Space-time Galerkin POD with application in optimal control of semi-linear parabolic partial differential equations

TL;DR

This paper develops a space-time Galerkin POD method for efficient reduced-order modeling of semi-linear parabolic PDEs, demonstrating its optimality and effectiveness in optimal control applications through numerical experiments.

Contribution

It introduces a generalized space-time Galerkin POD theory, proves its optimality, and applies it to nonlinear PDE optimal control, showing advantages over standard methods.

Findings

The space-time Galerkin POD is theoretically optimal in relevant function spaces.

The method effectively reduces computational complexity in optimal control of nonlinear PDEs.

Numerical results with Burgers' equation demonstrate competitive performance compared to traditional approaches.

Abstract

In the context of Galerkin discretizations of a partial differential equation (PDE), the modes of the classical method of Proper Orthogonal Decomposition (POD) can be interpreted as the ansatz and trial functions of a low-dimensional Galerkin scheme. If one also considers a Galerkin method for the time integration, one can similarly define a POD reduction of the temporal component. This has been described earlier but not expanded upon -- probably because the reduced time discretization globalizes time which is computationally inefficient. However, in finite-time optimal control systems, time \textit{is} a global variable and there is no disadvantage from using a POD reduced Galerkin scheme in time. In this paper, we provide a newly developed generalized theory for space-time Galerkin POD, prove its optimality in the relevant function spaces, show its application for the optimal control of nonlinear PDEs, and, by means of a numerical example with Burgers' equation, discuss the competitiveness by comparing to standard approaches.