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Parareal Algorithms for Stochastic Maxwell Equations Driven by Multiplicative Noise

Liying Zhang, Qi Zhang, Lihai Ji

TL;DR

The paper tackles numerical simulation of stochastic Maxwell equations driven by multiplicative noise using a parareal approach. It combines a stochastic exponential integrator as the coarse propagator with two possible fine propagators (the exact solution or the stochastic exponential scheme) to enable time-parallel computation. Theoretical results establish a uniform mean-square convergence rate proportional to $k/2$ with respect to the iteration count, and numerical experiments confirm improved convergence, stability, and computational efficiency for long-time simulations compared with traditional exponential methods. This work demonstrates that parareal methods can substantially reduce computational costs while maintaining high accuracy in stochastic electromagnetic models with state-dependent noise.

Abstract

This paper investigates the parareal algorithms for solving the stochastic Maxwell equations driven by multiplicative noise, focusing on their convergence, computational efficiency and numerical performance. The algorithms use the stochastic exponential integrator as the coarse propagator, while both the exact integrator and the stochastic exponential integrator are used as fine propagators. Theoretical analysis shows that the mean square convergence rates of the two algorithms selected above are proportional to $k/2$, depending on the iteration number of the algorithms. Numerical experiments validate these theoretical findings, demonstrating that larger iteration numbers $k$ improve convergence rates, while larger damping coefficients $σ$ accelerate the convergence of the algorithms. Furthermore, the algorithms maintain high accuracy and computational efficiency, highlighting their significant advantages over traditional exponential methods in long-term simulations.

Parareal Algorithms for Stochastic Maxwell Equations Driven by Multiplicative Noise

TL;DR

The paper tackles numerical simulation of stochastic Maxwell equations driven by multiplicative noise using a parareal approach. It combines a stochastic exponential integrator as the coarse propagator with two possible fine propagators (the exact solution or the stochastic exponential scheme) to enable time-parallel computation. Theoretical results establish a uniform mean-square convergence rate proportional to with respect to the iteration count, and numerical experiments confirm improved convergence, stability, and computational efficiency for long-time simulations compared with traditional exponential methods. This work demonstrates that parareal methods can substantially reduce computational costs while maintaining high accuracy in stochastic electromagnetic models with state-dependent noise.

Abstract

This paper investigates the parareal algorithms for solving the stochastic Maxwell equations driven by multiplicative noise, focusing on their convergence, computational efficiency and numerical performance. The algorithms use the stochastic exponential integrator as the coarse propagator, while both the exact integrator and the stochastic exponential integrator are used as fine propagators. Theoretical analysis shows that the mean square convergence rates of the two algorithms selected above are proportional to , depending on the iteration number of the algorithms. Numerical experiments validate these theoretical findings, demonstrating that larger iteration numbers improve convergence rates, while larger damping coefficients accelerate the convergence of the algorithms. Furthermore, the algorithms maintain high accuracy and computational efficiency, highlighting their significant advantages over traditional exponential methods in long-term simulations.

Paper Structure

This paper contains 15 sections, 7 theorems, 66 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Stochastic Maxwell equations driven by multiplicative noise
  3. Temporal semidiscretization by parareal algorithm
  4. Framework of parareal algorithm
  5. Stochastic exponential scheme
  6. Coarse and fine propagators
  7. Computational cost and efficiency analysis
  8. Computational cost of the parareal algorithm
  9. Computational cost of the exponential solution
  10. Computational efficiency
  11. Efficiency analysis
  12. Error estimate of exact integrator as the fine integrator
  13. Error estimate of stochastic exponential integrator as the fine integrator
  14. Numerical experiments
  15. Conclusion

Key Result

Lemma 1

Chenetal2023 The Maxwell operator defined in (eq1) 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 the Maxwell operator $M$ is: $\langle M u, u \rangle_{\mathbb{H}}

Figures (7)

  • Figure 1: Convergence with interation number $k$ for different values of $\sigma=0,2^1,2^3,2^5$
  • Figure 2: Convergence for different time $T=1,10,20$
  • Figure 3: Mean-square order for $k=3$ in the temporal direction in the cases of (left)$F(u)=u+cos(u),G(u)=sin(u)$ and (right)$F(u)=cos(u),G(u)=u$
  • Figure 4: Mean-square order for $k=4$ in the temporal direction in the cases of (left)$F(u)=u+cos(u),G(u)=sin(u)$ and (right)$F(u)=cos(u),G(u)=u$
  • Figure 5: Mean-square order for $k=5$ in the temporal direction in the cases of (left)$F(u)=u+cos(u),G(u)=sin(u)$ and (right)$F(u)=cos(u),G(u)=u$
  • ...and 2 more figures

Theorems & Definitions (13)

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