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Parareal algorithms for stochastic Maxwell equations with the damping term driven by additive noise

Liying Zhang, Qi Zhang

TL;DR

The paper analyzes strong (mean-square) convergence of two-level parareal algorithms for stochastic Maxwell equations with an additive-noise damping term. By using the stochastic exponential scheme as the coarse propagator and either the exact solver or a stochastic exponential fine propagator, it proves that the iteration-induced error decays with order $O(\triangle T^{k})$ and grows linearly in $k$, under suitable regularity assumptions. Numerical experiments in 1D and 2D confirm linear-in-$k$ convergence and show that increased damping accelerates convergence while larger noise scales perturb the solution. This work extends parareal analysis to stochastic Maxwell systems and highlights practical effects of damping and noise on temporal parallelization efficiency and accuracy.

Abstract

In this paper, we investigate the strong convergence analysis of parareal algorithms for stochastic Maxwell equations with the damping term driven by additive noise. The proposed parareal algorithms proceed as two-level temporal parallelizable integrators with the stochastic exponential integrator as the coarse propagator and both the exact solution integrator and the stochastic exponential integrator as the fine propagator. It is proved that the convergence order of the proposed algorithms linearly depends on the iteration number. Numerical experiments are performed to illustrate the convergence of the parareal algorithms for different choices of the iteration number and the damping coefficient.

Parareal algorithms for stochastic Maxwell equations with the damping term driven by additive noise

TL;DR

The paper analyzes strong (mean-square) convergence of two-level parareal algorithms for stochastic Maxwell equations with an additive-noise damping term. By using the stochastic exponential scheme as the coarse propagator and either the exact solver or a stochastic exponential fine propagator, it proves that the iteration-induced error decays with order and grows linearly in , under suitable regularity assumptions. Numerical experiments in 1D and 2D confirm linear-in- convergence and show that increased damping accelerates convergence while larger noise scales perturb the solution. This work extends parareal analysis to stochastic Maxwell systems and highlights practical effects of damping and noise on temporal parallelization efficiency and accuracy.

Abstract

In this paper, we investigate the strong convergence analysis of parareal algorithms for stochastic Maxwell equations with the damping term driven by additive noise. The proposed parareal algorithms proceed as two-level temporal parallelizable integrators with the stochastic exponential integrator as the coarse propagator and both the exact solution integrator and the stochastic exponential integrator as the fine propagator. It is proved that the convergence order of the proposed algorithms linearly depends on the iteration number. Numerical experiments are performed to illustrate the convergence of the parareal algorithms for different choices of the iteration number and the damping coefficient.
Paper Structure (15 sections, 10 theorems, 76 equations, 6 figures)

This paper contains 15 sections, 10 theorems, 76 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Parareal algorithm for stochastic Maxwell equations
  4. Parareal algorithm
  5. Stochastic exponential scheme
  6. Coarse and fine propagators
  7. Main results
  8. The exact integrator as the fine integrator $\mathcal{F}$
  9. The stochastic exponential integrator as the fine propagator
  10. Numerical experiments
  11. Convergence
  12. One-dimensional transverse magnetic wave
  13. Two-dimensional transverse magnetic waves
  14. Impact of the scale of noise
  15. Conclusion

Key Result

Lemma 1

Chenetal2023 The Maxwell operator defined in (eq3) with domain $\mathcal{D}(M)$ is closed and skew-adjoint, and generates a $C_{0}$-semigroup $S(t)=e^{tM}$on $\mathbb{H}$ for $t\in [0,T]$. Moreover, the frequently used property for Maxwell operator M is : $\langle Mu, u\rangle_{\mathbb{H}}=0$.

Figures (6)

  • Figure 1: Convergence of 1D case vs. interation number $k$ for different values of $\sigma=0,2^1,2^3,2^5$.
  • Figure 2: Mean-square order of 1D case with respect to $\Delta T = 2^{-i}, i = 5,6,7,8.$
  • Figure 3: Convergence of 2D case with interation number $k$ for different values of $\sigma=0,2^1,2^3,2^5$.
  • Figure 4: Mean-square order of 2D case with respect to $\Delta T = 2^{-i}, i = 3,4,5,6.$
  • Figure 5: 10 Contour of $E_{z}(x,y)$ with different sizes of noise $\lambda_{1}=\lambda_{2}=0, 2^1, 2^3, 2^5$ in the time $T=1$.
  • ...and 1 more figures

Theorems & Definitions (18)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Remark 1
  • Theorem 1
  • Definition 1
  • Lemma 5
  • Lemma 6
  • Definition 2
  • ...and 8 more