Numerical solution of partial differential equations with random coefficients: a stochastic finite element approach

Mathematical models of engineering systems and physical processes typically take the form of a partial differential equation (PDE). Variability or uncertainty on coefficients of a PDE can be expressed by introducing random variables, random fields or random processes into the PDE . Recently developed stochastic finite element methods enable the construction of high-order accurate solutions of a stochastic PDE , while reducing the high computational cost of more standard uncertainty quantification methods, such as the Monte Carlo simulation method. In this talk, we outline the methodology of the two main variants of the stochastic finite element method, i.e., the stochastic collocation and the stochastic Galerkin method. Both methods transform a stochastic PDE into a system of deterministic PDEs. In the latter case, the number of deterministic PDEs is generally smaller than for the stochastic collocation method, but the system is more complicated to solve. We will point out how these methods can be applied to efficiently solve a small flow problem through random porous media. Multigrid techniques lead to a fast solution of the resulting systems. These simulations can be extended to solve groundwater flow problems or to perform oil reservoir simulations.status: publishe

Optimal control problems constrained by elliptic stochastic partial differential equations

The optimal control of problems that are constrained by partial differential equations with uncertainties is addressed. The inclusion of the stochastic dimension provides additional freedom in the definition of cost functionals, for example by including statistics of the response in a cost functional. We formulate a one-shot approach to the stochastic optimal control problems and solve the resulting equations via a stochastic Galerkin or collocation finite element method. The stochastic collocation method is often preferred over the Galerkin approach as it converts a stochastic problem into a collection of decoupled deterministic problems. It is shown however that this so-called non-intrusivity property of the collocation method does not hold for a large class of stochastic PDE-constrained optimisation problems. The efficient solution of stochastic Galerkin finite element problems can hinge on the development and application of effective preconditioners. This aspect is addressed with two preconditioners that take the specific structure of the Galerkin one-shot systems into account. Numerical examples support the findings. The presented framework is sufficiently general to also consider a class of stochastic inverse problems and numerical examples of this type are also presented.status: publishe

A comparison of iterative solvers for the stochastic finite element method

The stochastic finite element method is an important technique for solving stochastic partial differential equations (PDEs). This method approximates the solution of the PDE by a generalized polynomial chaos expansion. By using a Galerkin projection in the stochastic dimension, the stochastic PDE is transformed into a coupled set of deterministic PDEs. A finite element discretization converts the deterministic PDEs into a high dimensional algebraic system. Specialized iterative solvers are required to solve this system. A number of specialized solvers have already been proposed, for example a conjugate gradient solver with multigrid preconditioning based on coarsening the spatial domain, or conjugate gradients combined with a Jacobi-based preconditioner. Here, we shall present an overview of solution approaches. We start from iterative methods based on a block splitting of the algebraic system matrices. Next, we extend these methods for use as preconditioner or for use in a multilevel context. Then, the various solvers are compared based on their convergence properties, computational cost and implementation effort. Our findings are illustrated on two numerical problems. The first is a steady-state diffusion problem with a discontinuous random field as diffusion coefficient. The second is a deterministic diffusion problem defined on a random domain.status: publishe

Iterative solvers for the stochastic finite element method

This paper presents an overview and comparison of iterative solvers for linear stochastic partial differential equations (PDEs). A stochastic Galerkin finite element discretization is applied to transform the PDE into a coupled set of deterministic PDEs . Specialized solvers are required to solve the very high-dimensional systems that result after a finite e lement discretization of the resulting set. This paper discusses one-level iterative methods, based on matrix splitting techniques; multigrid methods, which apply a coarsening in the spatial dimension; a nd multilevel methods, which make use of the hierarchical structure of the stochastic discretization. Also Krylov solvers with suitable preconditioning are addressed. A local Fourier analysis provides quantitative convergence properties. The efficiency and robustness of the methods are illustrated on two nontrivial numerical problems. The multigrid solver with block smoother yields the most robust convergence properties, though a cheaper point smoother performs as well in most cases. Mult ilevel methods based on coarsening the stochastic dimension perform in general poorly due to a lar ge computational cost per iteration. Moderate size problems can be solved very quickly by a Krylov method with a mean-based precon- ditioner. For larger spatial and stochastic discretizations, however, this approach suffers from its nonoptimal convergence properties. © 2010 Society for Industrial and Applied Mathematics.status: publishe

Iterative solvers for the stochastic finite element method

The stochastic finite element method is a recent technique for solving partial differential equations (PDE) that contain stochastic coefficients. It can result in high-order accurate stochastic solutions while avoiding the computation of numerous Monte Carlo simulations. Two prominent variants are the stochastic collocation and stochastic Galerkin technique. The former samples a stochastic PDE in a set of well-chosen collocation points and results in a number of decoupled deterministic PDEs. The latter transforms a stochastic PDE into a coupled system of deterministic PDEs. We shall illustrate the action of both techniques in the context of a nonlinear PDE with stochastic coefficients. The applicability of the stochastic Galerkin method strongly depends on efficient solvers for the corresponding high-dimensional algebraic systems. In this talk, we focus on a comparison and analysis of iterative solution methods for stochastic Galerkin discretizations of linear stochastic PDEs [1]. The tensor product structure of the algebraic systems is exploited in both one-level as well as multilevel iterative methods. Using multigrid techniques, a convergence rate independent of the stochastic and spatial discretization parameters can be obtained. This may lead however for certain types of problems to a larger solution time than, for example, a Krylov solver with mean-based preconditioner. Numerical experiments illustrate the performance of the iterative solvers for some non-trivial problems. [1] E. Rosseel and S. Vandewalle. Iterative solvers for the stochastic finite element method. SIAM J. Sci. Comput., 2009. Submittedstatus: publishe

Iterative solvers for stochastic Galerkin finite element discretizations

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Iterative solvers for the stochastic finite element method

The need for accurate simulations and reliability estimates of predictions has led to a variety of techniques to mathematically model and quantify uncertainty and variability. Depending on the problem characteristics, techniques based on probability theory or non-probabilistic techniques, for example based on fuzzy theory, are used. In the first category, the stochastic finite element method has received a lot of attention in recent years. This method transforms a stochastic partial differential equation (PDE) into a large coupled system of deterministic PDEs. It has been successfully applied to several engineering disciplines, for example structural mechanics, fluid dynamics and thermal engineering. Its applicability can further be enhanced by developing efficient numerical solution techniques for the resulting discretized systems. In this work, we will focus on multigrid solvers for the algebraic systems that result from stochastic finite element discretizations. As the stochastic finite element method increases the dimension of the original problem, specialized large-scale solvers are required. Multigrid methods are widely used to solve large-scale algebraic systems that result from discretized PDEs. We have developed a multigrid solution method for stochastic Galerkin finite element discretizations. The method is applicable to stationary and time-dependent problems, possibly combined with a high-order time discretization scheme. In addition, we constructed a multigrid approach that combines a hierarchy in the spatial and the stochastic dimension. We will point out the favorable convergence properties of our developed solvers and compare them to the state-of-art.status: publishe