206 research outputs found
A Subspace Shift Technique for Nonsymmetric Algebraic Riccati Equations
The worst situation in computing the minimal nonnegative solution of a
nonsymmetric algebraic Riccati equation associated with an M-matrix occurs when
the corresponding linearizing matrix has two very small eigenvalues, one with
positive and one with negative real part. When both these eigenvalues are
exactly zero, the problem is called critical or null recurrent. While in this
case the problem is ill-conditioned and the convergence of the algorithms based
on matrix iterations is slow, there exist some techniques to remove the
singularity and transform the problem to a well-behaved one. Ill-conditioning
and slow convergence appear also in close-to-critical problems, but when none
of the eigenvalues is exactly zero the techniques used for the critical case
cannot be applied.
In this paper, we introduce a new method to accelerate the convergence
properties of the iterations also in close-to-critical cases, by working on the
invariant subspace associated with the problematic eigenvalues as a whole. We
present a theoretical analysis and several numerical experiments which confirm
the efficiency of the new method
Two-Step Relaxation Newton Method for Nonsymmetric Algebraic Riccati Equations Arising from Transport Theory
We propose a new idea to construct an effective algorithm to compute the minimal positive solution of the nonsymmetric algebraic Riccati equations arising from transport theory. For a class of these equations, an important feature is that the minimal positive solution can be obtained by computing the minimal positive solution of a couple of fixed-point equations with vector form. Based on the fixed-point vector equations, we introduce a new algorithm, namely,
two-step relaxation Newton, derived by combining two different relaxation Newton methods to compute the minimal positive solution. The monotone convergence of the solution sequence generated by this new algorithm is established. Numerical results are given to show the advantages of the new algorithm for the nonsymmetric algebraic Riccati equations in vector form
A numerical comparison of solvers for large-scale, continuous-time algebraic Riccati equations and LQR problems
In this paper, we discuss numerical methods for solving large-scale
continuous-time algebraic Riccati equations. These methods have been the focus
of intensive research in recent years, and significant progress has been made
in both the theoretical understanding and efficient implementation of various
competing algorithms. There are several goals of this manuscript: first, to
gather in one place an overview of different approaches for solving large-scale
Riccati equations, and to point to the recent advances in each of them. Second,
to analyze and compare the main computational ingredients of these algorithms,
to detect their strong points and their potential bottlenecks. And finally, to
compare the effective implementations of all methods on a set of relevant
benchmark examples, giving an indication of their relative performance
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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