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

TL;DR

This paper analyzes the weak scalability of the time parallel Schwarz method for parabolic optimal control problems, providing theoretical bounds and confirming results through numerical experiments, highlighting its suitability for large-scale simulations.

Contribution

It introduces the first theoretical analysis of weak scalability for time domain decomposition methods applied to parabolic optimal control problems.

Findings

The spectral radius bounds are derived using matrix norm and Toeplitz theory.

Numerical experiments confirm the weak scalability of the method.

The analysis provides insights into the method's performance on high-performance computing architectures.

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.