Solving differential riccati equations: A nonlinear space-time method using tensor trains

Tobias Breiten, Sergey Dolgov, Martin Stoll

Research output: Contribution to journalArticlepeer-review

1 Citation (SciVal)

Abstract

Differential Riccati equations are at the heart of many applications in control theory. They are time-dependent, matrix-valued, and in particular nonlinear equations that require special methods for their solution. Low-rank methods have been used heavily for computing a low-rank solution at every step of a time-discretization. We propose the use of an all-at-once space-time solution leading to a large nonlinear space-time problem for which we propose the use of a Newton–Kleinman iteration. Approximating the space-time problem in a higher-dimensional low-rank tensor form requires fewer degrees of freedom in the solution and in the operator, and gives a faster numerical method. Numerical experiments demonstrate a storage reduction of up to a factor of 100.

Original languageEnglish
Pages (from-to)407-429
Number of pages23
JournalNumerical Algebra, Control and Optimization
Volume11
Issue number3
DOIs
Publication statusPublished - 30 Sep 2021

Keywords

  • Low-rank methods
  • Non-linear problems
  • Optimal Control
  • Riccati equations

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Control and Optimization
  • Applied Mathematics

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