
EP - 499
AB - 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.
LA - eng
KW - VALENCIA-IVP; consistency tests for the reduction of overestimation; identification of dynamical constraints; Hamiltonian systems; branch and prune algorithms; ValEncIA-IVP
UR - http://eudml.org/doc/207950
ER -

References

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