Inhalt des Dokuments
Numerical approximation of the stochastic Cahn-Hilliard equation and the (stochastic) Hele-Shaw problem
The Cahn-Hilliard equation is a fourth oder parabolic partial differential equation (PDE) that is widely used as a phenomenological model to describe the evolution of interfaces in many practical problems, such as, the microstructure formation in materials, fluid flow, etc. It has been observed in the engineering literature that the stochastic version of the Cahn-Hilliard equation provides a better description of the
experimentally observed evolution of complex microstructure. The equation belongs to a class of so-called phase-field models where the interface is replaced by a diffuse layer with small thickness proportional to an interfacial thickness parameter $eps$. It can be shown that for vanishing interfacial thickness the deterministic as well as the stochastic Cahn-Hilliard equation (with proper scaling of the noise) both converge to a sharp-interface limit which is given by the deterministic Hele-Shaw problem.
We propose a time implicit numerical approximation of the stochastic Cahn-Hilliard equation which is robust with respect to the interfacial thickness parameter. We show that, with suitable scaling of the noise, the sharp-interface limit of the proposed numerical approximation converges to the deterministic Hele-Shaw problem. In addition we present numerical evidence that without the scaling of the noise the sharp-interface limit of the stochastic Cahn-Hilliard equation is a stochastic version of the Hele-Shaw problem.
We propose a numerical approximation of the stochastic Hele-Shaw problem and present computational results which demonstrate the respective convergence of the stochastic Cahn-Hilliard equation to the deterministic or the stochastic version of the Hele-Shaw problem depending on scaling of the noise term.
This is a joint work with D. Antonopoulou, R. Nurnberg and A. Prohl.
On regularity results for some Kolmogorov equations in infinite dimension
We consider the function u(t,x)=E[φ(X(t,x))], defined for t≥0, x ∈ H, where X(t,x) is the solution of a parabolic, semilinear, SDPE dX(t)=AX(t)dt+F(X(t))dt+σ(X(t))dW(t), X(0)=x, with values in the infinite dimensional Hilbert space H.
At a formal level, u solves the Kolmogorov equation ∂tu=Lu, with the initial condition u(0,⋅)=φ, where L is the associated infinitesimal generator.
Stochastic Neural Mass Models through Structure-Preserving Approximate Bayesian Computation
Random initial conditions and noise in 2D Euler equations
After a review about a recent new approach to incompressible 2D Euler equations with random initial condition and transport noise, we show that a special scaling limit of the stochastic Euler equations leads to the stochastic Navier-Stokes equations with additive space-time white noise. Remarkable is the difference of the noise. And the inversion with respect to usual paradigm that considers Euler equations as limit of Navier-Stokes equations. It is based on joint work with Dejun Luo.
Generation of random dynamical systems for SPDE with nonlinear noise
In this talk we will revisit the problem of generation of random dynamical systems by solutions to SPDE. Despite being at the heart of a dynamical system approach to stochastic dynamics in infinite dimensions, most known results are restricted to SPDE driven by affine linear noise, which can be treated via transformation arguments. In contrast, in this talk we will address instances of SPDE with nonlinear noise, with particular emphasis on porous media equations driven by conservative noise.
Markov selection for the stochastic compressible Navier—Stokes system
We analyze the Markov property of solutions to the compressible Navier—Stokes system perturbed by a general multiplicative stochastic forcing. We show the existence of an almost sure Markov selection to the associated martingale problem. Our proof is based on the abstract framework introduced in [F. Flandoli, M. Romito: Markov selections for the 3D stochastic Navier—Stokes equations. Probab. Theory Relat. Fields 140, 407—458. (2008)]. A major difficulty arises from the fact, different from the incompressible case, that the velocity field is not continuous in time. In addition, it cannot be recovered from the variables whose time evolution is described by the Navier—Stokes system, namely, the density and the momentum. We overcome this issue by introducing an auxiliary variable into the Markov selection procedure.
On Recent Progress in Dynamics of Reaction-Diffusion Equations
In this talk, I am going to provide a survey of some recent progress on dynamics of certain (still relatively elementary) classes of reaction-diffusion SPDEs. Even the basic theory of what a "solution" should be is still under very active development and discussion; here I shall mention recent progress made for quasi-linear and nonlocal SPDEs [4,8]. Then I shall focus in more detail on how to capture local fluctuations near deterministically stationary solutions from an analytical [2,6] and a numerical standpoint [3,5,7]. Time permitting, I am going to provide a brief outlook towards the case of travelling waves . The talk is based upon a series of works with several co-authors [1-8].
 Warning signs for wave speed transitions of noisy Fisher-KPP invasion fronts, C. Kuehn, Theoretical Ecology, Vol. 6, No. 3, pp. 295-308, 2013.
 Early-warning signs for pattern formation in stochastic partial differential equations, K. Gowda and C. Kuehn, Communications in Nonlinear Science and Numerical Simulation, Vol. 22, No. 1, pp. 55-69, 2015.
 Numerical continuation and SPDE stability for the 2D cubic-quintic Allen-Cahn equation, C. Kuehn, SIAM/ASA Journal on Uncertainty Quantification, Vol. 3, No. 1, pp. 762-789, 2015.
 Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs in three space dimensions, N. Berglund and C. Kuehn, Electronic Journal of Probability, Vol. 21, No. 18, pp. 1-48, 2016.
 Continuation of probability density functions using a generalized Lyapunov approach, S. Baars, et al., Journal of Computational Physics, Vol. 336, No. 1, pp. 627643, 2017.
 Scaling laws and warning signs for bifurcations of SPDEs, C. Kuehn and F. Romano. European Journal of Applied Mathematics, accepted / to appear, 2018.
 Combined Error Estimates for Local Fluctuations of SPDEs, C. Kuehn and P. Kuerschner, arXiv:1611.04629, 2016.
 Pathwise mild solutions for quasilinear stochastic partial differential equations, C. Kuehn and A. Neamtu, arXiv:1802.10016, 2018.
SPDE simulation on spheres
The simulation of solutions to stochastic partial differential equations requires besides discretization in space and time the approximation of the driving noise. This problem can be transferred to the simulation of a sequence of random fields on the underlying domain. In this talk I will concentrate on domains that are spheres and review some recent developments.
Stochastic PDEs Driven by Noise with Memory
The talk is based on several recent papers by P. Coupek, B.
Maslowski and their collaborators. By processes with memory we
understand a broad class of so-called Volterra processes in infinite
dimensions which (beside the standard Brownian motion) include, for
example, fractional and multifractional Brownian motions and the
Rosenblatt process in infinite dimensions. After developing basic
tools of stochastic analysis with respect to these processes
(including also the Ito formula for Rosenblatt process) we study
regularity and basic properties of stochastic convolution integrals
and apply the results to SPDEs. We also study
the large time behaviour for linear SPDEs (and in Gaussian case of equations with bilinear noise term) and its dependence on the type of noise.
On strong convergence of time numerical schemes for the stochastic 2D Navier-Stokes equations
We prove that some time discretization schemes, such as the splitting, fully and semi-implicit ones, of the 2D Navier-Stokes equations subject to a random perturbation converge "strongly", that is in the set of square integrable random variables. The speed of convergence depends on the viscosity. The argument is based on convergence of a localized scheme, and on exponential moments of the solution to the stochastic 2D Navier-Stokes equations. This joint work with H. Bessaih improves previous results which only described the speed of convergence in probability of these numerical schemes.
On temporal regularity for SPDEs in Besov-Orlicz spaces
Space-time adaptivity for Stochastic PDEs
I propose a new adaptive time stepping method to numerically solve a general SPDE, where local step sizes are chosen in regard of the distance between empirical laws of time iterates and extrapolated data. Time adaptivity is then complemented by a local refinement/coarsening strategy of the spatial mesh via a stochastic version of the ZZ-estimator. Next to an improved accuracy, we observe a significantly reduced empirical variance of standard estimators, and therefore a reduced sampling effort. The performance of the adaptive strategies is studied for SPDEs with linear drift, including the convection-dominated case where the streamline diffusion method is adopted to attain a stable discretization, and a nonlinear SPDE where approximate solutions exhibit discrete blow-up dynamics. - This is joint work with C. Schellnegger (U Tuebingen).
Fluctuations for point vortex models
The first part of the presentation is a short review of the statistical mechanics theory for points vortex models for the 2D Euler equations. In the second part we outline a recent result obtained in collaboration with F. Grotto (Scuola Normale, Pisa) about the fluctuations of the mean field limit for point vortices. In the last part we outline an extension of the theory to a slightly more general class of models (generalized SQG) with more singular interaction. This is a work in collaboration with C. Geldhauser (Chebychev Laboratory, St. Petersburg).
Existence, uniqueness and stability of semi-linear rough partial differential equations
We prove well-posedness and rough path stability of a class of
linear and semi-linear rough PDE's using the variational approach.
This includes well-posedness of (possibly degenerate) linear rough
PDE's in Lp and then — based on a new method —
energy estimates for non-degenerate linear rough PDE's. We accomplish
this by controlling the energy in a properly chosen weighted
L2-space, where the weight is given as a solution of an
associated backward equation. These estimates then allow us to extend
well-posedness for linear rough PDE's to semi-linear perturbations.
As a particular example we consider the generalized viscous Burgers equation perturbed by a rough path transport noise.
The talk is based on joint work with P. Friz, A. Hocquet and T. Nielsen from TU Berlin.
P. Friz, T. Nilssen, W. Stannat: Existence, uniqueness and stability of semi-linear rough partial differential equations, arXiv:1809.00841
A. Hocquet, T. Nilssen, W. Stannat: Generalized Burgers equation with rough transport noise, arXiv:1804.01335
Theoretical study and numerical simulation of pattern formation in the deterministic and stochastic Gray-Scott equations
Mathematical models based on systems of reaction-diffusion equations provide fundamental tools for the description and investigation of various processes in biology, biochemistry, and chemistry; in specific situations, an appealing characteristic of the arising nonlinear partial differential equations is the formation of patterns, reminiscent of those found in nature. The deterministic Gray–Scott equations constitute an elementary two-component system that describes autocatalytic reaction processes; depending on the choice of the specific parameters, complex patterns of spirals, waves, stripes, or spots appear.
In the derivation of a macroscopic model such as the deterministic Gray–Scott equations from basic physical principles, certain aspects of microscopic dynamics, e.g. fluctuations of molecules, are disregarded; an expedient mathematical approach that accounts for significant microscopic effects relies on the incorporation of stochastic processes and the consideration of stochastic partial differential equations.
In this talk, I will study the stochastic Gray–Scott equations driven by independent spatially time-homogeneous Wiener processes. Under suitable regularity assumptions on the prescribed initial states, existence as well as the uniqueness of the solution processes, is proven. Numerical simulations based on the application of a time-adaptive first-order operator splitting method and the fast Fourier transform illustrate the formation of patterns in the deterministic case and their variation under the influence of stochastic noise.
On some stochastic nonlocal equations
In this talk, we will be interested in the well-posedness, and the stability of the entropy solution, for a stochastic fractional-parabolic problem with a first order nonlinear operator.
Hydrodynamic limit of kinetic equations with environmental noise
We give several results of hydrodynamic limits of stochastic kinetic equations. We are particularly interested by the regime of diffusion approximation, where the noise allows to consider terms with a much more singular amplitude than in the deterministic case. We will explain these differences. Works in collaboration with Caillerie, Debussche, Hofmanova, Rosello.
Time-fractional stochastic conservation laws (joint work with Martin Scholtes)