Dirk Blömker
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On the approximation of finite-time Lyapunov exponents for the stochastic Burgers equation Open
We analyze stochastic partial differential equations (SPDEs) with quadratic nonlinearities close to a change of stability. To this aim we compute finite-time Lyapunov exponents (FTLEs), observing a change of sign based on the interplay bet…
Effective dynamics of interfaces for nonlinear SPDEs driven by multiplicative white noise Open
In the present work, we investigate the dynamics of the infinite-dimensional stochastic partial differential equation (SPDE) with multiplicative white noise. We derive the effective equation on the approximate slow manifold in detail by ut…
Numerical approximation of nonlinear fourth-order SPDEs with additive space-time white noise Open
We consider the strong numerical approximation for a fourth-order stochastic nonlinear SPDE driven by space-time white noise on $2$-dimensional torus. We consider its full discretisation with a spectral Galerkin scheme in space and Euler s…
Finite-Time Lyapunov Exponents for SPDEs with Fractional Noise Open
We estimate the finite-time Lyapunov exponents for a stochastic partial differential equation driven by a fractional Brownian motion (fbm) with Hurst index $$H\in (0,1)$$ close to a bifurcation of pitchfork type. We characterize regions de…
Stabilization by rough noise for an epitaxial growth model Open
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 do…
Finite-time Lyapunov exponents for SPDEs with fractional noise Open
We estimate the finite-time Lyapunov exponents for a stochastic partial differential equation driven by a fractional Brownian motion (fbm) with Hurst index $H\in(0,1)$ close to a bifurcation of pitchfork type. We characterize regions depen…
Bifurcation theory for SPDEs: finite-time Lyapunov exponents and amplitude equations Open
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 b…
Continuous Time Limit of the Stochastic Ensemble Kalman Inversion: Strong Convergence Analysis Open
The ensemble Kalman inversion (EKI) method is a method for the estimation of \nunknown parameters in the context of (Bayesian) inverse problems. The method approximates the \nunderlying measure by an ensemble of particles and iteratively a…
Singular limits for stochastic equations Open
We study singular limits of stochastic evolution equations in the interplay of disappearing strength of the noise and insufficient regularity, where the equation in the limit with noise would not be defined due to lack of regularity. We re…
Stochastic turbulence for Burgers equation driven by cylindrical Lévy process Open
This work is devoted to investigating stochastic turbulence for the fluid flow in one-dimensional viscous Burgers equation perturbed by Lévy space-time white noise with the periodic boundary condition. We rigorously discuss the regularity …
Amplitude equations for SPDEs driven by fractional additive noise with small Hurst parameter Open
We study stochastic partial differential equations (SPDEs) with potentially very rough fractional noise with Hurst parameter $H\in(0,1)$. Close to a change of stability measured with a small parameter $\varepsilon$, we rely on the natural …
Continuous time limit of the stochastic ensemble Kalman inversion: Strong convergence analysis Open
The Ensemble Kalman inversion (EKI) method is a method for the estimation of unknown parameters in the context of (Bayesian) inverse problems. The method approximates the underlying measure by an ensemble of particles and iteratively appli…
Modulation and amplitude equations on bounded domains for nonlinear SPDEs driven by cylindrical α-stable Lévy processes Open
In the present work, we establish the approximation of nonlinear stochastic partial differential equation (SPDE) driven by cylindrical α-stable Lévy processes via modulation or amplitude equations. We study SPDEs with a cubic nonlinearity,…
Stochastic turbulence for Burgers equation driven by cylindrical Lévy process Open
This work is devoted to investigating stochastic turbulence for the fluid flow in one-dimensional viscous Burgers equation perturbed by Lévy space-time white noise with the periodic boundary condition. We rigorously discuss the regularity …
Kink motion for the one-dimensional stochastic Allen-Cahn equation Open
We study the kink motion for the one-dimensional stochastic Allen-Cahn equation and its mass conserving counterpart. Using a deterministic slow manifold, in the sharp interface limit for sufficiently small noise strength we derive an expli…
The impact of white noise on a supercritical bifurcation Open
We consider the impact of additive Gaussian white noise on a supercritical pitchfork bifurcation in an unbounded domain. As an example we focus on the stochastic Swift-Hohenberg equation with polynomial nonlinearity. Here we identify the o…
The impact of multiplicative noise in SPDEs close to bifurcation via amplitude equations Open
This article deals with the approximation of a stochastic partial differential equation (SPDE) via amplitude equations. We consider an SPDE with a cubic nonlinearity perturbed by a general multiplicative noise that preserves the constant t…
Stochastic Cahn-Hilliard equation in higher space dimensions: The motion of bubbles Open
We study the stochastic motion of a droplet in a stochastic Cahn-Hilliard equation in the sharp interface limit for sufficiently small noise. The key ingredient in the proof is a deterministic slow manifold, where we show its stability for…
Well posedness and convergence analysis of the ensemble Kalman inversion Open
The ensemble Kalman inversion is widely used in practice to estimate unknown\nparameters from noisy measurement data. Its low computational costs,\nstraightforward implementation, and non-intrusive nature makes the method\nappealing in var…
Random initial conditions for semi-linear PDEs Open
We analyse 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. In particular, in some cases, st…
Rigorous a-posteriori analysis using numerical eigenvalue bounds in a surface growth model Open
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 b…
A strongly convergent numerical scheme from Ensemble Kalman inversion Open
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 resu…
A strongly convergent numerical scheme from EnKF continuum analysis Open
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 resu…
Numerically Computable A Posteriori-Bounds for stochastic Allen-Cahn\n equation Open
The aim of this paper is the derivation of an a-posteriori error estimate for\na numerical method based on an exponential scheme in time and spectral Galerkin\nmethods in space. We obtain analytically a rigorous bound on the mean square\ne…
Numerically Computable A Posteriori-Bounds for stochastic Allen-Cahn equation Open
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 erro…