Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimal Control of a Sub-diffusion Model using Dirichlet-Neumann and Neumann-Neumann Waveform Relaxation Algorithms

Soura Sana, Bankim C. Mandal

TL;DR

This work analyzes the convergence of Dirichlet-Neumann and Neumann-Neumann waveform relaxation algorithms for distributed optimal control problems constrained by sub-diffusion PDEs on 1D domains. It derives explicit geometric convergence bounds for both DNWR (two-subdomain) and NNWR (multi-subdomain) using a spectral, RL-to-Caputo framework and introduces a novel both-sided graded time mesh to enhance accuracy and analysis. Theoretical results are complemented by 1D numerical experiments that confirm the optimal relaxation parameters and demonstrate robustness across fractional orders, subdomain counts, and mesh types. The findings offer practical, scalable solvers for fractional PDE-constrained optimization with clear guidance on parameter choices and mesh design.

Abstract

This paper explores the convergence behavior of two waveform relaxation algorithms, namely the Dirichlet-Neumann and Neumann-Neumann Waveform Relaxation algorithms, for an optimal control problem with a sub-diffusion partial differential equation (PDE) constraint. The algorithms are tested on regular 1D domains with multiple subdomains, and the analysis focuses on how different constant values of the generalized diffusion coefficient affect the convergence of these algorithms.

Optimal Control of a Sub-diffusion Model using Dirichlet-Neumann and Neumann-Neumann Waveform Relaxation Algorithms

TL;DR

This work analyzes the convergence of Dirichlet-Neumann and Neumann-Neumann waveform relaxation algorithms for distributed optimal control problems constrained by sub-diffusion PDEs on 1D domains. It derives explicit geometric convergence bounds for both DNWR (two-subdomain) and NNWR (multi-subdomain) using a spectral, RL-to-Caputo framework and introduces a novel both-sided graded time mesh to enhance accuracy and analysis. Theoretical results are complemented by 1D numerical experiments that confirm the optimal relaxation parameters and demonstrate robustness across fractional orders, subdomain counts, and mesh types. The findings offer practical, scalable solvers for fractional PDE-constrained optimization with clear guidance on parameter choices and mesh design.

Abstract

This paper explores the convergence behavior of two waveform relaxation algorithms, namely the Dirichlet-Neumann and Neumann-Neumann Waveform Relaxation algorithms, for an optimal control problem with a sub-diffusion partial differential equation (PDE) constraint. The algorithms are tested on regular 1D domains with multiple subdomains, and the analysis focuses on how different constant values of the generalized diffusion coefficient affect the convergence of these algorithms.
Paper Structure (12 sections, 15 theorems, 93 equations, 13 figures)

This paper contains 12 sections, 15 theorems, 93 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Optimal Control of a subdiffusion model
  3. The Dirichlet-Neumann Waveform Relaxation algorithm
  4. The Neumann-Neumann Waveform Relaxation algorithm
  5. Auxiliary Results
  6. Mesh Grading
  7. Convergence of Dirichlet-Neumann Waveform Relaxation
  8. Convergence of Neumann-Neumann Waveform Relaxation
  9. Numerical Experiments
  10. DNWR Algorithm
  11. NNWR Algorithm in 1D
  12. Conclusions

Key Result

Lemma 5.1

\newlabelrl_eqi_c The Riemann-Liouville time derivative $D^{\alpha}_{RL}y(t)$ with homogeneous initial condition $I^{1-\alpha}_{RL}y(0^{+}) = 0$ is equivalent to the Caputo fractional time derivative $_0D_t^{\alpha}y(t)$ with zero initial condition $y(0) = 0$.

Figures (13)

  • Figure 9.1: Monodomain solutions profile of solution state, control state, and target state.
  • Figure 9.2: DNWR: The numerical convergence rate varies across different values of $\theta$, while $h_1<h_2$ and $T=1$ remain fixed, considering a fractional order $\alpha = 0.3$. On the left, we employ a uniform mesh in time; in the middle, a one-sided graded mesh in time is utilized; and on the right, both-sided graded mesh in time is applied.
  • Figure 9.3: DNWR: The numerical convergence rate varies across different values of $\theta$, while $h_1<h_2$ and $T=1$ remain fixed, considering a fractional order $\alpha = 0.8$. On the left, we employ a uniform mesh in time; in the middle, a one-sided graded mesh in time is utilized; and on the right, both-sided graded mesh in time is applied.
  • Figure 9.4: DNWR: As the number of time nodes increases within a fixed time window of $T = 1$ and a constant regularization parameter of $\sigma = 10^{-6}$, the convergence factor varies. On the left side, with a fractional order of $\alpha = 0.3$, while on the right side, with a fractional order of $\alpha = 0.8$.
  • Figure 9.5: DNWR: As the regularization parameter $\sigma$ changes, the convergence factor varies for a constant number of nodes $N_t = 100$ and for a constant time window $T = 1$. On the left, we examine this phenomenon with a fractional order of $\alpha = 0.3$, while on the right, we consider a fractional order of $\alpha = 0.8$
  • ...and 8 more figures

Theorems & Definitions (28)

  • Lemma 5.1: Theorem 2.10 mophou2011optimal
  • proof
  • Lemma 5.2
  • proof
  • Lemma 5.3
  • proof
  • Lemma 5.4
  • proof
  • Corollary 5.5
  • Corollary 5.6
  • ...and 18 more