Parabolic Stochastic Partial Differential Equations Research Articles

Weak Convergence of the Rosenbrock Semi-implicit Method for Semilinear Parabolic SPDEs Driven by Additive Noise

Abstract This paper aims to investigate the weak convergence of the Rosenbrock semi-implicit method for semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive noise. We are interested in SPDEs where the nonlinear part is stronger than the linear part, also called stochastic reaction dominated transport equations. For such SPDEs, many standard numerical schemes lose their stability properties. Exponential Rosenbrock and Rosenbrock-type methods were proved to be efficient for such SPDEs, but only their strong convergence were recently analyzed. Here, we investigate the weak convergence of the Rosenbrock semi-implicit method. We obtain a weak convergence rate which is twice the rate of the strong convergence. Our error analysis does not rely on Malliavin calculus, but rather only uses the Kolmogorov equation and the smoothing properties of the resolvent operator resulting from the Rosenbrock semi-implicit approximation.

Localized Orthogonal Decomposition for a Multiscale Parabolic Stochastic Partial Differential Equation

Localized Orthogonal Decomposition for a Multiscale Parabolic Stochastic Partial Differential Equation

Global existence of dissipative solutions to the Camassa–Holm equation with transport noise

We consider a nonlinear stochastic partial differential equation (SPDE) that takes the form of the Camassa–Holm equation perturbed by a convective, position-dependent, noise term. We establish the first global-in-time existence result for dissipative weak martingale solutions to this SPDE, with general finite-energy initial data. The solution is obtained as the limit of classical solutions to parabolic SPDEs. The proof combines model-specific statistical estimates with stochastic propagation of compactness techniques, along with the systematic use of tightness and a.s. representations of random variables on specific quasi-Polish spaces. The spatial dependence of the noise function makes more difficult the analysis of a priori estimates and various renormalisations, giving rise to nonlinear terms induced by the martingale part of the equation and the second-order Stratonovich–Itô correction term.

Existence of smooth stable manifolds for a class of parabolic SPDEs with fractional noise

Little seems to be known about the invariant manifolds for stochastic partial differential equations (SPDEs) driven by nonlinear multiplicative noise. Here we contribute to this aspect and analyze the Lu-Schmalfuß conjecture [Garrido-Atienza, et al., (2010) [14]] on the existence of stable manifolds for a class of parabolic SPDEs driven by nonlinear multiplicative fractional noise. We emphasize that stable manifolds for SPDEs are infinite-dimensional objects, and the classical Lyapunov-Perron method cannot be applied, since the Lyapunov-Perron operator does not give any information about the backward orbit. However, by means of interpolation theory, we construct a suitable function space in which the discretized Lyapunov-Perron-type operator has a unique fixed point. Based on this we further prove the existence and smoothness of local stable manifolds for such SPDEs.

Malliavin calculus and densities for singular stochastic partial differential equations

We study Malliavin differentiability of solutions to sub-critical singular parabolic stochastic partial differential equations (SPDEs) and we prove the existence of densities for a class of singular SPDEs. Both of these results are implemented in the setting of regularity structures. For this we construct renormalized models in situations where some of the driving noises are replaced by deterministic Cameron–Martin functions, and we show Lipschitz continuity of these models with respect to the Cameron–Martin norm. In particular, in many interesting situations we obtain a convergence and stability result for lifts of L^2-functions to models, which is of independent interest. The proof also involves two separate algebraic extensions of the regularity structure which are carried out in rather large generality.

A multilevel Monte Carlo ensemble and hybridizable discontinuous Galerkin method for a stochastic parabolic problem

AbstractA second‐order, multilevel Monte Carlo ensemble, and hybridizable discontinuous Galerkin (MLMCE‐HDG) method is proposed to solve the stochastic parabolic partial differential equations (SPDEs). By introducing an ensemble average of the diffusion coefficient, the MLMCE‐HDG method results in a single discrete system with multiple right‐hand‐vectors, which can be solved more efficiently than a group of linear systems. A rigorous error estimate is obtained with a second‐order accuracy in time and optimal convergence rate in physical space. Comparing with the multilevel Monte Carlo and hybridizable discontinuous Galerkin (MLMC‐HDG) method, the MLMCE‐HDG method can reduce the computational cost. Finally, we provide several numerical experiments to illustrate the theoretical results.

Observer-Based Fuzzy Quantized Control for a Stochastic Third-Order Parabolic PDE System

The work addresses an observer-based fuzzy quantized control for stochastic third-order parabolic partial differential equations (PDEs) using discrete point measurements. For the first time, we contribute in introducing three types of quantizer—logarithmic quantizer, uniform quantizer, and hysteresis quantizer into the controller designs for the stochastic PDE system. The main advantage of the quantized control law lies in reducing the energy that the system spends. The stability analysis is nontrivial due to the complex stochastic PDE model. Different stability results are presented for each case. Sufficient LMI-based conditions are investigated to ensure the internal mean-square exponential stability and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{\infty }$ </tex-math></inline-formula> performance of the perturbed actuated system. Consistent simulation results that support the proposed theoretical statements are provided.

A Weighted Average Finite Difference Scheme for the Numerical Solution of Stochastic Parabolic Partial Differential Equations

In the present paper, the numerical solution of Itô type stochastic parabolic equation with a time white noise process is imparted based on a stochastic finite difference scheme. At the beginning, an implicit stochastic finite difference scheme is presented for this equation. Some mathematical analyses of the scheme are then discussed. Lastly, to ascertain the efficacy and accuracy of the suggested technique, the numerical results are discussed and compared with the exact solution.

Weak convergence of the finite element method for semilinear parabolic SPDEs driven by additive noise

This paper aims to investigate the finite element weak convergence rate for semilinear parabolic stochastic partial differential equations(SPDEs) driven by additive noise. In contrast to many results in the current scientific literature, we investigate the more general case where the nonlinearity is allowed to be of Nemytskii-type and the linear operator is not necessarily self-adjoint, which is more challenging and models more realistic phenomena such as convection–reaction–diffusion processes. Using Malliavin calculus, Kolmogorov equations and by splitting the linear operator into a self-adjoint and non self-adjoint parts, we prove the convergence of the finite element approximation and obtain a weak convergence rate that is twice the strong convergence rate.

A learning scheme by sparse grids and Picard approximations for semilinear parabolic PDEs

Abstract Relying on the classical connection between backward stochastic differential equations and nonlinear parabolic partial differential equations (PDEs), we propose a new probabilistic learning scheme for solving high-dimensional semilinear parabolic PDEs. This scheme is inspired by the approach coming from machine learning and developed using deep neural networks in Han et al. (2018, Solving high-dimensional partial differential equations using deep learning. Proc. Natl. Acad. Sci., 115, 8505–8510. Our algorithm is based on a Picard iteration scheme in which a sequence of linear-quadratic optimization problem is solved by means of stochastic gradient descent algorithm. In the framework of a linear specification of the approximation space, we manage to prove a convergence result for our scheme, under some smallness condition. In practice, in order to be able to treat high-dimensional examples, we employ sparse-grid approximation spaces. In the case of periodic coefficients and using pre-wavelet basis functions, we obtain an upper bound on the global complexity of our method. It shows, in particular, that the curse of dimensionality is tamed in the sense that in order to achieve a root mean squared error of order $\varepsilon $, for a prescribed precision $\varepsilon $, the complexity of the Picard algorithm grows polynomially in $\varepsilon ^{-1}$ up to some logarithmic factor $|\!\log (\varepsilon )|$, whose exponent grows linearly with respect to the PDE dimension. Various numerical results are presented to validate the performance of our method, and to compare them with some recent machine learning schemes proposed in E et al. (2017, Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations. Commun. Math. Stat., 5, 349–380) and Huré et al. (2020, Deep backward schemes for high-dimensional nonlinear PDEs. Math. Comput., 89, 1547–1579).

Well-Posedness for Nonlinear Parabolic Stochastic Differential Equations with Nonlinear Robin Conditions

In this paper, we present the existence and uniqueness of strong probabilistic solutions for nonlinear parabolic Stochastic Partial Differential Equations (SPDEs) with nonlinear Robin boundary conditions in a domain with holes. On the boundary of the holes, a nonlinear Robin condition is imposed, while a homogeneous Dirichlet condition is prescribed on the exterior boundary. The coefficient matrix is assumed to be symmetric, while the nonlinear random forces are assumed to satisfy some types of regularities. We use Galerkin’s approximation method, probabilistic compactness results and some results from stochastic calculus.

Explicit Stabilized Multirate Method for Stiff Stochastic Differential Equations

Stabilized explicit methods are particularly efficient for large systems of stiff stochastic differential equations (SDEs) due to their extended stability domain. However, they loose their efficiency when a severe stiffness is induced by very few fast degrees of freedom, as the stiff and nonstiff terms are evaluated concurrently. Therefore, inspired by [A. Abdulle, M. J. Grote, and G. Rosilho de Souza, Preprint (2020), arXiv:2006.00744] we introduce a stochastic modified equation whose stiffness depends solely on the slow terms. By integrating this modified equation with a stabilized explicit scheme we devise a multirate method which overcomes the bottleneck caused by a few severely stiff terms and recovers the efficiency of stabilized schemes for large systems of nonlinear SDEs. The scheme is not based on any scale separation assumption of the SDE and therefore it is employable for problems stemming from the spatial discretization of stochastic parabolic partial differential equations on locally refined grids. The multirate scheme has strong order 1/2, weak order 1 and its stability is proved on a model problem. Numerical experiments confirm the efficiency and accuracy of the scheme.

The Dirichlet Problem for Stochastic Degenerate Parabolic-Hyperbolic Equations

We consider the Dirichlet problem for a quasilinear degenerate parabolic stochastic partial differential equation with multiplicative noise and nonhomogeneous Dirichlet boundary condition. We introduce the definition of kinetic solution for this problem and prove existence and uniqueness of solutions. For the uniqueness of kinetic solutions we prove a new version of the doubling of variables method and use it to deduce a comparison principle between solutions. The proof requires a delicate analysis of the boundary values of the solutions for which we develop some techniques that enable the usage of the existence of normal weak traces for divergence measure fields in this stochastic setting. The existence of solutions, as usual, is obtained through a two-level approximation scheme consisting of nondegenerate regularizations of the equations which we show to be consistent with our definition of solutions. In particular, the regularity conditions that give meaning to the boundary values of the solutions are shown to be inherited by limits of nondegenerate parabolic approximations provided by the vanishing viscosity method.

Central limit theorems for parabolic stochastic partial differential equations

Soit {u(t,x)}t≥0,x∈Rd la solution d’une équation de la chaleur stochastique non-linéaire d-dimensionnelle, perturbée par un bruit gaussien, blanc en temps et avec une covariance homogène en espace donnée par une mesure de Borel finie qui satisfait la condition de Dalang. Nous démontrons deux théorèmes de la limite centrale fonctionnels pour des champs d’occupation de la forme N−d∫Rdg(u(t,x))ψ(x/N)dx quand N→∞, où g est une function lipschitzienne sur Rd et ψ∈L2(Rd). La preuve utilise des inegalités de type Poincaré, le calcul de Malliavin, des arguments de compacité et la caractérisation du mouvement brownien comme le seul processus de Lévy continu de moyenne nulle. Notre résultat généralise les théorèmes de la limite centrale de Huang et al (Stochastic Process. Appl. 131 (2020) 7170–7184 ; Stoch. Partial Differ. Equ., Anal. Computat. 8 (2020) 402–421) qui sont valables lorsque g(u)=u et ψ=1[0,1]d.

On the Small Time Asymptotics of Quasilinear Parabolic Stochastic Partial Differential Equations

In this paper, we establish small time large deviation principles for the quasilinear parabolic stochastic partial differential equations with multiplicative noise, which are neither monotone nor locally monotone.

A weak Galerkin method for nonlinear stochastic parabolic partial differential equations with additive noise

&lt;abstract&gt;&lt;p&gt;In this paper, a weak Galerkin (WG for short) finite element method is used to approximate nonlinear stochastic parabolic partial differential equations with spatiotemporal additive noises. We set up a semi-discrete WG scheme for the stochastic equations, and derive the optimal order for error estimates in the sense of strong convergence.&lt;/p&gt;&lt;/abstract&gt;

Approximation of the invariant distribution for a class of ergodic SPDEs using an explicit tamed exponential Euler scheme

We consider the long-time behavior of an explicit tamed exponential Euler scheme applied to a class of parabolic semilinear stochastic partial differential equations driven by additive noise, under a one-sided Lipschitz continuity condition. The setting encompasses nonlinearities with polynomial growth. First, we prove that moment bounds for the numerical scheme hold, with at most polynomial dependence with respect to the time horizon. Second, we apply this result to obtain error estimates, in the weak sense, in terms of the time-step size and of the time horizon, to quantify the error to approximate averages with respect to the invariant distribution of the continuous-time process. We justify the efficiency of using the explicit tamed exponential Euler scheme to approximate the invariant distribution, since the computational cost does not suffer from the at most polynomial growth of the moment bounds. To the best of our knowledge, this is the first result in the literature concerning the approximation of the invariant distribution for SPDEs with non-globally Lipschitz coefficients using an explicit tamed scheme.

A RBF partition of unity collocation method based on a finite difference scheme to solve parabolic stochastic partial differential equations

A RBF partition of unity collocation method based on a finite difference scheme to solve parabolic stochastic partial differential equations

The exponential behavior and stabilizability of quasilinear parabolic stochastic partial differential equation

In this paper, we study the stability of quasilinear parabolic stochastic partial differential equations with multiplicative noise, which are neither monotone nor locally monotone. The exponential mean square stability and pathwise exponential stability of the solutions are established. Moreover, under certain hypothesis on the stochastic perturbations, pathwise exponential stability can be derived, without utilizing the mean square stability.

Nonlinear stochastic parabolic partial differential equations with a monotone operator of the Ladyzenskaya-Smagorinsky type, driven by a Lévy noise

The aim of this article is to show the global existence of both martingale and pathwise solutions of stochastic equations with a monotone operator, of the Ladyzenskaya-Smagorinsky type, driven by a general Lévy noise. The classical approach based on using directly the Galerkin approximation is not valid. Instead, our approach is based on using appropriate approximations for the monotone operator, Galerkin approximations and on the theory of martingale solutions.