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JKO for Landau: a variational particle method for homogeneous Landau equation

Yan Huang, Li Wang

TL;DR

This work develops a novel implicit particle method in the framework of the JKO scheme that enjoys exact entropy dissipation and unconditional stability, therefore making it suitable for large-scale plasma simulations over extended time periods.

Abstract

Inspired by the gradient flow viewpoint of the Landau equation and corresponding dynamic formulation of the Landau metric in [arXiv:2007.08591], we develop a novel implicit particle method for the Landau equation in the framework of the JKO scheme. We first reformulate the Landau metric in a computationally friendly form, and then translate it into the Lagrangian viewpoint using the flow map. A key observation is that, while the flow map evolves according to a rather complicated integral equation, the unknown component is merely a score function of the corresponding density plus an additional term in the null space of the collision kernel. This insight guides us in designing and training the neural network for the flow map. Additionally, the objective function is in a double summation form, making it highly suitable for stochastic methods. Consequently, we design a tailored version of stochastic gradient descent that maintains particle interactions and significantly reduces the computational complexity. Compared to other deterministic particle methods, the proposed method enjoys exact entropy dissipation and unconditional stability, therefore making it suitable for large-scale plasma simulations over extended time periods.

JKO for Landau: a variational particle method for homogeneous Landau equation

TL;DR

This work develops a novel implicit particle method in the framework of the JKO scheme that enjoys exact entropy dissipation and unconditional stability, therefore making it suitable for large-scale plasma simulations over extended time periods.

Abstract

Inspired by the gradient flow viewpoint of the Landau equation and corresponding dynamic formulation of the Landau metric in [arXiv:2007.08591], we develop a novel implicit particle method for the Landau equation in the framework of the JKO scheme. We first reformulate the Landau metric in a computationally friendly form, and then translate it into the Lagrangian viewpoint using the flow map. A key observation is that, while the flow map evolves according to a rather complicated integral equation, the unknown component is merely a score function of the corresponding density plus an additional term in the null space of the collision kernel. This insight guides us in designing and training the neural network for the flow map. Additionally, the objective function is in a double summation form, making it highly suitable for stochastic methods. Consequently, we design a tailored version of stochastic gradient descent that maintains particle interactions and significantly reduces the computational complexity. Compared to other deterministic particle methods, the proposed method enjoys exact entropy dissipation and unconditional stability, therefore making it suitable for large-scale plasma simulations over extended time periods.
Paper Structure (20 sections, 9 theorems, 57 equations, 11 figures, 2 algorithms)

This paper contains 20 sections, 9 theorems, 57 equations, 11 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Variational formulation and the JKO scheme
  3. Gradient flow perspective
  4. Dynamical JKO scheme
  5. Lagrangian formulation
  6. A particle method
  7. Particle representation
  8. Neural network approximation
  9. Stochasticity accelerated JKO scheme
  10. Stochastic optimization
  11. Convergence analysis for optimization
  12. Random batch method
  13. Numerical examples
  14. BKW solution for Maxwellian molecules
  15. 2D bi-Maxwellian example with Coulomb interaction
  16. ...and 5 more sections

Key Result

Theorem 2.1

carrillo2024landau Fix $d=3$ and $\gamma \in (-3,0]$. Suppose that a curve $\mu: [0,T] \to \mathcal{P}(\mathbb{R}^3)$ has a density $f(t, {\boldsymbol{v}})$ that satisfies the following assumptions: Then $\mu$ is a curve of maximal slope for $\mathcal{H}$ with respect to its strong upper gradient $\sqrt{D_{\mathcal{H}}}$ if and only if its density $f$ is a weak solution of the homogeneous Landau

Figures (11)

  • Figure 1: Time evolution of macroscopical physical quantities for a 2D BKW solution with weak collision strength $C_\gamma=\frac{1}{16}$. Left: time evolution of the kinetic energy, where the exact kinetic energy is $2$. Center: time evolution of the entropy. Right: rate of decay of entropy with respect to time.
  • Figure 2: Comparison between implicit JKO-based, explicit JKO-based, and score-based particle methods with varying time step sizes for a 2D BKW solution with strong collision strength $C_\gamma=5$. Left: time evolution of the kinetic energy. Right: time evolution of the entropy.
  • Figure 3: Comparison between implicit JKO-based, explicit JKO-based, and score-based particle methods with varying time step sizes for a 3D BKW solution with strong collision strength $C_\gamma=3$. Left: time evolution of the kinetic energy. Right: time evolution of the entropy.
  • Figure 4: Slice plots of the reconstructed and blob solution for an example of 2D Coulomb interaction at $t=0$, $20$, and $40$. Left: $f(\cdot, y=-1)$. Right: $f(x=0, \cdot)$.
  • Figure 5: Time evolution of macroscopic physical quantities for an example of 2D Coulomb interaction. Left: time evolution of the kinetic energy, where the exact kinetic energy is $5$. Right: time evolution of the entropy.
  • ...and 6 more figures

Theorems & Definitions (21)

  • Definition 1
  • Theorem 2.1: Landau equation as a gradient flow
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 11 more