141 research outputs found
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
Random initial conditions for semi-linear PDEs
We analyze the effect of random initial conditions on the local
well--posedness of semi--linear PDEs, to investigate to what extent recent
ideas on singular stochastic PDEs can prove useful in this framework
Rigorous Numerical Verification of Uniqueness and Smoothness in a Surface Growth Model
Based on numerical data and a-posteriori analysis we verify rigorously the
uniqueness and smoothness of global solutions to a scalar surface growth model
with striking similarities to the 3D Navier--Stokes equations, for certain
initial data for which analytical approaches fail. The key point is the
derivation of a scalar ODE controlling the norm of the solution, whose
coefficients depend on the numerical data. Instead of solving this ODE
explicitly, we explore three different numerical methods that provide rigorous
upper bounds for its solutio
Galerkin approximations for the stochastic Burgers equation
Existence and uniqueness for semilinear stochastic evolution equations with
additive noise by means of finite dimensional Galerkin approximations is
established and the convergence rate of the Galerkin approximations to the
solution of the stochastic evolution equation is estimated. These abstract
results are applied to several examples of stochastic partial differential
equations (SPDEs) of evolutionary type including a stochastic heat equation, a
stochastic reaction diffusion equation and a stochastic Burgers equation. The
estimated convergence rates are illustrated by numerical simulations. The main
novelty in this article is to estimate the difference of the finite dimensional
Galerkin approximations and of the solution of the infinite dimensional SPDE
uniformly in space, i.e., in the L^{\infty}-topology, instead of the usual
Hilbert space estimates in the L^2-topology, that were shown before.Comment: 22 page
Stabilization by rough noise for an epitaxial growth model
In this article we study a model from epitaxial thin-film growth. It was
originally introduced as a phenomenological model of growth in the presence of
a Schwoebbel barrier, where diffusing particles on a terrace are not allowed to
jump down at the boundary.
Nevertheless, we show that the presence of arbitrarily small space-time white
noise due to fluctuations in the incoming particles surprisingly eliminates all
nonlinear interactions in the model and thus has the potential to stabilize the
dynamics and suppress the growth of hills in these models
Bifurcation theory for SPDEs: finite-time Lyapunov exponents and amplitude equations
We consider a stochastic partial differential equation close to bifurcation
of pitchfork type, where a one-dimensional space changes its stability. For
finite-time Lyapunov exponents we characterize regions depending on the
distance from bifurcation and the noise strength where finite-time Lyapunov
exponents are positive and thus detect bifurcations. One technical tool is the
reduction of the essential dynamics of the infinite dimensional stochastic
system to a simple ordinary stochastic differential equation, which is valid
close to the bifurcation.Comment: 32 page
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