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153
Semiclassical analysis for the KramersFokkerPlanck equation
 Comm. PDE
"... On étudie des estimations semiclassiques sur la résolvente d’opérateurs qui ne sont ni elliptiques ni autoadjoints, que l’on utilise pour étudier le problème de Cauchy. En particulier on obtient une description précise du spectre pres de l’axe imaginaire, et des estimations de résolvente à l’intérie ..."
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Cited by 37 (15 self)
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On étudie des estimations semiclassiques sur la résolvente d’opérateurs qui ne sont ni elliptiques ni autoadjoints, que l’on utilise pour étudier le problème de Cauchy. En particulier on obtient une description précise du spectre pres de l’axe imaginaire, et des estimations de résolvente à l’intérieur du pseudospectre. On applique ensuite les résultats à l’opérateur de KramersFokkerPlanck. We study some accurate semiclassical resolvent estimates for operators that are neither selfadjoint nor elliptic, and applications to the Cauchy problem. In particular we get a precise description of the spectrum near the imaginary axis and precise resolvent estimates inside the pseudospectrum. We apply our results to the KramersFokkerPlanck operator.
Two Numerical Methods for Optimizing Matrix Stability
 Linear Algebra Appl
, 2001
"... Consider the ane matrix family A(x) = A 0 + k=1 x k A k , mapping a design vector x 2 R into the space of n n real matrices. ..."
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Cited by 35 (8 self)
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Consider the ane matrix family A(x) = A 0 + k=1 x k A k , mapping a design vector x 2 R into the space of n n real matrices.
Optimization and pseudospectra, with applications to robust stability
 SIAM Journal on Matrix Analysis and Applications
, 2003
"... ..."
Linearized pipe flow to Reynolds number 10^7
, 2003
"... A Fourier–Chebyshev Petrov–Galerkin spectral method is described for highaccuracy computation of linearized dynamics for flow in an infinite circular pipe. Our code is unusual in being based on solenoidal velocity variables and in being written in MATLAB. Systematic studies are presented of the dep ..."
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Cited by 29 (1 self)
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A Fourier–Chebyshev Petrov–Galerkin spectral method is described for highaccuracy computation of linearized dynamics for flow in an infinite circular pipe. Our code is unusual in being based on solenoidal velocity variables and in being written in MATLAB. Systematic studies are presented of the dependence of eigenvalues, transient growth factors, and other quantities on the axial and azimuthal wave numbers and the Reynolds number R for R ranging from 10 2 to the idealized (physically unrealizable) value 10 7. Implications for transition to turbulence are considered in the light of recent theoretical results of S.J. Chapman.
Optimizing matrix stability
 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
, 2000
"... Given an affine subspace of square matrices, we consider the problem of minimizing the spectral abscissa (the largest real part of an eigenvalue). We give an example whose optimal solution has Jordan form consisting of a single Jordan block, and we show, using nonlipschitz variational analysis, th ..."
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Cited by 27 (15 self)
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Given an affine subspace of square matrices, we consider the problem of minimizing the spectral abscissa (the largest real part of an eigenvalue). We give an example whose optimal solution has Jordan form consisting of a single Jordan block, and we show, using nonlipschitz variational analysis, that this behaviour persists under arbitrary small perturbations to the example. Thus although matrices with nontrivial Jordan structure are rare in the space of all matrices, they appear naturally in spectral abscissa minimization.
Growth and Decay of Random Fibonacci Sequences
 London Proceedings, Series A, Mathematical, Physical and Engineering Sciences
, 1999
"... Introduction In a remarkable recent paper, Viswanath (1998) has considered the large n behaviour of solutions to the `random Fibonacci recurrence' x n+1 = \Sigma x n \Sigma x n\Gamma1 ; (1.1) where the signs are chosen independently and with equal probabilities, and x 0 = x 1 = 1. Computer ex ..."
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Cited by 22 (2 self)
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Introduction In a remarkable recent paper, Viswanath (1998) has considered the large n behaviour of solutions to the `random Fibonacci recurrence' x n+1 = \Sigma x n \Sigma x n\Gamma1 ; (1.1) where the signs are chosen independently and with equal probabilities, and x 0 = x 1 = 1. Computer experiments, as in figure 1, show exponential growth with n. The problem of large n behaviour of (1.1) has been mentioned at least since 1963, when Furstenberg (1963) established exponential growth with probability 1, but Viswanath's contribution represents an intriguing new development. By an ingenious application of a SternBrocot tree (Graham et al. 1994), he proved that solutions to (1.1) satisfy lim n!1 jx n j<F8.064
Nonselfadjoint harmonic oscillator, compact semigroups and pseudospectra
 J. Operator Theory
"... We provide new information concerning the pseudospectra of the complex harmonic oscillator. Our analysis illustrates two different techniques for getting resolvent norm estimates. The first uses the JWKB method and extends for this particular potential some results obtained recently by E.B. Davies. ..."
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Cited by 21 (2 self)
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We provide new information concerning the pseudospectra of the complex harmonic oscillator. Our analysis illustrates two different techniques for getting resolvent norm estimates. The first uses the JWKB method and extends for this particular potential some results obtained recently by E.B. Davies. The second relies on the fact that the bounded holomorphic semigroup generated by the complex harmonic oscillator is of HilbertSchmidt type in a maximal angular region. In order to show this last property, we deduce a nonselfadjoint version of the classical Mehler’s formula.
Numerical study of quantum resonances in chaotic scattering
 J. Comp. Phys
"... This paper presents numerical evidence that for quantum systems with chaotic classical dynamics, the number of scattering resonances near an energy E scales like ¯h − D(K E)+1 2 as ¯h → 0. Here, KE denotes the subset of the classical energy surface {H = E} which stays bounded for all time under the ..."
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This paper presents numerical evidence that for quantum systems with chaotic classical dynamics, the number of scattering resonances near an energy E scales like ¯h − D(K E)+1 2 as ¯h → 0. Here, KE denotes the subset of the classical energy surface {H = E} which stays bounded for all time under the flow generated by the Hamiltonian H and D (KE) denotes its fractal dimension. Since the number of bound states in a quantum system with n degrees of freedom scales like ¯h −n, this suggests that the quantity D(KE)+1 2 represents the effective number of degrees of freedom in scattering problems. 1
Numerical Linear Algebra And Solvability Of Partial Differential Equations
 Comm. Math. Phys
, 2001
"... . It was observed long ago that the obstruction to the accurate computation of eigenvalues of large nonselfadjoint matrices is inherent in the problem. The basic idea is that any algorithm for locating the eigenvalues will also nd some `false eigenvalues'. These false eigenvalues also explain ..."
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Cited by 20 (4 self)
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. It was observed long ago that the obstruction to the accurate computation of eigenvalues of large nonselfadjoint matrices is inherent in the problem. The basic idea is that any algorithm for locating the eigenvalues will also nd some `false eigenvalues'. These false eigenvalues also explain one of the most surprising phenomena in linear PDEs, namely the fact (discovered by Hans Lewy in 1957, here in Berkeley) that one cannot always locally solve the PDE Pu = f . Almost immediately after that discovery, Hormander provided an explanation of Lewy's example showing that almost all operators with nonconstanct complex valued coecients are not locally solvable. In modern language, that was done by considering the essentially dual problem of existence of nonpropagating singularities. The purpose of this talk is to review this work in the context of \almost eigenvalues" and from the point of view of semiclassical analysis. The purpose of this talk is to describe a connection between ...