Space-time galerkin pod with application in optimal control of semilinear partial differential equations

Manuel Baumann, Peter Benner, Jan Heiland

Research output: Contribution to journalArticleScientificpeer-review

3 Citations (Scopus)

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 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.

Original languageEnglish
Pages (from-to)A1611-A1641
Number of pages31
JournalSIAM Journal on Scientific Computing
Volume40
Issue number3
DOIs
Publication statusPublished - 2018

Keywords

  • Low-rank Galerkin
  • Model reduction
  • Nonlinear PDE
  • Optimal control

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