68 research outputs found
Langevin molecular dynamics derived from Ehrenfest dynamics
Stochastic Langevin molecular dynamics for nuclei is derived from the
Ehrenfest Hamiltonian system (also called quantum classical molecular dynamics)
in a Kac-Zwanzig setting, with the initial data for the electrons
stochastically perturbed from the ground state and the ratio, , of nuclei
and electron mass tending to infinity. The Ehrenfest nuclei dynamics is
approximated by the Langevin dynamics with accuracy on bounded
time intervals and by on unbounded time intervals, which makes the small
friction and diffusion terms visible. The
initial electron probability distribution is a Gibbs density at low temperture,
derived by a stability and consistency argument: starting with any equilibrium
measure of the Ehrenfest Hamiltonian system, the initial electron distribution
is sampled from the equilibrium measure conditioned on the nuclei positions,
which after long time leads to the nuclei positions in a Gibbs distribution
(i.e. asymptotic stability); by consistency the original equilibrium measure is
then a Gibbs measure.The diffusion and friction coefficients in the Langevin
equation satisfy the Einstein's fluctuation-dissipation relation.Comment: 39 pages: modeling and analysis in separate sections. Formulation of
initial data simplifie
Monte Carlo Euler approximations of HJM term structure financial models
We present Monte Carlo-Euler methods for a weak approximation problem related
to the Heath-Jarrow-Morton (HJM) term structure model, based on \Ito stochastic
differential equations in infinite dimensional spaces, and prove strong and
weak error convergence estimates. The weak error estimates are based on
stochastic flows and discrete dual backward problems, and they can be used to
identify different error contributions arising from time and maturity
discretization as well as the classical statistical error due to finite
sampling. Explicit formulas for efficient computation of sharp error
approximation are included. Due to the structure of the HJM models considered
here, the computational effort devoted to the error estimates is low compared
to the work to compute Monte Carlo solutions to the HJM model. Numerical
examples with known exact solution are included in order to show the behavior
of the estimates
An a posteriori error estimate for Symplectic Euler approximation of optimal control problems
This work focuses on numerical solutions of optimal control problems. A time
discretization error representation is derived for the approximation of the
associated value function. It concerns Symplectic Euler solutions of the
Hamiltonian system connected with the optimal control problem. The error
representation has a leading order term consisting of an error density that is
computable from Symplectic Euler solutions. Under an assumption of the pathwise
convergence of the approximate dual function as the maximum time step goes to
zero, we prove that the remainder is of higher order than the leading error
density part in the error representation. With the error representation, it is
possible to perform adaptive time stepping. We apply an adaptive algorithm
originally developed for ordinary differential equations. The performance is
illustrated by numerical tests
Langevin molecular dynamics derived from Ehrenfest dynamics
Abstract. Stochastic Langevin molecular dynamics for nuclei is derived from the Ehrenfest Hamiltonian system (also called quantum classical molecular dynamics) in a Kac-Zwanzig setting, with the initial data for the electrons stochastically perturbed from the ground state and the ratio, M , of nuclei and electron mass tending to infinity. The Ehrenfest nuclei dynamics is approximated by the Langevin dynamics with accuracy o(M −1/2 ) on bounded time intervals and by o(1) on unbounded time intervals, which makes the small O(M −1/2 ) friction and o(M −1/2 ) diffusion terms visible. The initial electron probability distribution is a Gibbs density at low temperture, derived by a stability and consistency argument: starting with any equilibrium measure of the Ehrenfest Hamiltonian system, the initial electron distribution is sampled from the equilibrium measure conditioned on the nuclei positions, which after long time leads to the nuclei positions in a Gibbs distribution (i.e. asymptotic stability); by consistency the original equilibrium measure is then a Gibbs measure. The diffusion and friction coefficients in the Langevin equation satisfy the Einstein's fluctuation-dissipation relation
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