10,698 research outputs found
Solution of polynomial Lyapunov and Sylvester equations
A two-variable polynomial approach to solve the one-variable polynomial Lyapunov and Sylvester equations is proposed. Lifting the problem from the one-variable to the two-variable context gives rise to associated lifted equations which live on finite-dimensional vector spaces. This allows for the design of an iterative solution method which is inspired by the method of Faddeev for the computation of matrix resolvents. The resulting algorithms are especially suitable for applications requiring symbolic or exact computation
Stability of the iterative solutions of integral equations as one phase freezing criterion
A recently proposed connection between the threshold for the stability of the
iterative solution of integral equations for the pair correlation functions of
a classical fluid and the structural instability of the corresponding real
fluid is carefully analyzed. Direct calculation of the Lyapunov exponent of the
standard iterative solution of HNC and PY integral equations for the 1D hard
rods fluid shows the same behavior observed in 3D systems. Since no phase
transition is allowed in such 1D system, our analysis shows that the proposed
one phase criterion, at least in this case, fails. We argue that the observed
proximity between the numerical and the structural instability in 3D originates
from the enhanced structure present in the fluid but, in view of the arbitrary
dependence on the iteration scheme, it seems uneasy to relate the numerical
stability analysis to a robust one-phase criterion for predicting a
thermodynamic phase transition.Comment: 11 pages, 2 figure
On the approximability of Koopman-based operator Lyapunov equations
Lyapunov functions play a vital role in the context of control theory for
nonlinear dynamical systems. Besides its classical use for stability analysis,
Lyapunov functions also arise in iterative schemes for computing optimal
feedback laws such as the well-known policy iteration. In this manuscript, the
focus is on the Lyapunov function of a nonlinear autonomous finite-dimensional
dynamical system which will be rewritten as an infinite-dimensional linear
system using the Koopman or composition operator. Since this
infinite-dimensional system has the structure of a weak-* continuous semigroup,
in a specially weighted -space one can establish a connection
between the solution of an operator Lyapunov equation and the desired Lyapunov
function. It will be shown that the solution to this operator equation attains
a rapid eigenvalue decay which justifies finite rank approximations with
numerical methods. The potential benefit for numerical computations will be
demonstrated with two short examples.Comment: Lyapunov equations, Koopman operator, infinite dimensional systems,
semigroup
Efficient Iterative Algorithms for Linear Stability Analysis of Incompressible Flows
Linear stability analysis of a dynamical system entails finding the rightmost eigenvalue for a series of eigenvalue problems. For large-scale systems, it is known that conventional iterative eigenvalue solvers are not reliable for computing this eigenvalue. A more robust method recently developed in Elman & Wu (2012) and Meerbergen & Spence (2010), Lyapunov inverse iteration, involves solving large-scale Lyapunov equations, which in turn requires the solution of large, sparse linear systems analogous to those arising from solving the underlying partial differential equations. This study explores the efficient implementation of Lyapunov inverse iteration when it is used for linear stability analysis of incompressible flows. Efficiencies are obtained from effective solution strategies for the Lyapunov equations and for the underlying partial differential equations. Existing solution strategies are tested and compared, and a modified version of a Lyapunov solver is proposed that achieves significant savings in computational cost
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
On the ADI method for the Sylvester Equation and the optimal- points
The ADI iteration is closely related to the rational Krylov projection
methods for constructing low rank approximations to the solution of Sylvester
equation. In this paper we show that the ADI and rational Krylov approximations
are in fact equivalent when a special choice of shifts are employed in both
methods. We will call these shifts pseudo H2-optimal shifts. These shifts are
also optimal in the sense that for the Lyapunov equation, they yield a residual
which is orthogonal to the rational Krylov projection subspace. Via several
examples, we show that the pseudo H2-optimal shifts consistently yield nearly
optimal low rank approximations to the solutions of the Lyapunov equations
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