Set propagation in dynamical systems with generalised polynomial algebra and its computational complexity

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Abstract

This paper presents an approach to propagate sets of initial conditions and model parameters through dynamical systems. It is assumed that the dynamics is dependent on a number of model parameters and that the state of the system evolves from some initial conditions. Both model parameters and initial conditions vary within a set Ω. The paper presents an approach to approximate the set Ω with a polynomial expansion and to propagate, under some regularity assumptions, the polynomial representation through the dynamical system. The approach is based on a generalised polynomial algebra that replaces algebraic operators between real numbers with operators between polynomials. The paper first introduces the concept of generalised polynomial algebra and its use to propagate sets through dynamical systems. Then it analyses, both theoretically and experimentally, its time complexity and compares it against the time complexity of a non-intrusive counterpart. Finally, the paper provides an empirical convergence analysis on two illustrative examples of linear and non-linear dynamical systems.

LanguageEnglish
Pages22-49
Number of pages28
JournalCommunications in Nonlinear Science and Numerical Simulation
Volume75
Early online date20 Mar 2019
DOIs
Publication statusPublished - 31 Aug 2019

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Polynomial Algebra
Generalized Polynomials
Set theory
Algebra
Computational complexity
Dynamical systems
Computational Complexity
Dynamical system
Polynomials
Propagation
Initial conditions
Time Complexity
Polynomial
Mathematical operators
Nonlinear Dynamical Systems
Empirical Analysis
Operator
Nonlinear dynamical systems
Convergence Analysis
Regularity

Keywords

  • uncertainty propagation
  • polynomial algebra
  • dynamical systems

Cite this

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abstract = "This paper presents an approach to propagate sets of initial conditions and model parameters through dynamical systems. It is assumed that the dynamics is dependent on a number of model parameters and that the state of the system evolves from some initial conditions. Both model parameters and initial conditions vary within a set Ω. The paper presents an approach to approximate the set Ω with a polynomial expansion and to propagate, under some regularity assumptions, the polynomial representation through the dynamical system. The approach is based on a generalised polynomial algebra that replaces algebraic operators between real numbers with operators between polynomials. The paper first introduces the concept of generalised polynomial algebra and its use to propagate sets through dynamical systems. Then it analyses, both theoretically and experimentally, its time complexity and compares it against the time complexity of a non-intrusive counterpart. Finally, the paper provides an empirical convergence analysis on two illustrative examples of linear and non-linear dynamical systems.",
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author = "Massimiliano Vasile and {Ortega Absil}, Carlos and Annalisa Riccardi",
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N2 - This paper presents an approach to propagate sets of initial conditions and model parameters through dynamical systems. It is assumed that the dynamics is dependent on a number of model parameters and that the state of the system evolves from some initial conditions. Both model parameters and initial conditions vary within a set Ω. The paper presents an approach to approximate the set Ω with a polynomial expansion and to propagate, under some regularity assumptions, the polynomial representation through the dynamical system. The approach is based on a generalised polynomial algebra that replaces algebraic operators between real numbers with operators between polynomials. The paper first introduces the concept of generalised polynomial algebra and its use to propagate sets through dynamical systems. Then it analyses, both theoretically and experimentally, its time complexity and compares it against the time complexity of a non-intrusive counterpart. Finally, the paper provides an empirical convergence analysis on two illustrative examples of linear and non-linear dynamical systems.

AB - This paper presents an approach to propagate sets of initial conditions and model parameters through dynamical systems. It is assumed that the dynamics is dependent on a number of model parameters and that the state of the system evolves from some initial conditions. Both model parameters and initial conditions vary within a set Ω. The paper presents an approach to approximate the set Ω with a polynomial expansion and to propagate, under some regularity assumptions, the polynomial representation through the dynamical system. The approach is based on a generalised polynomial algebra that replaces algebraic operators between real numbers with operators between polynomials. The paper first introduces the concept of generalised polynomial algebra and its use to propagate sets through dynamical systems. Then it analyses, both theoretically and experimentally, its time complexity and compares it against the time complexity of a non-intrusive counterpart. Finally, the paper provides an empirical convergence analysis on two illustrative examples of linear and non-linear dynamical systems.

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