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Fully adaptive structure-preserving hyper-reduction of parametric Hamiltonian systems

Cecilia Pagliantini, Federico Vismara

TL;DR

The resulting adaptive hyper-reduced models preserve the geometric structure of the Hamiltonian flow, do not rely on prior information on the dynamics, and can be solved at a cost that islinear in the dimension of the full order model and linear in the number of test parameters.

Abstract

Model order reduction provides low-complexity high-fidelity surrogate models that allow rapid and accurate solutions of parametric differential equations. The development of reduced order models for parametric \emph{nonlinear} Hamiltonian systems is challenged by several factors: (i) the geometric structure encoding the physical properties of the dynamics; (ii) the slowly decaying Kolmogorov $n$-width of conservative dynamics; (iii) the gradient structure of the nonlinear flow velocity; (iv) high variations in the numerical rank of the state as a function of time and parameters. We propose to address these aspects via a structure-preserving adaptive approach that combines symplectic dynamical low-rank approximation with adaptive gradient-preserving hyper-reduction and parameters sampling. Additionally, we propose to vary in time the dimensions of both the reduced basis space and the hyper-reduction space by monitoring the quality of the reduced solution via an error indicator related to the projection error of the Hamiltonian vector field. The resulting adaptive hyper-reduced models preserve the geometric structure of the Hamiltonian flow, do not rely on prior information on the dynamics, and can be solved at a cost that is linear in the dimension of the full order model and linear in the number of test parameters. Numerical experiments demonstrate the improved performances of the fully adaptive models compared to the original and reduced models.

Fully adaptive structure-preserving hyper-reduction of parametric Hamiltonian systems

TL;DR

The resulting adaptive hyper-reduced models preserve the geometric structure of the Hamiltonian flow, do not rely on prior information on the dynamics, and can be solved at a cost that islinear in the dimension of the full order model and linear in the number of test parameters.

Abstract

Model order reduction provides low-complexity high-fidelity surrogate models that allow rapid and accurate solutions of parametric differential equations. The development of reduced order models for parametric \emph{nonlinear} Hamiltonian systems is challenged by several factors: (i) the geometric structure encoding the physical properties of the dynamics; (ii) the slowly decaying Kolmogorov -width of conservative dynamics; (iii) the gradient structure of the nonlinear flow velocity; (iv) high variations in the numerical rank of the state as a function of time and parameters. We propose to address these aspects via a structure-preserving adaptive approach that combines symplectic dynamical low-rank approximation with adaptive gradient-preserving hyper-reduction and parameters sampling. Additionally, we propose to vary in time the dimensions of both the reduced basis space and the hyper-reduction space by monitoring the quality of the reduced solution via an error indicator related to the projection error of the Hamiltonian vector field. The resulting adaptive hyper-reduced models preserve the geometric structure of the Hamiltonian flow, do not rely on prior information on the dynamics, and can be solved at a cost that is linear in the dimension of the full order model and linear in the number of test parameters. Numerical experiments demonstrate the improved performances of the fully adaptive models compared to the original and reduced models.
Paper Structure (20 sections, 4 theorems, 51 equations, 13 figures, 1 table, 4 algorithms)

This paper contains 20 sections, 4 theorems, 51 equations, 13 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Symplectic dynamical low-rank approximation
  3. Reduced dynamics via Dirac--Frenkel variational principle
  4. Adaptive gradient-preserving empirical interpolation
  5. Adaptivity of the EIM basis and interpolation points
  6. Selection of sample indices and sample parameters for hyper-reduction
  7. Adaptivity of the approximation rank and of the EIM space dimension
  8. Rank adaptivity
  9. Adaptation of the dimension of the EIM space
  10. Parameters sampling for adaptivity
  11. Temporal discretization of the hyper-reduced model
  12. Summary of the algorithm
  13. Numerical experiments
  14. Test 1: Localized solution
  15. Non-rank-adaptive algorithm
  16. ...and 5 more sections

Key Result

Lemma 1

Let $\mathcal{O}(\alpha,\beta):=\lVert M_1+\alpha\beta^\top M_2\rVert_F^2$, with given $M_1\in\mathbb{R}^{q_1\times g}$ and $M_2\in\mathbb{R}^{q_2\times g}$. Then, for $1\leq r\leq q_2$, it holds where

Figures (13)

  • Figure 1: Test 1, localized solution. Left: modulus of the full order solution at four different time instants for $\eta=(0.8,2,-1.5)$. Right: $\epsilon$-rank of the full solution $\mathcal{W}(t)\in\mathbb{R}^{20000\times 125}$.
  • Figure 2: Test 1, localized solution. Dimension of the dynamical reduced basis space $2{n_\tau}=2{n_0}=24$. Left: evolution of the relative errors of the reduced and hyper-reduced systems for $m_\tau=m_0=80$ and with different choices of the sample parameters. Right: evolution of $p_U^*$ (solid lines) and of $p_A^*$ (dashed lines).
  • Figure 3: Test 1, localized solution. Non-rank-adaptive case, $2{n_\tau}=2{n_0}=24$, $m_\tau=m_0=80$. Left: computational times and relative errors at the final time. Right: evolution of the number ${m_s}$ of EIM sample indices selected according to \ref{['alg:DEIM_update']} with $\tau_{{m_s}}=10^{-4}$.
  • Figure 4: Test 1, localized solution. First row: $2{n_0}=24$, $\tau_m=\tau_{{m_s}}=10^{-4}$. Second row: $2{n_0}=40$, $\tau_m=\tau_{{m_s}}=10^{-6}$. Left: evolution of the relative errors of the reduced and hyper-reduced systems. Center: evolution of the dimension of the reduced space. Right: evolution of the dimension of the EIM space.
  • Figure 5: Test 1, localized solution. Relative errors at the final time vs. computational time.
  • ...and 8 more figures

Theorems & Definitions (8)

  • Lemma 1
  • Proof 1
  • Proposition 1
  • Proof 2
  • Proposition 2
  • Proof 3
  • Proposition 3
  • Proof 4