Weak Scalability of time parallel Schwarz methods for parabolic optimal control problems

Abstract

Parabolic optimal control problems arise in numerous scientific and engineering applications. They typically lead to large-scale coupled forward-backward systems that cannot be treated with classical time-stepping schemes and are computationally expensive to solve. Therefore, parallel methods are essential to reduce the computational time required. In this work, we investigate a time domain decomposition approach, namely the time parallel Schwarz method, applied to parabolic optimal control problems. We analyze the convergence behavior and focus on the weak scalability property of this method as the number of time intervals increases. To characterize the spectral radius of the iteration matrix, we present two analysis techniques: the construction of a tailored matrix norm and the application of block Toeplitz matrix theory. Our analyses yield both nonasymptotic bounds on the spectral radius and an asymptotic characterization of the eigenvalues as the number of time intervals tends to infinity. Numerical experiments further confirm our theoretical findings and demonstrate the weak scalability of the time parallel Schwarz method. This work introduces the first theoretical tool for analyzing the weak scalability of time domain decomposition methods, and our results shed light on the suitability of our algorithm for large-scale simulations on modern high-performance computing architectures.

Figures (8)

• Figure 1: Illustration of $N$ time intervals with fixed size $\Delta t$.
• Figure 2: Comparison of $\rho(T^{\text{PS}}_{N,m})$, $\|T^{\text{PS}}_{N,m}\|_{\infty}$ and $\widetilde{\rho}(m)$ for different numbers of time intervals $N$. Left panels refer to $m=1$ while right panels refer to $m=M$. The remaining parameters are: $M=128$, $\nu=10^{-2}$ and $\Delta t=1/M$.
• Figure 3: Plot of the map $m\rightarrow \widetilde{\rho}(m;\nu,\Delta t)$ for different values of $\nu$ and $\Delta t$. Parameters: $M=128$, $\Delta t=1/M$ (left panel), $M=128$, $\nu=10^{-2}$ (right panel).
• Figure 4: Clustering of the eigenvalues of $\{T_{N,m}^{\text{PS}}\}_{N}$ for $m=1$ (top row) and $m=M$ (bottom row). Parameters: $\Delta t=1/M$ and $\nu=10^{-2}$ (left panels) and $\nu=10^{-4}$ (right panels). In the legend, we omit the superscript $\text{PS}$ to improve readability.
• Figure 5: Second order convergence of the Crank--Nicolson method applied to solve the reduced optimality system \ref{['eq:reduced']}. The mesh size and time step satisfy $h_x=h_t=h$, and $h$ varies in the set $\{2^{-8}, 2^{-7}, \ldots, 2^{-3}\}$.

Theorems & Definitions (8)

• Theorem 1
• proof
• Definition 1: Eigenvalue cluster
• Theorem 2
• proof
• Theorem 3: Theorem 1.2, Ref. donatelli2012canonical
• Corollary 1
• proof