Stochastic Allen-Cahn Equation Research Articles

The POD–DEIM reduced-order method for stochastic Allen–Cahn equations with multiplicative noise

In this paper, we propose a reduced-order method (ROM) based on the Monte Carlo finite difference method (FDM) or Monte Carlo finite element method (FEM) for the stochastic Allen–Cahn (SAC) equation with multiplicative noise. The reduction method is a combination of proper orthogonal decomposition (POD) and discrete empirical interpolation method (DEIM). In the theoretical part, the error bounds of full-order method (FOM) and POD–DEIM are provided, as well as the computational complexity of FOM and POD–DEIM. In the numerical experiments section, several two-dimensional SAC examples with multiplicative noise are considered. Using the Monte Carlo FDM and Monte Carlo FEM as the reference methods, the POD–DEIM has obvious advantage in the computational efficiency. And with the encryption in the space or time direction, this advantage is more prominent.

Stochastic Allen–Cahn equation with logarithmic potential

We prove existence and uniqueness of a solution for the stochastic Allen–Cahn equation with logarithmic potential and multiplicative Wiener noise, under homogeneous Neumann boundary condition. The existence of a solution is obtained in the variational sense by means of an approximated equation involving a Yosida regularization of the nonlinearity. The noise is assumed to vanish at the extremal points of the physical relevant domain and to satisfy a suitable Lipschitz-continuity property, allowing to prove uniform estimates of the approximated solution. The passage to the limit is then carried out and continuous dependence on the initial datum of the solution is verified. Under an additional assumption, the existence of an analytically strong solution is proved. Finally, estimates for the derivatives of the logarithmic potential are derived, in view of the study of an optimal control problem associated to the stochastic Allen–Cahn equation.

An efficient explicit full-discrete scheme for strong approximation of stochastic Allen–Cahn equation

In Becker and Jentzen (2019) and Becker et al. (2017), an explicit temporal semi-discretization scheme and a space–time full-discretization scheme were, respectively, introduced and analyzed for the additive noise-driven stochastic Allen–Cahn type equations, with strong convergence rates recovered. The present work aims to propose a different explicit full-discrete scheme to numerically solve the stochastic Allen–Cahn equation with cubic nonlinearity, perturbed by additive space–time white noise. The approximation is easily implementable, performing the spatial discretization by a spectral Galerkin method and the temporal discretization by a kind of nonlinearity-tamed accelerated exponential integrator scheme. Error bounds in a strong sense are analyzed for both the spatial semi-discretization and the spatio-temporal full discretization, with convergence rates in both space and time explicitly identified. It turns out that the obtained convergence rate of the new scheme is, in the temporal direction, twice as high as existing ones in the literature. Numerical results are finally reported to confirm the previous theoretical findings.

Lower and upper bounds for strong approximation errors for numerical approximations of stochastic heat equations

Optimal upper and lower error estimates for strong full-discrete numerical approximations of the stochastic heat equation driven by space-time white noise are obtained. In particular, we establish the optimality of strong convergence rates for full-discrete approximations of stochastic Allen-Cahn equations with space-time white noise which have recently been obtained in [Becker, S., Gess, B., Jentzen, A., and Kloeden, P. E., Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations. arXiv:1711.02423 (2017)].

Convergence Rate for Galerkin Approximation of the Stochastic Allen—Cahn Equations on 2D Torus

In this paper we discuss the convergence rate for Galerkin approximation of the stochastic Allen-Cahn equations driven by space-time white noise on $$\mathbb{T}^2$$ . First we prove that the convergence rate for stochastic 2D heat equation is of order α — δ in Besov space $$\mathcal{C}^{-\alpha}$$ for α ∈ (0,1) and δ > 0 arbitrarily small. Then we obtain the convergence rate for Galerkin approximation of the stochastic Allen-Cahn equations of order α — δ in $$\mathcal{C}^{-\alpha}$$ for α ∈ (0,2/9) and δ > 0 arbitrarily small.

Weak convergence rates of splitting schemes for the stochastic Allen–Cahn equation

This article is devoted to the analysis of the weak rates of convergence of schemes introduced by the authors in a recent work, for the temporal discretization of the one-dimensional stochastic Allen–Cahn equation driven by space-time white noise. The schemes are based on splitting strategies and are explicit. We prove that they have a weak rate of convergence equal to $$\frac{1}{2}$$ , like in the more standard case of SPDEs with globally Lipschitz continuous nonlinearity. To deal with the polynomial growth of the nonlinearity, several new estimates and techniques are used. In particular, new regularity results for solutions of related infinite dimensional Kolmogorov equations are established. Our contribution is the first one in the literature concerning weak convergence rates for parabolic semilinear SPDEs with non globally Lipschitz nonlinearities.

Optimal Error Estimates of Galerkin Finite Element Methods for Stochastic Allen–Cahn Equation with Additive Noise

Strong approximation errors of both finite element semi-discretization and spatio-temporal full discretization are analyzed for the stochastic Allen-Cahn equation driven by additive noise in space dimension $d \leq 3$. The full discretization is realized by combining the standard finite element method with the backward Euler time-stepping scheme. Distinct from the globally Lipschitz setting, the error analysis becomes rather challenging and demanding, due to the presence of the cubic nonlinearity in the underlying model. By introducing two auxiliary approximation processes, we propose an appropriate decomposition of the considered error terms and introduce a novel approach of error analysis, to successfully recover the convergence rates of the numerical schemes. The approach is original and does not rely on high-order spatial regularity properties of the approximation processes. It is shown that the fully discrete scheme possesses convergence rates of order $ O(h^{\gamma} ) $ in space and order $ O( \tau^{ \frac{\gamma}{2} } )$ in time, subject to the spatial correlation of the noise process, characterized by $ \|A^{\frac{\gamma-1}2}Q^{\frac12}\|_{\mathcal{L}_2}<\infty, \, \gamma \in[\frac d3,2] $, $ d\in\{1,2,3\}$. In particular, a classical convergence rate of order $O(h^2 +\tau)$ is reachable, even in multiple space dimensions, when the aforementioned condition is fulfilled with $ \gamma = 2 $. Numerical examples confirm the previous findings.

Space–time discontinuous Galerkin methods for the ε-dependent stochastic Allen–Cahn equation with mild noise

Abstract We consider the $\varepsilon $-dependent stochastic Allen–Cahn equation with mild space–time noise posed on a bounded domain of $\mathbb{R}^2$. The positive parameter $\varepsilon $ is a measure for the inner layers width that are generated during evolution. This equation, when the noise depends only on time, has been proposed by Funaki (1999, Singular limit for stochastic reaction–diffusion equation and generation of random interfaces. Acta Math. Sin., 15, 407–438). The noise, although smooth, becomes white on the sharp interface limit $\varepsilon \rightarrow 0^+$. We construct a nonlinear discontinous Galerkin scheme with space–time finite elements of general type that are discontinuous in time. Existence of a unique discrete solution is proven by application of Brouwer’s Theorem. We first derive abstract error estimates and then, for the case of piecewise polynomial finite elements, we prove an error in expectation of optimal order. All the appearing constants are estimated in terms of the parameter $\varepsilon $. Finally, we present a linear approximation of the nonlinear scheme, for which we prove existence of solution and optimal error in expectation in piecewise linear finite element spaces. The novelty of this work is based on the use of a finite element formulation in space and in time in $2+1$-dimensional subdomains for a nonlinear parabolic problem. In addition this problem involves noise. These types of schemes avoid any Runge–Kutta-type discretization for the evolutionary variable, and seem to be very effective when applied to equations of such a difficulty.

Numerically computable a posteriori-bounds for the stochastic Allen–Cahn equation

The aim of this paper is the derivation of an a-posteriori error estimate for the numerical method based on an exponential scheme in time and spectral Galerkin methods in space. We obtain analytically a rigorous bound on the conditional mean square error, which is conditioned to the given realization of the data calculated by a numerical method. This bound is explicitly computable and uses only the computed numerical approximation. Thus one can check a-posteriori the error for a given numerical computation for a fixed discretization 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 and uniqueness 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.

Strong approximation of monotone stochastic partial differential equations driven by white noise

Abstract We establish an optimal strong convergence rate of a fully discrete numerical scheme for second-order parabolic stochastic partial differential equations with monotone drifts, including the stochastic Allen–Cahn equation, driven by an additive space-time white noise. Our first step is to transform the original stochastic equation into an equivalent random equation whose solution possesses more regularity than the original one. Then we use the backward Euler in time and spectral Galerkin in space to fully discretise this random equation. By the monotonicity assumption, in combination with the factorisation method and stochastic calculus in martingale-type 2 Banach spaces, we derive a uniform maximum norm estimation and a Hölder-type regularity for both stochastic and random equations. Finally, the strong convergence rate of the proposed fully discrete scheme is obtained. Several numerical experiments are carried out to verify the theoretical result.

Singular stochastic Allen–Cahn equations with dynamic boundary conditions

We prove a well-posedness result for stochastic Allen–Cahn type equations in a bounded domain coupled with generic boundary conditions. The (nonlinear) flux at the boundary aims at describing the interactions with the hard walls and is motivated by some recent literature in physics. The singular character of the drift part allows for a large class of maximal monotone operators, generalizing the usual double-well potentials. One of the main novelties of the paper is the absence of any growth condition on the drift term of the evolution, neither on the domain nor on the boundary. A well-posedness result for variational solutions of the system is presented using a priori estimates as well as monotonicity and compactness techniques. A vanishing viscosity argument for the dynamic on the boundary is also presented.

Strong convergence rates of semidiscrete splitting approximations for the stochastic Allen–Cahn equation

Abstract This article analyses an explicit temporal splitting numerical scheme for the stochastic Allen–Cahn equation driven by additive noise in a bounded spatial domain with smooth boundary in dimension $d\leqslant 3$. The splitting strategy is combined with an exponential Euler scheme of an auxiliary problem. When $d=1$ and the driving noise is a space–time white noise we first show some a priori estimates of this splitting scheme. Using the monotonicity of the drift nonlinearity we then prove that under very mild assumptions on the initial data this scheme achieves the optimal strong convergence rate $\mathcal{O}(\delta t^{\frac 14})$. When $d\leqslant 3$ and the driving noise possesses some regularity in space we study exponential integrability properties of the exact and numerical solutions. Finally, in dimension $d=1$, these properties are used to prove that the splitting scheme has a strong convergence rate $\mathcal{O}(\delta t)$.

Analysis of some splitting schemes for the stochastic Allen-Cahn equation

We introduce and analyze an explicit time discretization scheme for the one-dimensional stochastic Allen-Cahn, driven by space-time white noise. The scheme is based on a splitting strategy, and uses the exact solution for the nonlinear term contribution. We first prove boundedness of moments of the numerical solution. We then prove strong convergence results: first, L^2 ($\Omega$)-convergence of order almost 1/4, localized on an event of arbitrarily large probability, then convergence in probability of order almost 1/4. The theoretical analysis is supported by numerical experiments, concerning strong and weak orders of convergence.

Generation of interfaces for multi-dimensional stochastic Allen–Cahn equation with a noise smooth in space

In this paper, we study the generation of interfaces for a stochastic Allen–Cahn equation with general initial value in the multi-dimensional case that external noise is given by Q-Brownian motion. We prove that interfaces, for d-dimensional stochastic Allen–Cahn equation with scaling parameter , are generated at the time of order . Especially, in one-dimensional case, we give more detailed estimate and shape of the generated interface than that obtained in our previous result. Assuming that the Q-Brownian motion is smooth in space variable, we extend a comparison theorem for PDE to SPDE’s in order to prove the generation. Moreover, we connect the generated interface to the motion of interface in one-dimension. In this case, we consider the white noise only in time multiplied by as the noise term, where a is a smooth function which has a compact support. This is the special case of Q-Brownian motion. We take the time scale of order for studying the motion of interface.

On the discretisation in time of the stochastic Allen–Cahn equation

AbstractWe consider the stochastic Allen–Cahn equation perturbed by smooth additive Gaussian noise in a bounded spatial domain with smooth boundary in dimension , and study the semidiscretisation in time of the equation by an Euler type split‐step method with step size . We show that the method converges strongly with a rate . By means of a perturbation argument, we also establish the strong convergence of the standard backward Euler scheme with the same rate.

Particle representations for stochastic partial differential equations with boundary conditions

In this article, we study a weighted particle representation for a class of stochastic partial differential equations with Dirichlet boundary conditions. The locations and weights of the particles satisfy an infinite system of stochastic differential equations (SDEs). The evolution of the particles is modeled by an infinite system of stochastic differential equations with reflecting boundary condition and driven by independent finite dimensional Brownian motions. The weights of the particles evolve according to an infinite system of stochastic differential equations driven by a common cylindrical noise $W$ and interact through $V$, the associated weighted empirical measure. When the particles hit the boundary their corresponding weights are assigned a pre-specified value. We show the existence and uniqueness of a solution of the infinite dimensional system of stochastic differential equations modeling the location and the weights of the particles. We also prove that the associated weighted empirical measure $V$ is the unique solution of a nonlinear stochastic partial differential equation driven by $W$ with Dirichlet boundary condition. The work is motivated by and applied to the stochastic Allen-Cahn equation and extends the earlier of work of Kurtz and Xiong.

Small noise spectral gap asymptotics for a large system of nonlinear diffusions

We study the L^2 spectral gap of a large system of strongly coupled diffusions on unbounded state space and subject to a double-well potential. This system can be seen as a spatially discrete approximation of the stochastic Allen–Cahn equation on the one-dimensional torus. We prove upper and lower bounds for the leading term of the spectral gap in the small temperature regime with uniform control in the system size. The upper bound is given by an Eyring–Kramers-type formula. The lower bound is proven to hold also for the logarithmic Sobolev constant. We establish a sufficient condition for the asymptotic optimality of the upper bound and show that this condition is fullled under suitable assumptions on the growth of the system size. Our results can be reformulated in terms of a semiclassical Witten Laplacian in large dimension.

Stochastic Allen–Cahn equation with mobility

We introduce a class of stochastic Allen–Cahn equations with a mobility coefficient and colored noise. For initial data with finite free energy, we analyze the corresponding Cauchy problem on the d-dimensional torus in the time interval [0, T]. Assuming that \(d\le 3\) and that the potential has quartic growth, we prove existence and uniqueness of the solution as a process u in \(L^2\) with continuous paths, satisfying almost surely the regularity properties \(u\in C([0,T]; H^1)\) and \(u\in L^2([0,T];H^2)\).

Optimal Strong Rates of Convergence for a Space-Time Discretization of the Stochastic Allen–Cahn Equation with Multiplicative Noise

Abstract The stochastic Allen–Cahn equation with multiplicative noise involves the nonlinear drift operator 𝒜 ⁢ ( x ) = Δ ⁢ x - ( | x | 2 - 1 ) ⁢ x {{\mathscr{A}}(x)=\Delta x-(|x|^{2}-1)x} . We use the fact that 𝒜 ⁢ ( x ) = - 𝒥 ′ ⁢ ( x ) {{\mathscr{A}}(x)=-{\mathcal{J}}^{\prime}(x)} satisfies a weak monotonicity property to deduce uniform bounds in strong norms for solutions of the temporal, as well as of the spatio-temporal discretization of the problem. This weak monotonicity property then allows for the estimate sup 1 ≤ j ≤ J ⁡ 𝔼 ⁢ [ ∥ X t j - Y j ∥ 𝕃 2 2 ] ≤ C δ ⁢ ( k 1 - δ + h 2 ) \sup_{1\leq j\leq J}{\mathbb{E}}[\|X_{t_{j}}-Y^{j}\|_{{\mathbb{L}}^{2}}^{2}]% \leq C_{\delta}(k^{1-\delta}+h^{2}) for all small δ &gt; 0 {\delta&gt;0} , where X is the strong variational solution of the stochastic Allen–Cahn equation, while { Y j : 0 ≤ j ≤ J } {\{Y^{j}:0\leq j\leq J\}} solves a structure preserving finite element based space-time discretization of the problem on a temporal mesh { t j : 1 ≤ j ≤ J } {\{t_{j}:1\leq j\leq J\}} of size k &gt; 0 {k&gt;0} which covers [ 0 , T ] {[0,T]} .

A global existence and uniqueness result for a stochastic Allen–Cahn equation with constraint

This paper addresses the analysis of a time noise‐driven Allen–Cahn equation modelling the evolution of damage in continuum media in the presence of stochastic dynamics. The nonlinear character of the equation is mainly due to a multivoque maximal monotone operator representing a constraint on the damage variable, which is forced to take physically admissible values. By a Yosida approximation and a time‐discretization procedure, we prove a result of global‐in‐time existence and uniqueness of the solution to the stochastic problem. Copyright © 2017 John Wiley &amp; Sons, Ltd.