arXiv subjects

Dirk Blömker

Publications and source records attributed to Dirk Blömker.

35 records · Page 2Linked to original sources

arXiv2018

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.

math.PR↗

arXiv2017

Numerically Computable A Posteriori-Bounds for stochastic Allen-Cahn equation

The aim of this paper is the derivation of an a-posteriori error estimate for a numerical method based on an exponential scheme in time and spectral Galerkin methods in space. We obtain analytically a rigorous bound on the mean square error conditioned to the calculated data, which is numerically computable and uses the given numerical approximation. Thus one can check a-posteriori the error for a given numerical computation without relying on an asymptotic result. All estimates are only based on the numerical data and the structure of the equation, but they do not use any a-priori information of the solution, which makes the approach applicable to equations where global existence of solutions is not known. For simplicity of presentation, we develop the method here in a relatively simple situation of a stable one-dimensional Allen-Cahn equation with additive forcing.

math.NA↗

arXiv2017

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.

math.AP↗

arXiv2017

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.

math.PR↗

arXiv2016

The Sharp interface limit for the stochastic Cahn-Hilliard equation

We study the two and three dimensional stochastic Cahn-Hilliard equation in the sharp interface limit, where the positive parameter $ε$ tends to zero, which measures the width of transition layers generated during phase separation. We also couple the noise strength to this parameter. Using formal asymptotic expansions, we identify the limit. In the right scaling we indicate that the solutions of stochastic Cahn-Hilliard converge to a solution of a Hele-Shaw problem with stochastic forcing. In the case when the noise is sufficiently small, we rigorously prove that the limit is a deterministic Hele-Shaw problem. Finally, we discuss which estimates are necessary in order to extend the rigorous result to larger noise strength.

math.AP↗

arXiv2016

Modulation Equation for SPDEs in unbounded domains with space-time white noise -- Linear Theory

We study the approximation of SPDEs on the whole real line near a change of stability via modulation or amplitude equations, which acts as a replacement for the lack of random invariant manifolds on extended domains. Due to the unboundedness of the underlying domain a whole band of infinitely many eigenfunctions changes stability. Thus we expect not only a slow motion in time, but also a slow spatial modulation of the dominant modes, which is described by the modulation equation. As a first step towards a full theory of modulation equations for nonlinear SPDEs on unbounded domains, we focus, in the results presented here, on the linear theory for one particular example, the Swift-Hohenberg equation. These linear results are one of the key technical tools to carry over the deterministic approximation results to the stochastic case with additive forcing. One technical problem for establishing error estimates rises from the spatially translation invariant nature of space-time white noise on unbounded domains, which implies that at any time we can expect the error to be always very large somewhere in space.

math.PR↗

arXiv2016

Additive noise destroys the random attractor close to bifurcation

We provide an example for stabilization by noise. Our approach does not rely on monotonicity arguments due to the presence of higher order differential operators or mixing properties of the system as the noise might be highly degenerate. In the examples a scalar additive noise destroys a high-dimensional random attractor of a PDE on an unbounded domain. In the presence of small noise close to bifurcation all trajectories converge to a single stationary solution.

math.DS↗

arXiv2016

Probabilistic Estimates of the Maximum Norm of Random Neumann Fourier Series

We study the maximum norm behavior of $L^2$-normalized random Fourier cosine series with a prescribed large wave number. Precise bounds of this type are an important technical tool in estimates for spinodal decomposition, the celebrated phase separation phenomenon in metal alloys. We derive rigorous asymptotic results as the wave number converges to infinity, and shed light on the behavior of the maximum norm for medium range wave numbers through numerical simulations. Finally, we develop a simplified model for describing the magnitude of extremal values of random Neumann Fourier series. The model describes key features of the development of maxima and can be used to predict them. This is achieved by decoupling magnitude and sign distribution, where the latter plays an important role for the study of the size of the maximum norm. Since we are considering series with Neumann boundary conditions, particular care has to be placed on understanding the behavior of the random sums at the boundary.

math.PR↗

arXiv2015

Pattern size in Gaussian fields from spinodal decomposition

We study the two-dimensional snake-like pattern that arises in phase separation of alloys described by spinodal decomposition in the Cahn-Hilliard model. These are somewhat universal patterns due to an overlay of eigenfunctions of the Laplacian with a similar wave-number. Similar structures appear in other models like reaction-diffusion systems describing animal coats' patterns or vegetation patterns in desertification. Our main result studies random functions given by cosine Fourier series with independent Gaussian coefficients, that dominate the dynamics in the Cahn-Hilliard model. This is not a cosine process, as the sum is taken over domains in Fourier space that not only grow and scale with a parameter of order $1/\varepsilon$, but also move to infinity. Moreover, the model under consideration is neither stationary nor isotropic. To study the pattern size of nodal domains we consider the density of zeros on any straight line through the spatial domain. Using a theorem by Edelman and Kostlan and weighted ergodic theorems that ensure the convergence of the moving sums, we show that the average distance of zeros is asymptotically of order $\varepsilon$ with a precisely given constant.

math.PR↗

arXiv2015

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 solution

math.AP↗

arXiv2015

Motion of a droplet for the mass-conserving stochastic Allen-Cahn equation

We study the stochastic mass-conserving Allen-Cahn equation posed on a bounded two-dimensional domain with additive spatially smooth space-time noise. This equation associated with a small positive parameter describes the stochastic motion of a small almost semicircular droplet attached to domain's boundary and moving towards a point of locally maximum curvature. We apply Itô calculus to derive the stochastic dynamics of the droplet by utilizing the approximately invariant manifold introduced by Alikakos, Chen and Fusco for the deterministic problem. In the stochastic case depending on the scaling, the motion is driven by the change in the curvature of the boundary and the stochastic forcing. Moreover, under the assumption of a sufficiently small noise strength, we establish stochastic stability of a neighborhood of the manifold of droplets in $L^2$ and $H^1$, which means that with overwhelming probability the solution stays close to the manifold for very long time-scales.

math.DS↗

arXiv2014

Fast Diffusion Limit for Reaction-Diffusion Systems with Stochastic Neumann Boundary Conditions

We consider a class of reaction-diffusion equations with a stochastic perturbation on the boundary. We show that in the limit of fast diffusion, one can rigorously approximate solutions of the system of PDEs with stochastic Neumann boundary conditions by the solution of a suitable stochastic/deterministic differential equation for the average concentration that involves reactions only. An interesting effect occurs, if the noise on the boundary does not change the averaging concentration, but is sufficiently large. Then surprising additional effective reaction terms appear. We focus on systems with polynomial nonlinearities only and give applications to the two dimensional nonlinear heat equation and the cubic auto-catalytic reaction between two chemicals.

math.AP↗

arXiv2013

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.

math.NA↗

arXiv2013

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.

math.PR↗

arXiv2013

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.

math.NA↗

arXiv2006

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.

math.CA↗

arXiv2002

Stationary solutions for a model of amorphous thin-film growth

We consider a class of stochastic partial differential equations arising as a model for amorphous thin film growth. Using a spectral Galerkin method, we verify the existence of stationary mild solutions, although the specific nature of the nonlinearity prevents us from showing the uniqueness of the solutions as well as their boundedness (in time).

math-ph↗