239 research outputs found
Modulation Equations: Stochastic Bifurcation in Large Domains
We consider the stochastic Swift-Hohenberg equation on a large domain near
its change of stability. We show that, under the appropriate scaling, its
solutions can be approximated by a periodic wave, which is modulated by the
solutions to a stochastic Ginzburg-Landau equation. We then proceed to show
that this approximation also extends to the invariant measures of these
equations
Rigorous a-posteriori analysis using numerical eigenvalue bounds in a surface growth model
In order to prove numerically the global existence and uniqueness of smooth
solutions of a fourth order, nonlinear PDE, we derive rigorous a-posteriori
upper bounds on the supremum of the numerical range of the linearized operator.
These bounds also have to be easily computable in order to be applicable to our
rigorous a-posteriori methods, as we use them in each time-step of the
numerical discretization. The final goal is to establish global bounds on
smooth local solutions, which then establish global uniqueness.Comment: 19 pages, 9 figure
Local existence and uniqueness for a two-dimensional surface growth equation with space--time white noise
We study local existence and uniqueness for a surface growth model with
space-time white noise in 2D. Unfortunately, the direct fixed-point argument
for mild solutions fails here, as we do not have sufficient regularity for the
stochastic forcing. Nevertheless, one can give a rigorous meaning to the
stochastic PDE and show uniqueness of solutions in that setting. Using spectral
Galerkin method and any other types of regularization of the noise, we obtain
always the same solution
Numerical Solution of Stochastic Partial Differential Equations with Correlated Noise
In this paper we investigate the numerical solution of stochastic partial
differential equations (SPDEs) for a wider class of stochastic equations. We
focus on non-diagonal colored noise instead of the usual space-time white
noise. By applying a spectral Galerkin method for spatial discretization and a
numerical scheme in time introduced by Jentzen Kloeden, we obtain the rate
of path-wise convergence in the uniform topology. The main assumptions are
either uniform bounds on the spectral Galerkin approximation or uniform bounds
on the numerical data. Numerical examples illustrate the theoretically
predicted convergence rate
Predictability of the Burgers dynamics under model uncertainty
Complex systems may be subject to various uncertainties. A great effort has
been concentrated on predicting the dynamics under uncertainty in initial
conditions. In the present work, we consider the well-known Burgers equation
with random boundary forcing or with random body forcing. Our goal is to
attempt to understand the stochastic Burgers dynamics by predicting or
estimating the solution processes in various diagnostic metrics, such as mean
length scale, correlation function and mean energy. First, for the linearized
model, we observe that the important statistical quantities like mean energy or
correlation functions are the same for the two types of random forcing, even
though the solutions behave very differently. Second, for the full nonlinear
model, we estimate the mean energy for various types of random body forcing,
highlighting the different impact on the overall dynamics of space-time white
noises, trace class white-in-time and colored-in-space noises, point noises,
additive noises or multiplicative noises
A strongly convergent numerical scheme from Ensemble Kalman inversion
The Ensemble Kalman methodology in an inverse problems setting can be viewed
as an iterative scheme, which is a weakly tamed discretization scheme for a
certain stochastic differential equation (SDE). Assuming a suitable
approximation result, dynamical properties of the SDE can be rigorously pulled
back via the discrete scheme to the original Ensemble Kalman inversion.
The results of this paper make a step towards closing the gap of the missing
approximation result by proving a strong convergence result in a simplified
model of a scalar stochastic differential equation. We focus here on a toy
model with similar properties than the one arising in the context of Ensemble
Kalman filter. The proposed model can be interpreted as a single particle
filter for a linear map and thus forms the basis for further analysis. The
difficulty in the analysis arises from the formally derived limiting SDE with
non-globally Lipschitz continuous nonlinearities both in the drift and in the
diffusion. Here the standard Euler-Maruyama scheme might fail to provide a
strongly convergent numerical scheme and taming is necessary. In contrast to
the strong taming usually used, the method presented here provides a weaker
form of taming.
We present a strong convergence analysis by first proving convergence on a
domain of high probability by using a cut-off or localisation, which then
leads, combined with bounds on moments for both the SDE and the numerical
scheme, by a bootstrapping argument to strong convergence
Enstrophy Dynamics of Stochastically Forced Large-Scale Geophysical Flows
Enstrophy is an averaged measure of fluid vorticity. This quantity is
particularly important in {\em rotating} geophysical flows. We investigate the
dynamical evolution of enstrophy for large-scale quasi-geostrophic flows under
random wind forcing. We obtain upper bounds on the enstrophy, as well as
results establishing its H\"older continuity and describing the small-time
asymptotics
Multiscale Analysis for SPDEs with Quadratic Nonlinearities
In this article we derive rigorously amplitude equations for stochastic PDEs
with quadratic nonlinearities, under the assumption that the noise acts only on
the stable modes and for an appropriate scaling between the distance from
bifurcation and the strength of the noise. We show that, due to the presence of
two distinct timescales in our system, the noise (which acts only on the fast
modes) gets transmitted to the slow modes and, as a result, the amplitude
equation contains both additive and multiplicative noise.
As an application we study the case of the one dimensional Burgers equation
forced by additive noise in the orthogonal subspace to its dominant modes. The
theory developed in the present article thus allows to explain theoretically
some recent numerical observations from [Rob03]
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