Pseudospectra of semiclassical (pseudo‐) differential operators

Dencker, Nils; Sjöstrand, Johannes; Zworski, Maciej

doi:10.1002/cpa.20004

Cited by 142 publications

(287 citation statements)

References 29 publications

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“…is not self-adjoint, it is known (see [40,8,14,19,17,18]) that the information about the spectrum is a first step in estimating the exponential decay of the 29 semigroup, but that it has to be completed by estimates on the norm of the resolvent. This will be carried out by using a weighted L 2 -norm associated with the constructions of the matrices Q and J introduced in Section 3.…”

Section: General Initial Densitiesmentioning

confidence: 99%

“…so that ψ ∞ (x) dx is still the invariant measure of the dynamics (8). A general way to construct such a b is to consider…”

Section: Non-reversible Diffusionmentioning

confidence: 99%

“…For a given S, the question is thus how to choose J in order to optimize the rate of convergence to equilibrium for the dynamics (8), which in our case becomes:…”

Section: Outline Of the Papermentioning

confidence: 99%

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Optimal Non-reversible Linear Drift for the Convergence to Equilibrium of a Diffusion

2013

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We consider non-reversible perturbations of reversible diffusions that do not alter the invariant distribution and we ask whether there exists an optimal perturbation such that the rate of convergence to equilibrium is maximized. We solve this problem for the case of linear drift by proving the existence of such optimal perturbations and by providing an easily implementable algorithm for constructing them. We discuss in particular the role of the prefactor in the exponential convergence estimate. Our rigorous results are illustrated by numerical experiments.Keywords Non-reversible diffusion · Convergence to equilibrium · Wick calculus 1 Introduction MotivationThe problem of convergence to equilibrium for diffusion processes has attracted considerable attention in recent years. In addition to the relevance of this problem for the convergence to equilibrium of some systems in statistical T. Lelièvre at Université Paris-Est, Cermics, and INRIA, MicMac project team, Ecole des ponts, 6-8 avenue Blaise Pascal, 77455 Marne la Vallée Cedex 2, France. E-mail: lelievre@cermics.enpc.fr · F. Nier at IRMAR, Campus de Beaulieu, Université de Rennes 1, 35042 Rennes, France. E-mail: francis.nier@univ-rennes1.fr · G.A. Pavliotis at Imperial College London, Department of Mathematics, South Kensington Campus, London SW7 2AZ, England. E-mail: g.pavliotis@imperial.ac.uk 2 physics, see for example [29], such questions are also important in statistics, for example in the analysis of Markov Chain Monte Carlo (MCMC) algorithms [9]. Roughly speaking, one measure of efficiency of an MCMC algorithm is its rate of convergence to equilibrium, and increasing this rate is thus the aim of many numerical techniques (see for example [5]).Let us recall the basic approach for a reversible diffusion. Suppose that we are interested in sampling from a probability distribution functionwhere V : R N → R is a given smooth potential such that R N e −V dx < ∞ . A natural dynamics to use is the reversible dynamicswhere W t denotes a standard N -dimensional Brownian motion. Let us denote by ψ t the probability density function of the process X t at time t. It satisfies the Fokker-Planck equationUnder appropriate assumptions on the potential V (e.g. that 1 2 |∇V (x)| 2 − ∆V (x) → +∞ as |x| → +∞ , see [41, A.19]), the density ψ ∞ satisfies a Poincaré inequality: there exists λ > 0 such that for all probability density functions φ,The optimal parameter λ in (4) is the opposite of the smallest (in absolute value) non-zero eigenvalue of the Fokker-Planck operator ∇ · (∇V · +∇·), which is self-adjoint in L 2 (R N , ψ −1 ∞ dx) (see (7) below). Thus, λ is also called the spectral gap of the Fokker-Planck operator.It is then standard to show that (4) is equivalent to the following inequality, which shows exponential convergence to the equilibrium for (2): for all initial conditions ψ 0 ∈ L 2 (R N , ψ −1 ∞ dx), for all times t ≥ 0,where · L 2 (ψ −1 ∞ ) denotes the norm in L 2 (R N , ψ −1 ∞ ), namely f 2∞ (x) dx . This equivalence is a simple consequence of th...

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Section: General Initial Densitiesmentioning

confidence: 99%

“…so that ψ ∞ (x) dx is still the invariant measure of the dynamics (8). A general way to construct such a b is to consider…”

Section: Non-reversible Diffusionmentioning

confidence: 99%

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Optimal Non-reversible Linear Drift for the Convergence to Equilibrium of a Diffusion

2013

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“…This is certainly not a new field, as the nonselfadjoint spectral theory of Toeplitz and Wiener-Hopf operators has been studied for about a century by many mathematicians and physicists; however, it is an expanding area of mathematics. The growing interest in non-Hermitian quantum mechanics [8,7,36,35], nonselfadjoint differential operators [23,27] and in general nonnormal phenomena [22,49,50,48] has made nonselfadjoint operators and pseudospectral theory indispensable. Now returning to the main question, namely, can one compute spectra of arbitrary operators, we need to be more precise regarding the mathematical meaning of this.…”

Section: Introductionmentioning

confidence: 99%

On the Solvability Complexity Index, the 𝑛-pseudospectrum and approximations of spectra of operators

Hansen

2010

J. Amer. Math. Soc.

127

120

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We show that it is possible to compute spectra and pseudospectra of linear operators on separable Hilbert spaces given their matrix elements. The core in the theory is pseudospectral analysis and in particular the n n -pseudospectrum and the residual pseudospectrum. We also introduce a new classification tool for spectral problems, namely, the Solvability Complexity Index. This index is an indicator of the “difficultness” of different computational spectral problems.

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“…A solution to (1.2) will be called a quasimode for z. We restrict our attention to the case where X is constant so that there are no quasimodes given by the results of [4] and, moreover, the operator is normal when acting on L 2 (R d ).…”

Section: Introductionmentioning

confidence: 99%

Pseudospectra of semiclassical boundary value problems

Galkowski¹

2014

J. Inst. Math. Jussieu

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Abstract. We consider operators −∆ + X, where X is a constant vector field, in a bounded domain and show spectral instability when the domain is expanded by scaling. More generally, we consider semiclassical elliptic boundary value problems which exhibit spectral instability for small values of the semiclassical parameter h, which should be thought of as the reciprocal of the Peclet constant. This instability is due to the presence of the boundary: just as in the case of −∆ + X, some of our operators are normal when considered on R d . We characterize the semiclassical pseudospectrum of such problems as well as the areas of concentration of quasimodes. As an application, we prove a result about exit times for diffusion processes in bounded domains. We also demonstrate instability for a class of spectrally stable nonlinear evolution problems that are associated to these elliptic operators.

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Pseudospectra of semiclassical (pseudo‐) differential operators

Cited by 142 publications

References 29 publications

Optimal Non-reversible Linear Drift for the Convergence to Equilibrium of a Diffusion

Optimal Non-reversible Linear Drift for the Convergence to Equilibrium of a Diffusion

On the Solvability Complexity Index, the 𝑛-pseudospectrum and approximations of spectra of operators

Pseudospectra of semiclassical boundary value problems

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