## Linear Stability Analysis Using Lyapunov Inverse Iteration

##### Abstract

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.

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