574 research outputs found
Adaptive high-order splitting schemes for large-scale differential Riccati equations
We consider high-order splitting schemes for large-scale differential Riccati
equations. Such equations arise in many different areas and are especially
important within the field of optimal control. In the large-scale case, it is
critical to employ structural properties of the matrix-valued solution, or the
computational cost and storage requirements become infeasible. Our main
contribution is therefore to formulate these high-order splitting schemes in a
efficient way by utilizing a low-rank factorization. Previous results indicated
that this was impossible for methods of order higher than 2, but our new
approach overcomes these difficulties. In addition, we demonstrate that the
proposed methods contain natural embedded error estimates. These may be used
e.g. for time step adaptivity, and our numerical experiments in this direction
show promising results.Comment: 23 pages, 7 figure
Peer Methods for the Solution of Large-Scale Differential Matrix Equations
We consider the application of implicit and linearly implicit
(Rosenbrock-type) peer methods to matrix-valued ordinary differential
equations. In particular the differential Riccati equation (DRE) is
investigated. For the Rosenbrock-type schemes, a reformulation capable of
avoiding a number of Jacobian applications is developed that, in the autonomous
case, reduces the computational complexity of the algorithms. Dealing with
large-scale problems, an efficient implementation based on low-rank symmetric
indefinite factorizations is presented. The performance of both peer approaches
up to order 4 is compared to existing implicit time integration schemes for
matrix-valued differential equations.Comment: 29 pages, 2 figures (including 6 subfigures each), 3 tables,
Corrected typo
Order reduction methods for solving large-scale differential matrix Riccati equations
We consider the numerical solution of large-scale symmetric differential
matrix Riccati equations. Under certain hypotheses on the data, reduced order
methods have recently arisen as a promising class of solution strategies, by
forming low-rank approximations to the sought after solution at selected
timesteps. We show that great computational and memory savings are obtained by
a reduction process onto rational Krylov subspaces, as opposed to current
approaches. By specifically addressing the solution of the reduced differential
equation and reliable stopping criteria, we are able to obtain accurate final
approximations at low memory and computational requirements. This is obtained
by employing a two-phase strategy that separately enhances the accuracy of the
algebraic approximation and the time integration. The new method allows us to
numerically solve much larger problems than in the current literature.
Numerical experiments on benchmark problems illustrate the effectiveness of the
procedure with respect to existing solvers
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