Parabolic Stochastic Partial Differential Equations Research Articles

Degenerate parabolic stochastic partial differential equations

We study the Cauchy problem for a scalar semilinear degenerate parabolic partial differential equation with stochastic forcing. In particular, we are concerned with the well-posedness in any space dimension. We adapt the notion of kinetic solution which is well suited for degenerate parabolic problems and supplies a good technical framework to prove the comparison principle. The proof of existence is based on the vanishing viscosity method: the solution is obtained by a compactness argument as the limit of solutions of nondegenerate approximations.

Covariance structure of parabolic stochastic partial differential equations

In this paper parabolic random partial differential equations and parabolic stochastic partial differential equations driven by a Wiener process are considered. A deterministic, tensorized evolution equation for the second moment and the covariance of the solutions of the parabolic stochastic partial differential equations is derived. Well-posedness of a space–time weak variational formulation of this tensorized equation is established.

Preface to BIT 53:1

This is the fiftythird volume since the start of BIT, and the eleventh that I have the honor to present to you. We get a healthy number of new contributions from all corners of the World. The papers in this issue have been published on the Springer home page since last Summer and Fall, so many of you are already working on, making new progress on the problems treated here and cultivating new directions. Let me take this opportunity to say thank you to Per Lotstedt, who has been on our editorial board all since 1999, and made a great share of the work to attract good manuscripts and help authors to keep up the standard of those that get printed. I wish you good luck in your future endeavors, and hope that BIT will continue to be present in the dynamic atmosphere in the Uppsala group of Scientific Computing! In this issue, we continue to give new contributions to the numerical solution of differential equations, approximation, and linear algebra. One paper by di Pietro et al. may look unfamiliar to you. It is on a programming language construction, usable in finite element computations. There is a lively research going on to put knowledge from application areas into a computer programming language. Let BIT take part in these developments! These are the papers: Andrea Barth, Annika Lang, and Christoph Schwab describe a multilevel Monte Carlo method for parabolic stochastic partial differential equations. They balance accuracy in following solution paths to sampling errors, by following few paths in a fine grid, and making a large sample of coarse grid solutions. Begona Cano studies the conservation of invariants by symmetric multistep cosine methods for second-order partial differential equations. These methods give exact

Higher Order Spatial Approximations for Degenerate Parabolic Stochastic Partial Differential Equations

We consider an implicit finite difference scheme on uniform grids in time and space for the Cauchy problem for a second order parabolic stochastic partial differential equation where the parabolicity condition is allowed to degenerate. Such equations arise in the nonlinear filtering theory of partially observable diffusion processes. We show that the convergence of the spatial approximation can be accelerated to an arbitrarily high order, under suitable regularity assumptions, by applying an extrapolation technique.

Representation theorems for SPDEs via backward doubly

In this paper we establish a probabilistic representation for the spatial gradient ofthe viscosity solution to a quasilinear parabolic stochastic partial differential equations(SPDE, for short) in the spirit of the Feynman-Kac formula, without using thederivatives of the coefficients of the corresponding backward doubly stochastic differentialequations (FBDSDE, for short).

Weak Second Order Explicit Stabilized Methods for Stiff Stochastic Differential Equations

We introduce a new family of explicit integrators for stiff Ito stochastic differential equa- tions (SDEs) of weak order two. These numerical methods belong to the class of one-step stabilized methods with extended stability domains and do not suffer from the stepsize re- duction faced by standard explicit methods. The family is based on the standard second or- der orthogonal Runge-Kutta Chebyshev methods (ROCK2) for deterministic problems. The convergence, and the mean-square and asymptotic stability properties of the methods are analyzed. Numerical experiments, including applications to nonlinear SDEs and parabolic stochastic partial differential equations are presented and confirm the theoretical results.

Analytic Regularity and GPC Approximation for Control Problems Constrained by Linear Parametric Elliptic and Parabolic PDEs

This paper deals with linear-quadratic optimal control problems constrained by a parametric or stochastic elliptic or parabolic partial differential equation (PDE). We address the (difficult) case that the state equation depends on a countable number of parameters i.e., on $\sigma_j$ with $j\in\mathbb{N}$, and that the PDE operator may depend nonaffinely on the parameters. We consider tracking-type functionals and distributed as well as boundary controls. Building on recent results in [A. Cohen, R. DeVore, and Ch. Schwab, Found. Comput. Math., 10 (2010), pp. 615--646; Anal. Appl., 9 (2011), pp. 1--37], we show that the state and the control are analytic as functions depending on these parameters $\sigma_j$. We establish sparsity of generalized polynomial chaos (gpc) expansions of both state and control in terms of the stochastic coordinate sequence $\sigma = (\sigma_j)_{j\geq 1}$ of the random inputs, and we prove convergence rates of best $N$-term truncations of these expansions. Such truncations are the key for subsequent computations since they do not assume that the stochastic input data has a finite expansion. In a follow-up paper [A. Kunoth and Ch. Schwab, Sparse adaptive tensor Galerkin approximations of stochastic PDE-constrained control problems, in preparation], we explain two methods for how such best $N$-term truncations can be computed practically: by greedy-type algorithms as in [Ch. Schwab and C. J. Gittelson, Acta Numer., 20 (2011), pp. 291--467; C. J. Gittelson, Report 2011-12, Seminar for Applied Mathematics, ETH Zürich, Zürich, Switzerland, 2011], or by multilevel Monte-Carlo methods as in [F. Y. Kuo, Ch. Schwab, and I. H. Sloan, SIAM J. Numer. Anal., 50 (2012), pp. 3351--3374]. The sparsity result allows, in conjunction with adaptive wavelet Galerkin schemes, for sparse, adaptive tensor discretizations of control problems constrained by linear elliptic and parabolic PDEs developed in [W. Dahmen and A. Kunoth, SIAM J. Control Optim., 43 (2005), pp. 1640--1675; M. D. Gunzburger and A. Kunoth, SIAM J. Control. Optim., 49 (2011), pp. 1150--1170; A. Kunoth, Numer. Algorithms, 39 (2005), pp. 199--220]; see [A. Kunoth and Ch. Schwab, Sparse adaptive tensor Galerkin approximations of stochastic PDE-constrained control problems, in preparation].

Sharp estimates for metastable lifetimes in parabolic SPDEs: Kramers' law and beyond

We prove a Kramers-type law for metastable transition times for a class of one-dimensional parabolic stochastic partial differential equations (SPDEs) with bistable potential. The expected transition time between local minima of the potential energy depends exponentially on the energy barrier to overcome, with an explicit prefactor related to functional determinants. Our results cover situations where the functional determinants vanish owing to a bifurcation, thereby rigorously proving the results of formal computations announced in [Berglund and Gentz, J. Phys. A 42:052001 (2009)]. The proofs rely on a spectral Galerkin approximation of the SPDE by a finite-dimensional system, and on a potential-theoretic approach to the computation of transition times in finite dimension.

Spatial Besov regularity for semilinear stochastic partial differential equations on bounded Lipschitz domains

We study the spatial regularity of semilinear parabolic stochastic partial differential equations on bounded Lipschitz domains 𝒪⊆ ℝ d in the scale , 1/τ=α/d+1/p, p≥2 fixed. The Besov smoothness in this scale determines the order of convergence that can be achieved by adaptive numerical algorithms and other nonlinear approximation schemes. The proofs are performed by establishing weighted Sobolev estimates and combining them with wavelet characterizations of Besov spaces.

Numerical approximation of multiplicative SPDEs

We prove convergence of the stochastic exponential time differencing scheme for parabolic stochastic partial differential equations (SPDEs) with one-dimensional multiplicative noise. We examine convergence for fourth-order SPDEs and consider as an example the Swift–Hohenberg equation. After examining convergence, we present preliminary evidence of a shift in the deterministic pinning region [J. Burke and E. Knobloch, Localized states in the generalized Swift–Hohenberg equation, Phys. Rev. E. 73 (2006), pp. 056211-1–15; J. Burke and E. Knobloch, Snakes and ladders: Localized states in the Swift–Hohenberg equation, Phys. Lett. A 360 (2007), pp. 681–688; Y.-P. Ma, J. Burke, and E. Knobloch, Snaking of radial solutions of the multi-dimensional Swift–Hohenberg equation: A numerical study, Physica D 239 (2010), pp. 1867–1883; S. McCalla and B. Sandstede, Snaking of radial solutions of the multi-dimensional Swift–Hohenberg equation: A numerical study, Physica D 239 (2010), pp. 1581–1592] with space–time white noise.

Multilevel Monte Carlo method with applications to stochastic partial differential equations

In this work, the approximation of Hilbert-space-valued random variables is combined with the approximation of the expectation by a multilevel Monte Carlo (MLMC) method. The number of samples on the different levels of the multilevel approximation are chosen such that the errors are balanced. The overall work then decreases in the optimal case to O(h −2) if h is the error of the approximation. The MLMC method is applied to functions of solutions of parabolic and hyperbolic stochastic partial differential equations as needed, for example, for option pricing. Simulations complete the paper.

Mean-square stability and error analysis of implicit time-stepping schemes for linear parabolic SPDEs with multiplicative Wiener noise in the first derivative

In this article, we extend a Milstein finite difference scheme introduced in 8 for a certain linear stochastic partial differential equation (SPDE) to semi-implicit and fully implicit time-stepping as introduced by Szpruch 32 for stochastic differential equations (SDEs). We combine standard finite difference Fourier analysis for partial differential equations with the linear stability analysis in 3 for SDEs to analyse the stability and accuracy. The results show that Crank–Nicolson time-stepping for the principal part of the drift with a partially implicit but negatively weighted double Itô integral gives unconditional stability over all parameter values and converges with the expected order in the mean-square sense. This opens up the possibility of local mesh refinement in the spatial domain, and we show experimentally that this can be beneficial in the presence of reduced regularity at boundaries.

A Weighted Sobolev Space Theory of Parabolic Stochastic PDEs on Non-smooth Domains

In this paper we study parabolic stochastic partial differential equations defined on arbitrary bounded domain $\cO \subset \bR^d$ allowing Hardy inequality: $$ \int_{\cO}|\rho^{-1}g|^2\,dx\leq C\int_{\cO}|g_x|^2 dx, \quad \forall g\in C^{\infty}_0(\cO), $$ where $\rho(x)=\text{dist}(x,\partial \cO)$. Existence and uniqueness results are given in weighted Sobolev spaces $\frH^{\gamma}_{p,\theta}(\cO,T)$, where $p\in [2,\infty)$, $\gamma\in \bR$ is the number of derivatives of solutions and $\theta$ controls the boundary behavior of solutions. Furthermore several H\"older estimates of the solutions are also obtained. It is allowed that the coefficients of the equations blow up near the boundary.

Multilevel Monte Carlo method for parabolic stochastic partial differential equations

We analyze the convergence and complexity of multilevel Monte Carlo discretizations of a class of abstract stochastic, parabolic equations driven by square integrable martingales. We show under low regularity assumptions on the solution that the judicious combination of low order Galerkin discretizations in space and an Euler-Maruyama discretization in time yields mean square convergence of order one in space and of order 1/2 in time to the expected value of the mild solution. The complexity of the multilevel estimator is shown to scale log-linearly with respect to the corresponding work to generate a single path of the solution on the finest mesh, resp. of the corresponding deterministic parabolic problem on the finest mesh. © 2012 Springer Science + Business Media B.V.

Stochastic exponential integrators for the finite element discretization of SPDEs for multiplicative and additive noise

We consider the numerical approximation of a general second-order semilinear parabolic stochastic partial differential equation driven by multiplicative and additive space–time noise. We examine convergence of exponential integrators for multiplicative and additive noise. We consider noise that is in a trace class and give a convergence proof in the root-mean-square L2 norm. We discretize in space with the finite element method and in our implementation we examine both the finite element and the finite volume methods. We present results for a linear reaction–diffusion equation in two dimensions as well as a nonlinear example of a two-dimensional stochastic advection–diffusion–reaction equation motivated from realistic porous media flow.

Weak Order for the Discretization of the Stochastic Heat Equation Driven by Impulsive Noise

We study the approximation of the distribution of X T , where (X t ) t ∈ [0, T] is a Hilbert space valued stochastic process that solves a linear parabolic stochastic partial differential equation driven by an impulsive space time noise, $$ dX_t+AX_t\,dt= Q^{1/2}\,dZ_t,\quad X_0=x_0\in H,\quad t\in [0,T]. $$ Here (Z t ) t ∈ [0, T] is an impulsive cylindrical process and the operator Q describes the spatial covariance structure of the noise; we assume that A − α has finite trace for some α > 0 and that A β Q is bounded for some β ∈ (α − 1, α]. A discretized solution $(X_h^n)_{n\in\{0,1,\ldots,N\}}$ is defined via the finite element method in space (parameter h > 0) and a θ-method in time (parameter Δt = T/N). For $\varphi \in C^2_b(H;{\mathbb R})$ we show an integral representation for the error $|{\mathbb E}\varphi(X^N_h)-{\mathbb E}\varphi(X_T)|$ and prove that $$ \left|{\mathbb E}\varphi\left(X^N_h\right)-{\mathbb E}\varphi(X_T)\right|=O\left(h^{2\gamma}+\left(\Delta t\right)^{\gamma}\right) $$ where γ < 1 − α + β. This is the same order of convergence as in the case of a Gaussian space time noise, which has been obtained in a paper by Debussche and Printems (Math Comput 78:845–863, 2009). Our result also holds for a combination of impulsive and Gaussian space time noise.

Analysis of an HMM Time-Discretization Scheme for a System of Stochastic PDEs

We consider the discretization in time of a system of parabolic stochastic partial differential equations with slow and fast components; the fast equation is driven by an additive space-time white noise. The numerical method is inspired by the averaging principle satisfied by this system and fits to the framework of heterogeneous multiscale methods. The slow and the fast components are approximated with two coupled numerical semi-implicit Euler schemes depending on two different time step sizes. We derive bounds of the approximation error on the slow component in the strong sense---approximation of trajectories---and in the weak sense---approximation of the laws. The estimates generalize the results of [W. E, D. Liu, and E. Vanden-Eijnden, Comm. Pure Appl. Math., 58 (2005), pp. 1544--1585] in the case of infinite dimensional processes.

Approximate Euler Method for Parabolic Stochastic Partial Differential Equations Driven by Space-Time Lévy Noise

We consider parabolic stochastic partial differential equations driven by space-time Levy noise. Different discretization methods to accurately simulate jumps are proposed and analyzed in the context of an implicit time-discretization. Computational studies based on a finite element discretization are provided to illustrate combined truncation and time-discretization effects.

A Multistage Wiener Chaos Expansion Method for Stochastic Advection-Diffusion-Reaction Equations

Using Wiener chaos expansion (WCE), we develop numerical algorithms for solving second-order linear parabolic stochastic partial differential equations (SPDEs). We propose a deterministic WCE-based algorithm for computing moments of the SPDE solutions without any use of the Monte Carlo technique. We also compare the proposed deterministic algorithm with two other numerical methods based on the Monte Carlo technique and demonstrate that the new method is more efficient for highly accurate solutions. Numerical tests verify that the scheme is of mean-square order $O( \frac{\Delta^{N/2}}{\sqrt{(N+1)!}})$ for diffusion and for diffusion-reaction SPDEs with constant or variable coefficients, where $\Delta$ is the time step, and $N$ is the Wiener chaos order.

Accelerated Spatial Approximations for Time Discretized Stochastic Partial Differential Equations

The present article investigates the convergence of a class of space-time discretization schemes for the Cauchy problem for linear parabolic stochastic partial differential equations defined on the whole space. Sufficient conditions are given for accelerating the convergence of the scheme with respect to the spatial approximation to higher order accuracy by an application of Richardson's method. This work extends the results of Gyöngy and Krylov [SIAM J. Math. Anal., 42 (2010), pp. 2275--2296] to schemes that discretize in time as well as space.