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Optimization and pseudospectra, with applications to robust stability
 SIAM Journal on Matrix Analysis and Applications
, 2003
"... Abstract. The ɛpseudospectrum of amatrix A is the subset of the complex plane consisting of all eigenvalues of all complex matrices within a distance ɛ of A. We are interested in two aspects of “optimization and pseudospectra. ” The first concerns maximizing the function “real part ” over an ɛpseu ..."
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Cited by 19 (6 self)
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Abstract. The ɛpseudospectrum of amatrix A is the subset of the complex plane consisting of all eigenvalues of all complex matrices within a distance ɛ of A. We are interested in two aspects of “optimization and pseudospectra. ” The first concerns maximizing the function “real part ” over an ɛpseudospectrum of afixed matrix: this defines afunction known as the ɛpseudospectral abscissa of amatrix. We present abisection algorithm to compute this function. Our second interest is in minimizing the ɛpseudospectral abscissa over a set of feasible matrices. A prerequisite for local optimization of this function is an understanding of its variational properties, the study of which is the main focus of the paper. We show that, in a neighborhood of any nonderogatory matrix, the ɛpseudospectral abscissa is a nonsmooth but locally Lipschitz and subdifferentially regular function for sufficiently small ɛ; in fact, it can be expressed locally as the maximum of a finite number of smooth functions. Along the way we obtain an eigenvalue perturbation result: near a nonderogatory matrix, the eigenvalues satisfy a Hölder continuity property on matrix space—a property that is well known when only a single perturbation parameter is considered. The pseudospectral abscissa is a powerful modeling tool: not only is it a robust measure of stability, but it also reveals the transient (as opposed to asymptotic) behavior of associated dynamical systems.
Spectral properties of random nonselfadjoint matrices and operators
, 2001
"... We describe some numerical experiments which determine the degree of spectral instability of medium size randomly generated matrices which are far from selfadjoint. The conclusion is that the eigenvalues are likely to be intrinsically uncomputable for similar matrices of a larger size. We also desc ..."
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Cited by 7 (4 self)
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We describe some numerical experiments which determine the degree of spectral instability of medium size randomly generated matrices which are far from selfadjoint. The conclusion is that the eigenvalues are likely to be intrinsically uncomputable for similar matrices of a larger size. We also describe a stochastic family of bounded operators in infinite dimensions for almost all of which the eigenvectors generate a dense linear subspace, but the eigenvalues do not determine the spectrum. Our results imply that the spectrum of the nonselfadjoint Anderson model changes suddenly as one passes to the infinite volume limit.
Pseudospectra for Matrix Pencils and Stability of Equilibria
, 1996
"... . The concept of "pseudospectra for matrices, introduced by Trefethen and his coworkers, has been studied extensively since 1990. In this paper, " pseudospectra for matrix pencils, which are relevant in connection with generalized eigenvalue problems, are considered. Some properties as well as ..."
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Cited by 6 (0 self)
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. The concept of "pseudospectra for matrices, introduced by Trefethen and his coworkers, has been studied extensively since 1990. In this paper, " pseudospectra for matrix pencils, which are relevant in connection with generalized eigenvalue problems, are considered. Some properties as well as the practical computation of "pseudospectra for matrix pencils will be discussed. As an application, we demonstrate how this concept can be used for investigating the asymptotic stability of stationary solutions to timedependent ordinary or partial differential equations; two cases, based on Burgers' equation, will be shown. Key words: "pseudospectra, "pseudospectra for matrix pencils, (generalized) eigenvalue problems, equilibria of differential equations, stability of equilibria. AMS subject classification: 65H17, 65L07, 15A18. 1 Introduction In 1990, Trefethen and his coworkers introduced the concept op "pseudospectra (see, e.g., [12, 13, 18]). For a given N \Theta N matrix A...
Computing eigenmodes of elliptic operators using radial basis functions
 Comput. Math. Appl
, 2004
"... Radial basis function (RBF) approximations have been successfully used to solve boundaryvalue problems numerically. We show that RBFs can also be used to compute eigenmodes of elliptic operators. Particular attention is given to the Laplacian operator in two dimensions, including techniques to avoi ..."
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Cited by 4 (2 self)
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Radial basis function (RBF) approximations have been successfully used to solve boundaryvalue problems numerically. We show that RBFs can also be used to compute eigenmodes of elliptic operators. Particular attention is given to the Laplacian operator in two dimensions, including techniques to avoid degradation of the solution near the boundaries. For regions with corner singularities, special functions must be added to the basis to maintain good convergence. Numerical results compare favorably to basic finite element methods.
Convexity and Lipschitz behavior of small pseudospectra
, 2006
"... Foundation Grant DMS0412049 The ɛpseudospectrum of a matrix A is the subset of the complex plane consisting of all eigenvalues of complex matrices within a distance ɛ of A, measured by the operator 2norm. Given a nonderogatory matrix A0, for small ɛ> 0, we show that the ɛpseudospectrum of any ma ..."
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Cited by 1 (0 self)
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Foundation Grant DMS0412049 The ɛpseudospectrum of a matrix A is the subset of the complex plane consisting of all eigenvalues of complex matrices within a distance ɛ of A, measured by the operator 2norm. Given a nonderogatory matrix A0, for small ɛ> 0, we show that the ɛpseudospectrum of any matrix A near A0 consists of compact convex neighborhoods of the eigenvalues of A0. Furthermore, the dependence of each of these neighborhoods on A is Lipschitz. 1