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

Tobias Breiten, ,Institute of Mathematics, Technical University of Berlin, 10623 Berlin, Germany, Sergey Dolgov, Martin Stoll, ,University of Bath, Department of Mathematical Sciences, BA2 7AY Bath, United Kingdom, ,Technische Universität Chemnitz, Department of Mathematics, Scientific Computing Group, 09107 Chemnitz, Germany
2019 Numerical Algebra, Control and Optimization  
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.
more » ... ximating 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. 2010 Mathematics Subject Classification. Primary: 15A24, 65H10, 15A69, 93A15.
doi:10.3934/naco.2020034 fatcat:t7qjnxwiojebvf5anxi5hqdcje