278 research outputs found
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
KRYLOV SUBSPACE METHODS FOR SOLVING LARGE LYAPUNOV EQUATIONS
Published versio
Approximate solutions to large nonsymmetric differential Riccati problems with applications to transport theory
In the present paper, we consider large scale nonsymmetric differential
matrix Riccati equations with low rank right hand sides. These matrix equations
appear in many applications such as control theory, transport theory, applied
probability and others. We show how to apply Krylov-type methods such as the
extended block Arnoldi algorithm to get low rank approximate solutions. The
initial problem is projected onto small subspaces to get low dimensional
nonsymmetric differential equations that are solved using the exponential
approximation or via other integration schemes such as Backward Differentiation
Formula (BDF) or Rosenbrok method. We also show how these technique could be
easily used to solve some problems from the well known transport equation. Some
numerical experiments are given to illustrate the application of the proposed
methods to large-scale problem
A Hamiltonian Krylov-Schur-type method based on the symplectic Lanczos process
We discuss a Krylov-Schur like restarting technique applied within the symplectic Lanczos algorithm for the Hamiltonian eigenvalue problem. This allows to easily implement a purging and locking strategy in order to improve the convergence properties of the symplectic Lanczos algorithm. The Krylov-Schur-like restarting is based on the SR algorithm. Some ingredients of the latter need to be adapted to the structure of the symplectic Lanczos recursion. We demonstrate the efficiency of the new method for several Hamiltonian eigenproblems
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