Browsing by Author "Wu, Minghao"

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    Linear Stability Analysis Using Lyapunov Inverse Iteration
    (2012) Wu, Minghao; Elman, Howard; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    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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    Lyapunov Inverse Iteration for Computing a few Rightmost Eigenvalues of Large Generalized Eigenvalue Problems
    (2012-04-20) Elman, Howard C.; Wu, Minghao
    In linear stability analysis of a large-scale dynamical system, we need to compute the rightmost eigenvalue(s) for a series of large generalized eigenvalue problems. Existing iterative eigenvalue solvers are not robust when no estimate of the rightmost eigenvalue(s) is available. In this study, we show that such an estimate can be obtained from Lyapunov inverse iteration applied to a special eigenvalue problem of Lyapunov structure. We also show that Lyapunov inverse iteration will always converge in only two steps if the Lyapunov equation in the first step is solved accurately enough. Furthermore, we generalize the analysis to a deflated version of this Lyapunov eigenvalue problem and propose an algorithm that computes a few rightmost eigenvalues for the eigenvalue problems arising from linear stability analysis. Numerical experiments demonstrate the robustness of the algorithm.
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    Lyapunov Inverse Iteration for Identifying Hopf Bifurcations in Models of Incompressible Flow
    (2011-03-07) Elman, Howard C.; Meerbergen, Karl; Spence, Alastair; Wu, Minghao
    The identification of instability in large-scale dynamical systems caused by Hopf bifurcation is difficult because of the problem of identifying the rightmost pair of complex eigenvalues of large sparse generalized eigenvalue problems. A new method developed in [Meerbergen and Spence, SIAM J. Matrix Anal. Appl., 31 (2010), pp. 1982- 1999] avoids this computation, instead performing an inverse iteration for a certain set of real eigenvalues and that requires the solution of a large-scale Lyapunov equation at each iteration. In this study, we refine the Lyapunov inverse iteration method to make it more robust and efficient, and we examine its performance on challenging test problems arising from fluid dynamics. Various implementation issues are discussed, including the use of inexact inner iterations and the impact of the choice of iterative solution for the Lyapunov equations, and the effect of eigenvalue distribution on performance. Numerical experiments demonstrate the robustness of the algorithm.