12 research outputs found
Convergence of adaptive stochastic Galerkin FEM
We propose and analyze novel adaptive algorithms for the numerical solution
of elliptic partial differential equations with parametric uncertainty. Four
different marking strategies are employed for refinement of stochastic Galerkin
finite element approximations. The algorithms are driven by the energy error
reduction estimates derived from two-level a posteriori error indicators for
spatial approximations and hierarchical a posteriori error indicators for
parametric approximations. The focus of this work is on the mathematical
foundation of the adaptive algorithms in the sense of rigorous convergence
analysis. In particular, we prove that the proposed algorithms drive the
underlying energy error estimates to zero
A posteriori error estimation and adaptivity in stochastic Galerkin FEM for parametric elliptic PDEs: beyond the affine case
We consider a linear elliptic partial differential equation (PDE) with a
generic uniformly bounded parametric coefficient. The solution to this PDE
problem is approximated in the framework of stochastic Galerkin finite element
methods. We perform a posteriori error analysis of Galerkin approximations and
derive a reliable and efficient estimate for the energy error in these
approximations. Practical versions of this error estimate are discussed and
tested numerically for a model problem with non-affine parametric
representation of the coefficient. Furthermore, we use the error reduction
indicators derived from spatial and parametric error estimators to guide an
adaptive solution algorithm for the given parametric PDE problem. The
performance of the adaptive algorithm is tested numerically for model problems
with two different non-affine parametric representations of the coefficient.Comment: 32 pages, 4 figures, 6 table
A posteriori error estimation and adaptivity in stochastic Galerkin FEM for parametric elliptic PDEs: beyond the affine case
We consider a linear elliptic partial differential equation (PDE) with a
generic uniformly bounded parametric coefficient. The solution to this PDE
problem is approximated in the framework of stochastic Galerkin finite element
methods. We perform a posteriori error analysis of Galerkin approximations and
derive a reliable and efficient estimate for the energy error in these
approximations. Practical versions of this error estimate are discussed and
tested numerically for a model problem with non-affine parametric
representation of the coefficient. Furthermore, we use the error reduction
indicators derived from spatial and parametric error estimators to guide an
adaptive solution algorithm for the given parametric PDE problem. The
performance of the adaptive algorithm is tested numerically for model problems
with two different non-affine parametric representations of the coefficient.Comment: 32 pages, 4 figures, 6 table
Guaranteed quasi-error reduction of adaptive Galerkin FEM for parametric PDEs with lognormal coefficients
Solving high-dimensional random parametric PDEs poses a challenging
computational problem. It is well-known that numerical methods can greatly
benefit from adaptive refinement algorithms, in particular when functional
approximations in polynomials are computed as in stochastic Galerkin and
stochastic collocations methods. This work investigates a residual based
adaptive algorithm used to approximate the solution of the stationary diffusion
equation with lognormal coefficients. It is known that the refinement procedure
is reliable, but the theoretical convergence of the scheme for this class of
unbounded coefficients remains a challenging open question. This paper advances
the theoretical results by providing a quasi-error reduction results for the
adaptive solution of the lognormal stationary diffusion problem. A
computational example supports the theoretical statement
On the convergence of adaptive stochastic collocation for elliptic partial differential equations with affine diffusion
Convergence of an adaptive collocation method for the stationary parametric diffusion equation with finite-dimensional affine coefficient is shown. The adaptive algorithm relies on a recently introduced residual-based reliable a posteriori error estimator. For the convergence proof, a strategy recently used for a stochastic Galerkin method with an hierarchical error estimator is transferred to the collocation setting
Truncation preconditioners for stochastic Galerkin finite element discretizations
Stochastic Galerkin finite element method (SGFEM) provides an efficient
alternative to traditional sampling methods for the numerical solution of
linear elliptic partial differential equations with parametric or random
inputs. However, computing stochastic Galerkin approximations for a given
problem requires the solution of large coupled systems of linear equations.
Therefore, an effective and bespoke iterative solver is a key ingredient of any
SGFEM implementation. In this paper, we analyze a class of truncation
preconditioners for SGFEM. Extending the idea of the mean-based preconditioner,
these preconditioners capture additional significant components of the
stochastic Galerkin matrix. Focusing on the parametric diffusion equation as a
model problem and assuming affine-parametric representation of the diffusion
coefficient, we perform spectral analysis of the preconditioned matrices and
establish optimality of truncation preconditioners with respect to SGFEM
discretization parameters. Furthermore, we report the results of numerical
experiments for model diffusion problems with affine and non-affine parametric
representations of the coefficient. In particular, we look at the efficiency of
the solver (in terms of iteration counts for solving the underlying linear
systems) and compare truncation preconditioners with other existing
preconditioners for stochastic Galerkin matrices, such as the mean-based and
the Kronecker product ones.Comment: 27 pages, 6 table
Convergence of adaptive stochastic collocation with finite elements
We consider an elliptic partial differential equation with a random diffusion parameter discretized by a stochastic collocation method in the parameter domain and a finite element method in the spatial domain. We prove for the first time convergence of a stochastic collocation algorithm which adaptively enriches the parameter space as well as refines the finite element meshes
Convergence of adaptive stochastic collocation with finite elements
We consider an elliptic partial differential equation with a random diffusion
parameter discretized by a stochastic collocation method in the parameter
domain and a finite element method in the spatial domain. We prove convergence
of an adaptive algorithm which adaptively enriches the parameter space as well
as refines the finite element meshes
On the convergence of adaptive stochastic collocation for elliptic partial differential equations with affine diffusion
Convergence of an adaptive collocation method for the stationary parametric
diffusion equation with finite-dimensional affine coefficient is shown. The
adaptive algorithm relies on a recently introduced residual-based reliable a
posteriori error estimator. For the convergence proof, a strategy recently used
for a stochastic Galerkin method with an hierarchical error estimator is
transferred to the collocation setting. Extensions to other variants of
adaptive collocation methods (including the classical one proposed in the paper
"Dimension-adaptive tensor-product quadratuture" Computing (2003) by T.
Gerstner and M. Griebel) is explored.Comment: 24 pages, 1 figur