Special Session 92: Numerical Methods for SPDEs: Bridging Theory and Applications

Higher-order discretisation of the stochastic Navier-Stokes equations with additive noise

Lubomir Banas Bielefeld University Germany Co-Author(s):    Dominic Breit, Abhishek Chaudhary, Andreas Prohl

Abstract: We propose a higher-order time discretization scheme for the stochastic Navier-Stokes equation with additive noise, with convergence order 3/2 in probability for the velocity and pressure approximations. The construction rests on the reformulation of the Navier-Stokes SPDE as a random PDE, and the use of suitable higher order numerical quadrature rules for the diffusion and the drift part.

Nils Berglund IDP, University of Orleans France Co-Author(s):

Abstract:

Numerical Study of a Surface Growth Model with Singular Noise

Dirk Bl\"omker Universit\"at Augsburg Germany Co-Author(s):    David Buchberger, Johannes Rimmele

Abstract: We investigate an SPDE for epitaxial thin-film growth. Remarkably, it has been shown that even arbitrarily small space-time white noise, representing fluctuations in the incoming particles, can stabilize the dynamics and prevent hill formation, which is present for less regular noise. In order to see the transition via numerical simulations, we apply a spectral Galerkin approach together with an exponential Euler method to approximate the solution of the regularised SPDE and give rigorous error estimates.

Optimal Control of Fluid Flows

Fernanda Cipriano NOVA University Lisbon Portugal Co-Author(s):

Abstract: This talk is devoted to the study of optimal control problems in fluid dynamics. The fluid is assumed to occupy a bounded domain of R^d, d=2,3, and may be subject to external Gaussian random forces. Various types of control actions will be considered to achieve a prescribed velocity profile while minimizing the associated cost. Results concerning the existence and uniqueness of optimal solutions will be presented, and appropriate strategies for determining the optimal control will be developed.

Asymptotic error distribution of the Euler method for stochastic differential equations with irregular drifts

Konstantinos Dareiotis University of Leeds England Co-Author(s):    M\`at\`e Gerencs\`er, Khoa L\^e

Abstract: In this talk we will establish a central limit theorem for the Euler-Maruyama scheme approximating multidimensional SDEs with elliptic Brownian diffusion, under very mild regularity requirements on the drift coefficients. When the drift is H\older continuous, we show that the limiting law of the rescaled fluctuations around the true solution is characterised by the solution of a hybrid Young-It\^o differential equation. When the drift has positive Sobolev regularity, this limit is characterised by the solution of a transformed SDE. Our result is an extension of the result of Jacod--Protter (1998) in which SDEs with bounded differentiable coefficients were considered. To compensate for the lack of regularity of the drifts, we utilize the regularization effect from the non-degenerate noise.

Discrete Stochastic Maximal Regularity

Foivos F Evangelopoulos-Ntemiris TU DELFT Netherlands Co-Author(s):    M. Veraar

Abstract: Maximal $L^p$-regularity is a central tool in the analysis of deterministic and stochastic parabolic evolution equations, providing a framework for studying nonlinear problems via linearization techniques. In the deterministic case, a discrete-time theory of maximal $\ell^p$-regularity was recently developed for numerical schemes, and its equivalence with the continuous-time theory was established. In this talk, I will extend these ideas to the stochastic setting, introducing discrete stochastic maximal $\ell^p$-regularity and exploring its connection to the continuous-time counterpart. The talk is based on joint work with Mark Veraar (TU Delft).

Finite Volume scheme for the Heat equation with transport noise

Ludovic Goudenege CNRS France Co-Author(s):    Anne de Bouard, Flore Nabet

Abstract: We investigate a finite volume scheme for the heat equation perturbed by transport noise, a stochastic forcing that models random advection effects and arises naturally in fluid dynamics. The equation is interpreted in the Stratonovich sense, preserving key structural properties such as energy balance at the continuous level. We introduce a finite volume approximation coupled with an Euler-Maruyama time-splitting discretization with a consistent treatment of the stochastic transport term which is designed to maintain stability and discrete conservation properties. We establish well-posedness of the numerical scheme and prove convergence towards a solution of the stochastic partial differential equation. Finally, numerical experiments illustrate the behavior of the method, highlighting the impact of transport noise on the solution, in particular on its long-time dynamics. These results demonstrate the robustness and effectiveness of the proposed approach for more complex stochastic diffusion models.

Strong convergence rates for stochastic Burgers equations

Martin Hutzenthaler University of Duisburg-Essen Germany Co-Author(s):    Arnulf Jentzen, Felix Lindner, Robert Link, Primoz Pusnik

Abstract: Subject of this talk are strong convergence rates on the whole probability space for explicit full-discrete approximations of stochastic Burgers equations with multiplicative trace-class noise. The nonlinearity of this benchmark SPDE is not globally monotone. Many classical approximation methods do not converge in the strong sense. In this talk we discuss methods which do converge in the strong sense. The key step in the convergence proof are uniform exponential moment estimates for the numerical approximations.

Temporal approximation of semilinear hyperbolic SPDEs with polynomial nonlinearities

Katharina Klioba TU Delft Netherlands Co-Author(s):

Abstract: When approximating hyperbolic SPDEs such as the stochastic Schroedinger equation in time, challenges arise due to the lack of regularising behaviour of the underlying semigroup. In this talk, we present convergence rates for time discretization schemes for such equations, where the leading operator is the generator of a $C_0$-semigroup. The first main result are optimal bounds for the uniform strong error \[ E_k^\infty:= \Big(\mathbb{E}\max_{1\le j \le M}\|U(t_j)-U_j\|_X^p\Big)^{1/p} \] on a Hilbert space $X$ for $p\in [2,\infty)$, a time step $k>0$ and $T=Mk>0$. Under conditions on the globally Lipschitz nonlinearity and multiplicative noise, we show $E_k^\infty\lesssim\sqrt{k\log(T/k)}$ for a large class of time discretisation schemes. For equations such as Maxwell`s or Schroedinger, our results provide the first results known for rational approximations of $(S(t))_{t\ge 0}$ and improve mean-square error estimates to pathwise uniform ones. The second main result concerns polynomially growing nonlinearities and noise, which imposes additional challenges since only a local Lipschitz condition is satisfied. For a tamed exponential Euler scheme, stability and convergence $E_k^\infty\lesssim\sqrt{k}$ are shown under a coercivity assumption using a stochastic Gronwall inequality. This extension of the abstract framework developed in the first part allows to consider, e.g., the nonlinear Schroedinger equation. This is based partly on joint work with Mark Veraar and partly on ongoing work by the author.

Regularization by regular noise: a numerical result

Chengcheng Ling University of Augsburg Germany Co-Author(s):    Ke Song, Haiyi Wang

Abstract: We study a singular stochastic equation driven by regular noise of fractional Brownian type with Hurst index $H \in (1,\infty)\setminus\mathbb{Z}$ and drift coefficient $b \in \mathcal{C}^\alpha$, where $\alpha > 1 - \tfrac{1}{2H}$. The strong well-posedness of this equation was first established in [Gerencs\`er, 23], a phenomenon known as {\it regularization by regular noise}. In this note, we provide a numerical analysis of the equation. Specifically, we prove that the Euler--Maruyama approximation $X^n$ converges strongly to the unique solution $X$ at rate $n^{-1}$. Moreover, we show that $n(X - X^n)$ converges in probability to a non-trivial limit as $n \to \infty$, which confirms that the rate $n^{-1}$ is optimal for this scheme. In this sense, this provides a first-order numerical method for equations with non-Lipschitz drift while still achieving the rate $n^{-1}$.

Averaging Principle for the 3D Stochastic Primitive Equations

Vincent R Martinez CUNY Hunter College & Graduate Center USA Co-Author(s):    Quyuan Lin, Rongchang Liu, Vincent R. Martinez

Abstract: This talk will present recent results on the 3D stochastic primitive equations. In particular, we show that the system is nearly uniquely ergodic in the sense that the limit resonant system in the infinite rotation limit is uniquely ergodic and the law of the finite rotation system converges in probability to the law of the limit resonant system on any finite time horizon. The passage of limits is facilitated by weak moment bounds, an averaging procedure developed by Flandoli and Mahalov, as well as a stochastic control argument to establish the existence of a spectral gap.

A stochastic interpretation of the Water Waves model

Antoine Moneyron Inria Rennes France Co-Author(s):    Arnaud Debussche, Etienne Memin, Antoine Moneyron

Abstract: I will present a stochastic interpretation of the Water Waves equations, based on the LU formulation of the irrotational Euler s equations, which typically involves transport noises. The model itself is derived first, then the associated shallow water type models are formally inferred by neglecting some terms the limit of vanishing aspect ratios. Next, I will introduce some numerical schemes associated to the models above, which are intermediate between the deterministic 4th Runge Kutta method and a stochastic Euler Heun method. Eventually, some numerical simulations will be shown to assess the properties of these stochastic wave models.

Nonparametric Estimation of Noise Covariance in Parabolic SPDEs

Andreas Petersson Linnaeus University Sweden Co-Author(s):    Andreas Petersson

Abstract: We develop an asymptotic limit theory for nonparametric estimation of the noise covariance kernel in linear parabolic SPDEs with additive colored noise, using space-time infill asymptotics. The method employs discretized infinite-dimensional realized covariations and requires only mild regularity assumptions on the kernel to ensure consistent estimation and asymptotic normality of the estimator. On this basis, we construct omnibus goodness-of-fit tests for the noise covariance that are independent of the SPDE`s differential operator. Our framework accommodates a variety of spatial sampling schemes and allows for reliable inference even when spatial resolution is coarser than temporal resolution.

Convergence in law for SPDEs with additive noise

Lluis Quer Sardanyons Universitat Autonoma de Barcelona Spain Co-Author(s):

Abstract: We consider the quasi-linear stochastic wave and heat equations in $\mathbb{R}^d$ with $d\in \{1,2,3\}$ and $d\geq 1$, respectively, and perturbed by an additive Gaussian noise which is white in time and has a homogeneous spatial correlation with spectral measure $\mu_n$. We allow the Fourier transform of $\mu_n$ to be a genuine distribution. Let $u^n$ be the mild solution to these equations. We provide sufficient conditions on the measures $\mu_n$ and the initial data to ensure that $u^n$ converges in law, in the space of continuous functions, to the solution of our equations driven by a noise with spectral measure $\mu$, where $\mu_n\to\mu$ in some sense. We apply our main result to various types of noises, such as the anisotropic fractional noise. We also show that we cover existing results in the literature, such as the case of Riesz kernels and the fractional noise with $d=1$. The talk is based on joint work with Maria Jolis (Barcelona) and Salvador Ortiz (Oslo).

Uniform A Priori Estimates and Compatcness for a Stochastic Ferrohydrodynamic system: A Foundation for Numerical Analysis

Paul Razafimandimby Dublin City University Ireland Co-Author(s):    Aristide Ndongmo Ngana

Abstract: We consider the stochastic Rosensweig system, a complex model for the dynamics of ferrofluids under the influence of a magnetic field and thermal fluctuations. The system couples the Navier--Stokes equations for the fluid velocity with equations for the fluid`s magnetization and the magnetic field, leading to a highly nonlinear SPDE with non-convex constraints. This talk presents a compactness and convergence analysis for a bi-layer approximation of the stochastic Rosensweig system: a Bloch--Torrey regularization to handle the magnetization constraint, coupled with a Galerkin approximation for spatial discretization. This approach yields the existence of a global renormalized weak martingale solution to the system. From a numerical analysis perspective, the core contribution is the derivation of a series of \textbf{uniform a priori estimates} that are independent of both the regularization parameter and the Galerkin dimension. These estimates---including bounds on energy, dissipation, and higher-order moments---are precisely the \textbf{stability estimates required to prove convergence of fully discrete numerical schemes}, such as finite element or spectral methods. By establishing these foundational analytical results, this work lays the necessary groundwork for the future development and rigorous error analysis of computational methods for stochastic ferrohydrodynamic flows.

An Allen-Cahn model with jump-diffusion noise for biological damage and repair

Luca Scarpa Politecnico di Milano Italy Co-Author(s):    Andrea Di Primio, Marvin Fritz, Margherita Zanella

Abstract: We present a stochastic Allen-Cahn equation for the dynamics of biomolecular damage and repair. The system is driven by two noise processes: a multiplicative cylindrical Wiener process, modelling continuous background stochastic fluctuations, and a jump-type noise, modelling the abrupt, localised damage induced by external radiotherapy shocks. The drift of the equation is singular and covers the typical logarithmic Flory-Huggins potential required in phase-separation dynamics. We prove well-posedness of the model in a strong probabilistic sense, analyse its long-time behaviour, and present some simulations to illustrate how it captures fundamental biological phenomena, such as damage clustering, stress-induced topology perturbations, and damage dynamics. The works presented in the talk are based on joint collaborations with Andrea Di Primio (Scuola Normale Superiore, Pisa, Italy), Marvin Fritz (University of Vienna, Austria), and Margherita Zanella (Politecnico di Milano, Italy).

Convex integration on stochastic PDEs

Kazuo Yamazaki University of Nebraska-Lincoln USA Co-Author(s):    Kazuo Yamazaki

Abstract: Convex integration dates back to the work of Nash and has developed recently as a way to construct non-unique weak solutions to vaious PDEs. We discuss recent developments on convex integration on stochastic PDEs with examples including the Navier-Stokes equations, surface quasi-geostrophic equations, and others.