5,665 research outputs found
Weak order for the discretization of the stochastic heat equation driven by impulsive noise
Considering a linear parabolic stochastic partial differential equation
driven by impulsive space time noise, dX_t+AX_t dt= Q^{1/2}dZ_t, X_0=x_0\in H,
t\in [0,T], we approximate the distribution of X_T. (Z_t)_{t\in[0,T]} is an
impulsive cylindrical process and Q describes the spatial covariance structure
of the noise; Tr(A^{-\alpha})0 and A^\beta Q is bounded
for some \beta\in(\alpha-1,\alpha]. A discretization
(X_h^n)_{n\in\{0,1,...,N\}} is defined via the finite element method in space
(parameter h>0) and a \theta-method in time (parameter \Delta t=T/N). For
\phi\in C^2_b(H;R) we show an integral representation for the error
|E\phi(X^N_h)-E\phi(X_T)| and prove that
|E\phi(X^N_h)-E\phi(X_T)|=O(h^{2\gamma}+(\Delta t)^{\gamma}) where
\gamma<1-\alpha+\beta.Comment: 29 pages; Section 1 extended, new results in Appendix
Fast convex optimization via inertial dynamics with Hessian driven damping
We first study the fast minimization properties of the trajectories of the
second-order evolution equation where
is a smooth convex function acting on a real
Hilbert space , and , are positive parameters. This
inertial system combines an isotropic viscous damping which vanishes
asymptotically, and a geometrical Hessian driven damping, which makes it
naturally related to Newton's and Levenberg-Marquardt methods. For , , along any trajectory, fast convergence of the values
is
obtained, together with rapid convergence of the gradients
to zero. For , just assuming that has minimizers, we show that
any trajectory converges weakly to a minimizer of , and . Strong convergence is
established in various practical situations. For the strongly convex case,
convergence can be arbitrarily fast depending on the choice of . More
precisely, we have . We extend the results to the case of a general
proper lower-semicontinuous convex function . This is based on the fact that the inertial
dynamic with Hessian driven damping can be written as a first-order system in
time and space. By explicit-implicit time discretization, this opens a gate to
new possibly more rapid inertial algorithms, expanding the field of
FISTA methods for convex structured optimization problems
Weak order for the discretization of the stochastic heat equation
In this paper we study the approximation of the distribution of
Hilbert--valued stochastic process solution of a linear parabolic stochastic
partial differential equation written in an abstract form as driven by a Gaussian
space time noise whose covariance operator is given. We assume that
is a finite trace operator for some and that is
bounded from into for some . It is not required
to be nuclear or to commute with . The discretization is achieved thanks to
finite element methods in space (parameter ) and implicit Euler schemes in
time (parameter ). We define a discrete solution and for
suitable functions defined on , we show that |\E \phi(X^N_h) - \E
\phi(X_T) | = O(h^{2\gamma} + \Delta t^\gamma) \noindent where . Let us note that as in the finite dimensional case the rate of
convergence is twice the one for pathwise approximations
Lattice-Boltzmann simulations of the drag force on a sphere approaching a superhydrophobic striped plane
By means of lattice-Boltzmann simulations the drag force on a sphere of
radius R approaching a superhydrophobic striped wall has been investigated as a
function of arbitrary separation h. Superhydrophobic (perfect-slip vs. no-slip)
stripes are characterized by a texture period L and a fraction of the gas area
. For very large values of h/R we recover the macroscopic formulae for a
sphere moving towards a hydrophilic no-slip plane. For h/R=O(1) and smaller the
drag force is smaller than predicted by classical theories for hydrophilic
no-slip surfaces, but larger than expected for a sphere interacting with a
uniform perfectly slipping wall. At a thinner gap, the force reduction
compared to a classical result becomes more pronounced, and is maximized by
increasing . In the limit of very small separations our simulation data
are in quantitative agreement with an asymptotic equation, which relates a
correction to a force for superhydrophobic slip to texture parameters. In
addition, we examine the flow and pressure field and observe their oscillatory
character in the transverse direction in the vicinity of the wall, which
reflects the influence of the heterogeneity and anisotropy of the striped
texture. Finally, we investigate the lateral force on the sphere, which is
detectable in case of very small separations and is maximized by stripes with
.Comment: 9 pages, 7 figure
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