133 research outputs found
Computing the diffusivity of a particle subject to dry friction with colored noise
This paper considers the motion of an object subjected to dry friction and an
external random force. The objective is to characterize the role of the
correlation time of the external random force. We develop efficient stochastic
simulation methods for computing the diffusivity (the linear growth rate of the
variance of the displacement) and other related quantities of interest when the
external random force is white or colored. These methods are based on original
representation formulas for the quantities of interest which make it possible
to build unbiased and consistent estimators. The numerical results obtained
with these original methods are in perfect agreement with known closed-form
formulas valid in the white noise regime. In the colored noise regime the
numerical results show that the predictions obtained from the white-noise
approximation are reasonable for quantities such as the histograms of the
stationary velocity but can be wrong for the diffusivity unless the correlation
time is extremely small
Behavior of the plastic deformation of an elasto-perfectly-plastic oscillator with noise.
International audienceEarlier works in engineering, partly experimental, partly computational have revealed that asymptotically, when the excitation is a white noise, plastic deformation and total deformation for an elasto-perfectly-plastic oscillator have a variance which increases linearly with time with the same coefficient. In this work, we prove this result and we characterize the corresponding drift coefficient. Our study relies on a stochastic variational inequality governing the evolution between the velocity of the oscillator and the non-linear restoring force. We then define long cycles behavior of the Markov process solution of the stochastic variational inequality which is the key concept to obtain the result. An important question in engineering is to compute this coefficient. Also, we provide numerical simulations which show successful agreement with our theoretical prediction and previous empirical studies made by engineers
Membrane imaging by simultaneous second-harmonic generation and two-photon microscopy
International audienceWe demonstrate that simultaneous second-harmonic generation (SHG) and two-photon-excited fluorescence (TPEF) can be used to rapidly image biological membranes labeled with a styryl dye. The SHG power is made compatible with the TPEF power by use of near-resonance excitation, in accord with a model based on the theory of phased-array antennas, which shows that the SHG radiation is highly structured. Because of its sensitivity to local asymmetry, SHG microscopy promises to be a powerful tool for the study of membrane dynamics
An analytic approach to the ergodic theory of a stochastic variational inequality
International audienceIn an earlier work made by the first author with J. Turi (Degenerate Dirichlet Problems Related to the Invariant Measure of Elasto-Plastic Oscillators, AMO, 2008), the solution of a stochastic variational inequality modeling an elasto-perfectly-plastic oscillator has been studied. The existence and uniqueness of an invariant measure have been proven. Nonlocal problems have been introduced in this context. In this work, we present a new characterization of the invariant measure. The key finding is the connection between nonlocal PDEs and local PDEs which can be interpreted with short cycles of the Markov process solution of the stochastic variational inequality
Mortensen Observer for a class of variational inequalities -Lost equivalence with stochastic filtering approaches
We address the problem of deterministic sequential estimation for a nonsmooth
dynamics in R governed by a variational inequality, as illustrated by the
Skorokhod problem with a reflective boundary condition at 0. For smooth
dynamics, Mortensen introduced an energy for the likelihood that the state
variable produces-up to perturbations disturbances-a given observation in a
finite time interval, while reaching a given target state at the final time.
The Mortensen observer is the minimiser of this energy. For dynamics given by a
variational inequality and therefore not reversible in time, we study the
definition of a Mortensen estimator. On the one hand, we address this problem
by relaxing the boundary constraint of the synthetic variable and then
proposing an approximated variant of the Mortensen estimator that uses the
resulting nonlinear smooth dynamics. On the other hand, inspired by the smooth
dynamics approach, we study the vanishing viscosity limit of the
Hamilton-Jacobi equation satisfied by the Hopf-Cole transform of the solution
of the robust Zakai equation. We prove a stability result that allows us to
interpret the limiting solution as the value function associated with a control
problem rather than an estimation problem. In contrast to the case of smooth
dynamics, here the zero-noise limit of the robust form of the Zakai equation
cannot be understood from the Bellman equation of the value function arising in
Mortensen's deterministic estimation. This may unveil a violation of
equivalence for non-reversible dynamics between the Mortensen approach and the
low noise stochastic approach for nonsmooth dynamics
Deformation of geometry and bifurcation of vortex rings
We construct a smooth family of Hamiltonian systems, together with a family
of group symmetries and momentum maps, for the dynamics of point vortices on
surfaces parametrized by the curvature of the surface. Equivariant bifurcations
in this family are characterized, whence the stability of the Thomson heptagon
is deduced without recourse to the Birkhoff normal form, which has hitherto
been a necessary tool.Comment: 26 page
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