Linear Stability Analysis Using Lyapunov Inverse Iteration

In this dissertation, we develop robust and efficient methods for linear stability analysis of large-scale dynamical systems, with emphasis on the incompressible Navier-Stokes equations. Linear stability analysis is a widely used approach for studying whether a steady state of a dynamical system is sensitive to small perturbations. The main mathematical tool that we consider in this dissertation is Lyapunov inverse iteration, a recently developed iterative method for computing the eigenvalue with smallest modulus of a special eigenvalue problem that can be specified in the form of a Lyapunov equation. It has the following "inner-outer" structure: the outer iteration is the eigenvalue computation and the inner iteration is solving a large-scale Lyapunov equation. This method has two applications in linear stability analysis: it can be used to estimate the critical value of a physical parameter at which the steady state becomes unstable (i.e., sensitive to small perturbations), and it can also be applied to compute a few rightmost eigenvalues of the Jacobian matrix. We present numerical performance of Lyapunov inverse iteration in both applications, analyze its convergence in the second application, and propose strategies of implementing it efficiently for each application. In previous work, Lyapunov inverse iteration has been used to estimate the critical parameter value at which a parameterized path of steady states loses stability. We refine this method by proposing an adaptive stopping criterion for the Lyapunov solve (inner iteration) that depends on the accuracy of the eigenvalue computation (outer iteration). The use of such a criterion achieves dramatic savings in computational cost and does not affect the convergence of the target eigenvalue. The method of previous work has the limitation that it can only be used at a stable point in the neighborhood of the critical point. We further show that Lyapunov inverse iteration can also be used to generate a few rightmost eigenvalues of the Jacobian matrix at any stable point. These eigenvalues are crucial in linear stability analysis, and existing approaches for computing them are not robust. A convergence analysis of this method leads to a way of implementing it that only entails one Lyapunov solve. In addition, we explore the utility of various Lyapunov solvers in both applications of Lyapunov inverse iteration. We observe that different Lyapunov solvers should be used for the Lyapunov equations arising from the two applications. Applying a Lyapunov solver entails solving a number of large and sparse linear systems. We explore the use of sparse iterative methods for this task and construct a new variant of the Lyapunov solver that significantly reduces the costs of the sparse linear solves

On the ADI method for the Sylvester Equation and the optimal-H2\mathcal{H}_2 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

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 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

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-$1$ 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