Stochastic Partial Differential Equations

In this paper, we propose a numerical method to solve the elliptic stochastic partial differential equations (SPDEs) obtained by Gaussian noises using an element free Galerkin method based on stabilized interpolating moving least square shape functions. The error estimates of the method is presented. The method is tested via several problems. The numerical results show the usefulness and accuracy of the new method.

An optimization based domain decomposition method for PDEs with random inputs

Since Pardoux and Peng firstly studied the following nonlinear backward stochastic differential equations in 1990. The theory of BSDE has been widely studied and applied, especially in the stochastic control, stochastic differential games, financial mathematics and partial differential equations. In 1994, Pardoux and Peng came up with backward doubly stochastic differential equations to give the probabilistic interpretation for stochastic partial differential equations. Backward doubly stochastic differential equations theory has been widely studied because of its importance in stochastic partial differential equations and stochastic control problems. In this article, we will study the theory of doubly stochastic systems and related topics further.

In this work, we develop a method for dynamic output feedback covariance control of the state covariance of linear dissipative stochastic partial differential equations (PDEs) using spatially distributed control actuation and sensing with noise. Such stochastic PDEs arise naturally in the modeling of surface height profile evolution in thin film growth and sputtering processes. We begin with the formulation of the stochastic PDE into a system of infinite stochastic ordinary differential equations (ODEs) by using modal decomposition. A finite-dimensional approximation is then obtained to capture the dominant mode contribution to the surface roughness profile (i.e., the covariance of the surface height profile). Subsequently, a state feedback controller and a Kalman-Bucy filter are designed on the basis of the finite-dimensional approximation. The dynamic output feedback covariance controller is subsequently obtained by combining the state feedback controller and the state estimator. The steady-state expected surface covariance under the dynamic output feedback controller is then estimated on the basis of the closed-loop finite-dimensional system. An analysis is performed to obtain a theoretical estimate of the expected surface covariance of the closed-loop infinite-dimensional system. Applications of the linear dynamic output feedback controller to the linearized stochastic Kuramoto-Sivashinsky equation are presented.

In this paper we prove the existence of a unique solution for a class of stochastic parabolic partial differential equations in bounded domains, with Dirichlet boundary conditions. The main tool is an equivalence result, provided by the stochastic characteristics method, between the stochastic equations under investigation and a class of deterministic parabolic equations with moving boundaries, depending on random coefficients. We show the existence of the solution to this last problem, thus providing a solution to the former.

The present review discusses recent developments in numerical techniques for the solution of systems with stochastic uncertainties. Such systems are modelled by stochastic partial differential equations (SPDEs), and techniques for their discretisation by stochastic finite elements (SFEM) are reviewed. Also, short overviews of related fields are given, e.g. of mathematical properties of random fields and SPDEs and of techniques for high-dimensional integration. After a summary of aspects of stochastic analysis, models and representations of random variables are presented. Then mathematical theories for SPDEs with stochastic operator are reviewed. Discretisation-techniques for random fields and for SPDEs are summarised and solvers for the resulting discretisations are reviewed, where the main focus lies on series expansions in the stochastic dimensions with an emphasis on Galerkin-schemes.

Stochastic partial differential equations (SPDE) are used for stochastic modelling, for instance, in the study of neuronal behaviour in neurophysiology and in building stochastic models for turbulence. Huebner, Khasminskii and Rozovskii (1993) started the investigation of the maximum likelihood estimation of the parameters involved in two types of SPDE's and extended their results for a class of parabolic SPDE's in Huebner and Rozovskii (1995). Prakasa Rao (1998,2000) obtained Bernstein - von Mises type theorems for a class of parabolic SPDE's and investigated the properties of Bayes estimators of parameters involved in such SPDE's. In all the papers cited earlier, it was assumed that a continuous observation of a random field u ε ( x, t) satisfying the SPDE over the region [0, 1] × [0, T] is available. It is obvious that this assumption is not tenable in practice and the problem of interest is to develop methods of estimation of parameters from the random field u ε ( x, t) observed at discrete times t and at discrete positions x or from the Fourier coefficients u iε ( t) observed at discrete time instants. We construct consistent and asymptotically normal estimators of the parameter based on the Fourier coefficients u iε ( t) observed at discrete times [Formula: see text] tends to infinity. AMS (2000) Subject classification: Primary 62 M 40; Secondary 60 H 15.

In the present work, we consider an optimal control for a three-factor stochastic factor model. We assume that one of the factors is not observed and use classical filtering technique to transform the partial observation control problem for stochastic differential equation (SDE) to a full observation control problem for stochastic partial differential equation (SPDE). We then give a sufficient maximum principle for a system of controlled SDEs and degenerate SPDE. We also derive an equivalent stochastic maximum principle. We apply the obtained results to study a pricing and hedging problem of a commodity derivative at a given location, when the convenience yield is not observable.

In this article the pathwise numerical approximation of semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive noise is considered. A new numerical scheme for the time and space discretization of such SPDEs is proposed, and this scheme is shown to converge for such SPDEs faster than standard numerical schemes such as the linear implicit Euler scheme or the linear-implicit Crank-Nicholson scheme. The suggested scheme takes advantage of the smoothing effect of the dominant linear operator and of two linear functionals of the noise process of the SPDE. The abstract setting under which the scheme is analyzed includes SPDEs driven by fractional Brownian motions too.

We are interested in solving stochastic elliptic partial differential equations (SPDE) by sampling. Approximating statistics of the stochastic solution requires computing a large number of samples, typically using an iterative method, and accelerating the resolution of these computations presents a considerable interest. In this context, we focus on the preconditioning of domain decomposition methods. We propose a stochastic preconditioner adapted to the sampled parameter field. To prevent having to determine the preconditioner for each sample, we use its truncated Karhunen-Loève (KL) expansion. The stochastic preconditioner is constructed as a surrogate of the original system based on Polynomial Chaos expansions in the dominant KL random variables. The preconditioner is computed off-line, in a preprocessing stage, and subsequently evaluated during the sampling stage (in-line). In this work, we present a review of two main contributions, which construct stochastic preconditioners in different contexts. In the first contribution (Reis et al., Preconditioning the Schwartz Method for Stochastic Elliptic Equations, Submitted), we consider an overlapping setting and construct a stochastic preconditioner for accelerating the additive Schwarz method. In the second contribution (Reis et al., Acceleration of Domain Decomposition Methods for Stochastic Elliptic Equations, Submitted), we consider a nonoverlapping setting and construct a Schur complement system. The stochastic preconditioner is used to accelerate a Conjugate Gradient method used to solve the Schur system. In both cases, numerical experiments demonstrate the significant acceleration provided by the stochastic preconditioner, compared to deterministic preconditioners such as the one based on the mean or median of the coefficient parameter.

We study the Allen--Cahn equation with a cubic-quintic nonlinear term and a $Q$-trace-class stochastic forcing in two spatial dimensions. This stochastic partial differential equation (SPDE) is used as a test case to understand how numerical continuation methods can be carried over to the SPDE setting. First, we compute the deterministic bifurcation diagram for the PDE, i.e., without stochastic forcing. In this case, two locally asymptotically stable steady state solution branches exist upon variation of the linear damping term. Then we consider the Lyapunov operator equation for the locally linearized system around steady states for the SPDE. We discretize the full SPDE using a combination of finite differences and spectral noise approximation obtaining a finite-dimensional system of stochastic ordinary differential equations (SODEs). The large system of SODEs is used to approximate the Lyapunov operator equation via covariance matrices. The covariance matrices are numerically continued along the two bifurcation branches. We show that we can quantify the stochastic fluctuations along the branches. We also demonstrate scaling laws near branch and fold bifurcation points. Furthermore, we perform computational tests to show that, even with a suboptimal computational setup, we can quantify the subexponential-timescale fluctuations near the deterministic steady states upon stochastic forcing on a standard desktop computer setup. Hence, the proposed method for numerical continuation of SPDEs has the potential to allow for rapid parametric uncertainty quantification of spatio-temporal stochastic systems.

SummaryIncreasingly larger data sets of processes in space and time ask for statistical models and methods that can cope with such data. We show that the solution of a stochastic advection–diffusion partial differential equation provides a flexible model class for spatiotemporal processes which is computationally feasible also for large data sets. The Gaussian process defined through the stochastic partial differential equation has, in general, a non-separable covariance structure. Its parameters can be physically interpreted as explicitly modelling phenomena such as transport and diffusion that occur in many natural processes in diverse fields ranging from environmental sciences to ecology. To obtain computationally efficient statistical algorithms, we use spectral methods to solve the stochastic partial differential equation. This has the advantage that approximation errors do not accumulate over time, and that in the spectral space the computational cost grows linearly with the dimension, the total computational cost of Bayesian or frequentist inference being dominated by the fast Fourier transform. The model proposed is applied to post-processing of precipitation forecasts from a numerical weather prediction model for northern Switzerland. In contrast with the raw forecasts from the numerical model, the post-processed forecasts are calibrated and quantify prediction uncertainty. Moreover, they outperform the raw forecasts, in the sense that they have a lower mean absolute error.

Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos

Linear‐implicit versions of strong Taylor numerical schemes for finite dimensional Itô stochastic differential equations (SDEs) are shown to have the same order as the original scheme. The combined truncation and global discretization error of an γ strong linear‐implicit Taylor scheme with time‐step Δ applied to the N dimensional Itô‐Galerkin SDE for a class of parabolic stochastic partial differential equation (SPDE) with a strongly monotone linear operator with eigenvalues λ1 ≤ λ2 ≤ ⋯ in its drift term is then estimated by where the constant K depends on the initial value, bounds on the other coefficients in the SPDE and the length of the time interval under consideration.

Realistic mathematical models of physical processes contain uncertainties. These models are often described by stochastic differential equations (SDEs) or stochastic partial differential equations (SPDEs) with multiplicative noise. The uncertainties in the right-hand side or the coefficients are represented as random fields. To solve a given SPDE numerically one has to discretise the deterministic operator as well as the stochastic fields. The total dimension of the SPDE is the product of the dimensions of the deterministic part and the stochastic part. To approximate random fields with as few random variables as possible, but still retaining the essential information, the Karhunen–Loève expansion (KLE) becomes important. The KLE of a random field requires the solution of a large eigenvalue problem. Usually it is solved by a Krylov subspace method with a sparse matrix approximation. We demonstrate the use of sparse hierarchical matrix techniques for this. A log-linear computational cost of the matrix-vector product and a log-linear storage requirement yield an efficient and fast discretisation of the random fields presented.