100 research outputs found
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
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
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
The geometry of low-rank Kalman filters
An important property of the Kalman filter is that the underlying Riccati
flow is a contraction for the natural metric of the cone of symmetric positive
definite matrices. The present paper studies the geometry of a low-rank version
of the Kalman filter. The underlying Riccati flow evolves on the manifold of
fixed rank symmetric positive semidefinite matrices. Contraction properties of
the low-rank flow are studied by means of a suitable metric recently introduced
by the authors.Comment: Final version published in Matrix Information Geometry, pp53-68,
Springer Verlag, 201
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