Efficient Adaptive Multilevel Stochastic Galerkin Approximation Using Implicit A Posteriori Error Estimation

Abstract

Partial differential equations (PDEs) with inputs that depend on infinitely many parameters pose serious theoretical and computational challenges. Sophisticated numerical algorithms that automatically determine which parameters need to be activated in the approximation space in order to estimate a quantity of interest to a prescribed error tolerance are needed. For elliptic PDEs with parameter-dependent coefficients, stochastic Galerkin finite element methods (SGFEMs) have been well studied. Under certain assumptions, it can be shown that there exists a sequence of SGFEM approximation spaces for which the energy norm of the error decays to zero at a rate that is independent of the number of input parameters. However, it is not clear how to adaptively construct these spaces in a practical and computationally efficient way. We present a new adaptive SGFEM algorithm that tackles elliptic PDEs with parameter-dependent coefficients quickly and efficiently. We consider approximation spaces with a multilevel structure-where each solution mode is associated with a finite element space on a potentially different mesh-and use an implicit a posteriori error estimation strategy to steer the adaptive enrichment of the space. At each step, the components of the error estimator are used to assess the potential benefits of a variety of enrichment strategies, including whether or not to activate more parameters. No marking or tuning parameters are required. Numerical experiments for a selection of test problems demonstrate that the new method performs optimally in that it generates a sequence of approximations for which the estimated energy error decays to zero at the same rate as the error for the underlying finite element method applied to the associated parameter-free problem.

Adaptive stochastic Galerkin FEM for lognormal coefficients in hierarchical tensor representations

“…In order to achieve this, different paths have been pursued successfully. As a first approach, sparse approximations as in [15,16,19] or [4,5,10] with either a residual based or a hierarchical a posteriori error estimators can be computed. Here, the aim is to restrict an exponentially large discrete basis to the most relevant functions explictly by iteratively constructing a quasi-optimal subset.…”

Block Preconditioning of Stochastic Galerkin Problems: New Two-sided Guaranteed Spectral Bounds

“…If U and W are close to orthogonal, the preconditioning based on the splitting (4.15) enables to estimate the error reduction when the approximation space U is enriched by W in the Galerkin method. This can be exploited in adaptive algorithms, where W is sometimes called the 'detail' space; see Remark 4.13 and, e.g., [5,7,27].…”

“…Having a good preconditioning method or, in other words, a good and feasible approximation of A −1 , we may also efficiently estimate a posteriori the energy norm of the error during iterative solution processes [1,5,6,9,17]. This estimate can be used in adaptive algorithms [5,7,8]. In practice, matrix A is never built explicitly, only matrix-vector products are evaluated ( [25]).…”

Robust a posteriori error estimation for parameter-dependent linear elasticity equations

“…The theoretical basis for automatic refinement algorithms for parametric PDE problems has been established in the last decade. Building on the pioneering work of Cohen et al, see [9], recent work by Bachmayr et al [2] and by Crowder et al [11] opens up the possibility of designing optimal algorithms from a best approximation perspective. A priori analysis for so-called best N -term approximations of standard mixed formulations of stochastic and multiscale elasticity problems can be found in [30,19].…”