4,955 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
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
From low-rank approximation to an efficient rational Krylov subspace method for the Lyapunov equation
We propose a new method for the approximate solution of the Lyapunov equation
with rank- right-hand side, which is based on extended rational Krylov
subspace approximation with adaptively computed shifts. The shift selection is
obtained from the connection between the Lyapunov equation, solution of systems
of linear ODEs and alternating least squares method for low-rank approximation.
The numerical experiments confirm the effectiveness of our approach.Comment: 17 pages, 1 figure
Order reduction approaches for the algebraic Riccati equation and the LQR problem
We explore order reduction techniques for solving the algebraic Riccati
equation (ARE), and investigating the numerical solution of the
linear-quadratic regulator problem (LQR). A classical approach is to build a
surrogate low dimensional model of the dynamical system, for instance by means
of balanced truncation, and then solve the corresponding ARE. Alternatively,
iterative methods can be used to directly solve the ARE and use its approximate
solution to estimate quantities associated with the LQR. We propose a class of
Petrov-Galerkin strategies that simultaneously reduce the dynamical system
while approximately solving the ARE by projection. This methodology
significantly generalizes a recently developed Galerkin method by using a pair
of projection spaces, as it is often done in model order reduction of dynamical
systems. Numerical experiments illustrate the advantages of the new class of
methods over classical approaches when dealing with large matrices
Least Squares Ranking on Graphs
Given a set of alternatives to be ranked, and some pairwise comparison data,
ranking is a least squares computation on a graph. The vertices are the
alternatives, and the edge values comprise the comparison data. The basic idea
is very simple and old: come up with values on vertices such that their
differences match the given edge data. Since an exact match will usually be
impossible, one settles for matching in a least squares sense. This formulation
was first described by Leake in 1976 for rankingfootball teams and appears as
an example in Professor Gilbert Strang's classic linear algebra textbook. If
one is willing to look into the residual a little further, then the problem
really comes alive, as shown effectively by the remarkable recent paper of
Jiang et al. With or without this twist, the humble least squares problem on
graphs has far-reaching connections with many current areas ofresearch. These
connections are to theoretical computer science (spectral graph theory, and
multilevel methods for graph Laplacian systems); numerical analysis (algebraic
multigrid, and finite element exterior calculus); other mathematics (Hodge
decomposition, and random clique complexes); and applications (arbitrage, and
ranking of sports teams). Not all of these connections are explored in this
paper, but many are. The underlying ideas are easy to explain, requiring only
the four fundamental subspaces from elementary linear algebra. One of our aims
is to explain these basic ideas and connections, to get researchers in many
fields interested in this topic. Another aim is to use our numerical
experiments for guidance on selecting methods and exposing the need for further
development.Comment: Added missing references, comparison of linear solvers overhauled,
conclusion section added, some new figures adde
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