962 research outputs found
Ergodicity for SDEs and approximations: Locally Lipschitz vector fields and degenerate noise
The ergodic properties of SDEs, and various time discretizations for SDEs, are studied. The ergodicity of SDEs is established by using techniques from the theory of Markov chains on general state spaces, such as that expounded by Meyn-Tweedie. Application of these Markov chain results leads to straightforward proofs of geometric ergodicity for a variety of SDEs, including problems with degenerate noise and for problems with locally Lipschitz vector fields. Applications where this theory can be usefully applied include damped-driven Hamiltonian problems (the Langevin equation), the Lorenz equation with degenerate noise and gradient systems. The same Markov chain theory is then used to study time-discrete approximations of these SDEs. The two primary ingredients for ergodicity are a minorization condition and a Lyapunov condition. It is shown that the minorization condition is robust under approximation. For globally Lipschitz vector fields this is also true of the Lyapunov condition. However in the locally Lipschitz case the Lyapunov condition fails for explicit methods such as Euler-Maruyama; for pathwise approximations it is, in general, only inherited by specially constructed implicit discretizations. Examples of such discretization based on backward Euler methods are given, and approximation of the Langevin equation studied in some detail
Quasi-Monte Carlo and Multilevel Monte Carlo Methods for Computing Posterior Expectations in Elliptic Inverse Problems
We are interested in computing the expectation of a functional of a PDE solution under a Bayesian posterior distribution. Using Bayes's rule, we reduce the problem to estimating the ratio of two related prior expectations. For a model elliptic problem, we provide a full convergence and complexity analysis of the ratio estimator in the case where Monte Carlo, quasi-Monte Carlo, or multilevel Monte Carlo methods are used as estimators for the two prior expectations. We show that the computational complexity of the ratio estimator to achieve a given accuracy is the same as the corresponding complexity of the individual estimators for the numerator and the denominator. We also include numerical simulations, in the context of the model elliptic problem, which demonstrate the effectiveness of the approach
Adiabatic limit and the slow motion of vortices in a Chern-Simons-Schr\"odinger system
We study a nonlinear system of partial differential equations in which a
complex field (the Higgs field) evolves according to a nonlinear Schroedinger
equation, coupled to an electromagnetic field whose time evolution is
determined by a Chern-Simons term in the action. In two space dimensions, the
Chern-Simons dynamics is a Galileo invariant evolution for A, which is an
interesting alternative to the Lorentz invariant Maxwell evolution, and is
finding increasing numbers of applications in two dimensional condensed matter
field theory. The system we study, introduced by Manton, is a special case (for
constant external magnetic field, and a point interaction) of the effective
field theory of Zhang, Hansson and Kivelson arising in studies of the
fractional quantum Hall effect. From the mathematical perspective the system is
a natural gauge invariant generalization of the nonlinear Schroedinger
equation, which is also Galileo invariant and admits a self-dual structure with
a resulting large space of topological solitons (the moduli space of self-dual
Ginzburg-Landau vortices). We prove a theorem describing the adiabatic
approximation of this system by a Hamiltonian system on the moduli space. The
approximation holds for values of the Higgs self-coupling constant close to the
self-dual (Bogomolny) value of 1. The viability of the approximation scheme
depends upon the fact that self-dual vortices form a symplectic submanifold of
the phase space (modulo gauge invariance). The theorem provides a rigorous
description of slow vortex dynamics in the near self-dual limit.Comment: Minor typos corrected, one reference added and DOI give
Periodic Homogenization for Inertial Particles
We study the problem of homogenization for inertial particles moving in a
periodic velocity field, and subject to molecular diffusion. We show that,
under appropriate assumptions on the velocity field, the large scale, long time
behavior of the inertial particles is governed by an effective diffusion
equation for the position variable alone. To achieve this we use a formal
multiple scale expansion in the scale parameter. This expansion relies on the
hypo-ellipticity of the underlying diffusion. An expression for the diffusivity
tensor is found and various of its properties studied. In particular, an
expansion in terms of the non-dimensional particle relaxation time (the
Stokes number) is shown to co-incide with the known result for passive
(non-inertial) tracers in the singular limit . This requires the
solution of a singular perturbation problem, achieved by means of a formal
multiple scales expansion in Incompressible and potential fields are
studied, as well as fields which are neither, and theoretical findings are
supported by numerical simulations.Comment: 31 pages, 7 figures, accepted for publication in Physica D. Typos
corrected. One reference adde
Extracting Br(omega->pi^+ pi^-) from the Time-like Pion Form-factor
We extract the G-parity-violating branching ratio Br(omega->pi^+ pi^-) from
the effective rho-omega mixing matrix element Pi_{rho omega}(s), determined
from e^+e^- -> pi^+ pi^- data. The omega->pi^+ pi^- partial width can be
determined either from the time-like pion form factor or through the constraint
that the mixed physical propagator D_{rho omega}^{mu nu}(s) possesses no poles.
The two procedures are inequivalent in practice, and we show why the first is
preferred, to find finally Br(omega->pi^+ pi^-) = 1.9 +/- 0.3%.Comment: 12 pages (published version
Orbital stability: analysis meets geometry
We present an introduction to the orbital stability of relative equilibria of
Hamiltonian dynamical systems on (finite and infinite dimensional) Banach
spaces. A convenient formulation of the theory of Hamiltonian dynamics with
symmetry and the corresponding momentum maps is proposed that allows us to
highlight the interplay between (symplectic) geometry and (functional) analysis
in the proofs of orbital stability of relative equilibria via the so-called
energy-momentum method. The theory is illustrated with examples from finite
dimensional systems, as well as from Hamiltonian PDE's, such as solitons,
standing and plane waves for the nonlinear Schr{\"o}dinger equation, for the
wave equation, and for the Manakov system
Posterior Consistency via Precision Operators for Bayesian Nonparametric Drift Estimation in SDEs
We study a Bayesian approach to nonparametric estimation of the periodic
drift function of a one-dimensional diffusion from continuous-time data.
Rewriting the likelihood in terms of local time of the process, and specifying
a Gaussian prior with precision operator of differential form, we show that the
posterior is also Gaussian with precision operator also of differential form.
The resulting expressions are explicit and lead to algorithms which are readily
implementable. Using new functional limit theorems for the local time of
diffusions on the circle, we bound the rate at which the posterior contracts
around the true drift function
Synthesizing attractors of Hindmarsh-Rose neuronal systems
In this paper a periodic parameter switching scheme is applied to the
Hindmarsh-Rose neuronal system to synthesize certain attractors. Results show
numerically, via computer graphic simulations, that the obtained synthesized
attractor belongs to the class of all admissible attractors for the
Hindmarsh-Rose neuronal system and matches the averaged attractor obtained with
the control parameter replaced with the averaged switched parameter values.
This feature allows us to imagine that living beings are able to maintain vital
behavior while the control parameter switches so that their dynamical behavior
is suitable for the given environment.Comment: published in Nonlinear Dynamic
Global patterns of nitrate storage in the vadose zone
Global-scale nitrogen budgets developed to quantify anthropogenic impacts on the nitrogen cycle do not explicitly consider nitrate stored in the vadose zone. Here we show that the vadose zone is an important store of nitrate that should be considered in future budgets for effective policymaking. Using estimates of groundwater depth and nitrate leaching for 1900â2000, we quantify the peak global storage of nitrate in the vadose zone as 605â1814 Teragrams (Tg). Estimates of nitrate storage are validated using basin-scale and national-scale estimates and observed groundwater nitrate data. Nitrate storage per unit area is greatest in North America, China and Europe where there are thick vadose zones and extensive historical agriculture. In these areas, long travel times in the vadose zone may delay the impact of changes in agricultural practices on groundwater quality. We argue that in these areas use of conventional nitrogen budget approaches is inappropriate
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