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Adaptive hyper-reduction of non-sparse operators: application to parametric particle-based kinetic plasma models

Cecilia Pagliantini, Federico Vismara

Abstract

This paper proposes an adaptive hyper-reduction method to reduce the computational cost associated with the simulation of parametric particle-based kinetic plasma models, specifically focusing on the Vlasov-Poisson equation. Conventional model order reduction and hyper-reduction techniques are often ineffective for such models due to the non-sparse nature of the nonlinear operators arising from the interactions between particles. To tackle this issue, we propose an adaptive, structure-preserving hyper-reduction method that leverages a decomposition of the discrete reduced Hamiltonian into a linear combination of terms, each depending on a few components of the state. The proposed approximation strategy allows to: (i) preserve the Hamiltonian structure of the problem; (ii) evaluate nonlinear non-sparse operators in a computationally efficient way; (iii) overcome the Kolmogorov barrier of transport-dominated problems via evolution of the approximation space and adaptivity of the rank of the solution. The proposed method is validated on numerical benchmark simulations, demonstrating stable and accurate performance with substantial runtime reductions compared to the full order model.

Adaptive hyper-reduction of non-sparse operators: application to parametric particle-based kinetic plasma models

Abstract

This paper proposes an adaptive hyper-reduction method to reduce the computational cost associated with the simulation of parametric particle-based kinetic plasma models, specifically focusing on the Vlasov-Poisson equation. Conventional model order reduction and hyper-reduction techniques are often ineffective for such models due to the non-sparse nature of the nonlinear operators arising from the interactions between particles. To tackle this issue, we propose an adaptive, structure-preserving hyper-reduction method that leverages a decomposition of the discrete reduced Hamiltonian into a linear combination of terms, each depending on a few components of the state. The proposed approximation strategy allows to: (i) preserve the Hamiltonian structure of the problem; (ii) evaluate nonlinear non-sparse operators in a computationally efficient way; (iii) overcome the Kolmogorov barrier of transport-dominated problems via evolution of the approximation space and adaptivity of the rank of the solution. The proposed method is validated on numerical benchmark simulations, demonstrating stable and accurate performance with substantial runtime reductions compared to the full order model.

Paper Structure

This paper contains 20 sections, 2 theorems, 87 equations, 22 figures, 6 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Hyper-reduction of non-sparse operators
  3. The Vlasov-Poisson equation and its numerical discretization
  4. Semi-discrete approximation
  5. Numerical time integration
  6. Symplectic dynamical low-rank approximation
  7. Partitioned Runge-Kutta scheme for the reduced dynamics
  8. Rank adaptivity
  9. Update of the coefficients of the reduced solution
  10. Hyper-reduction of nonlinear terms
  11. Error bound on the EIM approximation
  12. Construction of the EIM space
  13. Summary of the algorithm
  14. Numerical experiments
  15. Nonlinear Landau damping
  16. ...and 5 more sections

Key Result

Proposition 1

Let If $\gamma=1$ in eq:thetanew and then

Figures (22)

  • Figure 1: NLLD. Numerical distribution function $f_h(t,x,v;\eta)$ at times $t=0$ (left), $t=20$ (center) and $t=40$ (right) for $\eta=(\alpha,\sigma)=(0.4644, 0.9867)$. Comparison between the FOM (first row), the ROM with $n=3$ (second row), and the hROM with $n=3$ and $m$ varying (third row).
  • Figure 2: NLLD. Relative errors \ref{['eq:rel_err']} of the ROM and hROM with respect to the FOM solution: $n=2$ (left) and $n=3$ (right).
  • Figure 3: NLLD. Evolution of the dimension of the EIM approximation space.
  • Figure 4: NLLD. Evolution of the electric energy $h$ evaluated at the FOM, ROM, and hROM solutions with $n=3$ and for two choices of the parameter $\eta$.
  • Figure 5: NLLD. Evolution of the error in the conservation of the Hamiltonian \ref{['eq:err_ham']}. Comparison between the FOM, the ROM, and the hROM with $n=2$ and $n=3$.
  • ...and 17 more figures

Theorems & Definitions (5)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Remark