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Goal-oriented time adaptivity for port-Hamiltonian systems

Andreas Bartel, Manuel Schaller

TL;DR

This paper addresses the challenge of faithfully resolving energy balance in port-Hamiltonian systems during time discretization. It introduces goal-oriented, a posteriori error estimation with adjoint sensitivities, and defines a local energy-balance QoI $I_{\mathrm{loc}}$ (and a weighted variant $I_{\mathrm{loc},\rho}$) to drive adaptive time grids. A variational formulation for the state and adjoint equations is developed, with a non-conforming Galerkin time discretization and a parallelizable block-Jacobi–style adjoint approximation that exploits dissipativity. Numerical tests on toy and circuit models show that the proposed GO-adaptive scheme yields smaller energy-balance violations than uniform refinement and standard step-size control, illustrating its potential for energy-aware pH simulations and multirate extensions. Overall, the approach provides a flexible framework to couple energy considerations with adaptive discretization, enabling more accurate and efficient simulations of energy-dissipative, modular systems.

Abstract

Port-Hamiltonian systems provide an energy-based modeling paradigm for dynamical input-state-output systems. At their core, they fulfill an energy balance relating stored, dissipated and supplied energy. To accurately resolve this energy balance in time discretizations, we propose an adaptive grid refinement technique based on a posteriori error estimation. The evaluation of the error estimator includes the computation of adjoint sensitivities. To interpret this adjoint equation as a backwards-in-time equation, we show piecewise weak differentiability of the dual variable. Then, leveraging dissipativity of the port-Hamiltonian dynamics, we present a parallelizable approximation of the underlying adjoint system in the spirit of a block-Jacobi method to efficiently compute error indicators. We illustrate the performance of the proposed scheme by means of numerical experiments showing that it yields a smaller violation of the energy balance when compared to uniform refinements and traditional step-size controlled time stepping.

Goal-oriented time adaptivity for port-Hamiltonian systems

TL;DR

This paper addresses the challenge of faithfully resolving energy balance in port-Hamiltonian systems during time discretization. It introduces goal-oriented, a posteriori error estimation with adjoint sensitivities, and defines a local energy-balance QoI (and a weighted variant ) to drive adaptive time grids. A variational formulation for the state and adjoint equations is developed, with a non-conforming Galerkin time discretization and a parallelizable block-Jacobi–style adjoint approximation that exploits dissipativity. Numerical tests on toy and circuit models show that the proposed GO-adaptive scheme yields smaller energy-balance violations than uniform refinement and standard step-size control, illustrating its potential for energy-aware pH simulations and multirate extensions. Overall, the approach provides a flexible framework to couple energy considerations with adaptive discretization, enabling more accurate and efficient simulations of energy-dissipative, modular systems.

Abstract

Port-Hamiltonian systems provide an energy-based modeling paradigm for dynamical input-state-output systems. At their core, they fulfill an energy balance relating stored, dissipated and supplied energy. To accurately resolve this energy balance in time discretizations, we propose an adaptive grid refinement technique based on a posteriori error estimation. The evaluation of the error estimator includes the computation of adjoint sensitivities. To interpret this adjoint equation as a backwards-in-time equation, we show piecewise weak differentiability of the dual variable. Then, leveraging dissipativity of the port-Hamiltonian dynamics, we present a parallelizable approximation of the underlying adjoint system in the spirit of a block-Jacobi method to efficiently compute error indicators. We illustrate the performance of the proposed scheme by means of numerical experiments showing that it yields a smaller violation of the energy balance when compared to uniform refinements and traditional step-size controlled time stepping.
Paper Structure (15 sections, 5 theorems, 69 equations, 6 figures, 1 algorithm)

This paper contains 15 sections, 5 theorems, 69 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Goal-oriented error estimation for port-Hamiltonian systems
  3. Refinements targeting the energy balance
  4. Fundamentals of goal-oriented adapativity
  5. Variational formulation and time discretization
  6. State equation
  7. Adjoint equation as piecewise backwards-in-time equation.
  8. Discretization
  9. Implementation of goal-oriented adaptivity
  10. A dissipativity exploiting parallelizable strategy for the adjoints
  11. Numerical results
  12. Approximation of the sensitivities
  13. Performance of the goal-oriented scheme: Toy example
  14. Performance of the goal-oriented scheme: Electrical circuit
  15. Conclusion and outlook

Key Result

Proposition 1

If for all $\varphi\in H^1([0,T],\mathbb{R}^n)$, Then $\dot x = Ax+f$ a.e. on $[0,T]$ and $x(0)=x^0$.

Figures (6)

  • Figure 1: Norm of the sensitivities $\lambda$ and $\tilde{\lambda}$ solving \ref{['eq:system']} and \ref{['eq:system_approx']} over time. Left: $A=J-R_1$, middle: $A=J-R_2$, right: $A=J-10R_2$.
  • Figure 2: Trajectories (top) and violation of energy balance (bottom) with QoI $I_{\mathrm{loc}}$. Left column: sensitivities computed form \ref{['eq:system']}, right column: sensitivities computed from \ref{['eq:system_approx']}
  • Figure 3: Trajectories (top) and violation of energy balance (top) with QoI $I_{\mathrm{loc},\rho}$. Left column: sensitivities computed form \ref{['eq:system']}, right column: sensitivities computed from \ref{['eq:system_approx']}
  • Figure 4: Violation of energy balance and comparison to implicit midpoint rule with uniform refinement with no control input: Left: $R$ from \ref{['eq:sysmat']}, Right: Dissipation given by $0.6\cdot R$.
  • Figure 5: Depiction of the considered electrical circuit
  • ...and 1 more figures

Theorems & Definitions (10)

  • Proposition 1
  • proof
  • Lemma 2
  • proof
  • Corollary 3
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • Remark 6