Related papers: Efficient adaptive multilevel stochastic Galerkin …
We use the ideas of goal-oriented error estimation and adaptivity to design and implement an efficient adaptive algorithm for approximating linear quantities of interest derived from solutions to elliptic partial differential equations…
This paper is concerned with the numerical approximation of quantities of interest associated with solutions to parametric elliptic partial differential equations (PDEs). The key novelty of this work is in its focus on the quantities of…
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…
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…
The paper considers a class of parametric elliptic partial differential equations (PDEs), where the coefficients and the right-hand side function depend on infinitely many (uncertain) parameters. We introduce a two-level a posteriori…
This work analyzes the overall computational complexity of the stochastic Galerkin finite element method (SGFEM) for approximating the solution of parameterized elliptic partial differential equations with both affine and non-affine random…
The focus of this work is a posteriori error estimation for stochastic Galerkin approximations of parameter-dependent linear elasticity equations. The starting point is a three-field PDE model in which the Young's modulus is an affine…
This is a survey on the theory of adaptive finite element methods (AFEMs), which are fundamental in modern computational science and engineering but whose mathematical assessment is a formidable challenge. We present a self-contained and…
The subject of this work is an adaptive stochastic Galerkin finite element method for parametric or random elliptic partial differential equations, which generates sparse product polynomial expansions with respect to the parametric…
Numerical methods for random parametric PDEs can greatly benefit from adaptive refinement schemes, in particular when functional approximations are computed as in stochastic Galerkin and stochastic collocations methods. This work is…
Near-optimal computational complexity of an adaptive stochastic Galerkin method with independently refined spatial meshes for elliptic partial differential equations is shown. The method takes advantage of multilevel structure in expansions…
The paper addresses a numerical method for solving second order elliptic partial differential equations that describe fields inside heterogeneous media. The scope is general and treats the case of rough coefficients, i.e. coefficients with…
A general adaptive refinement strategy for solving linear elliptic partial differential equation with random data is proposed and analysed herein. The adaptive strategy extends the a posteriori error estimation framework introduced by…
We propose and analyze a general goal-oriented adaptive strategy for approximating quantities of interest (QoIs) associated with solutions to linear elliptic partial differential equations with random inputs. The QoIs are represented by…
A multilevel adaptive refinement strategy for solving linear elliptic partial differential equations with random data is recalled in this work. The strategy extends the a posteriori error estimation framework introduced by Guignard and…
We consider Galerkin finite element methods for semilinear stochastic partial differential equations (SPDEs) with multiplicative noise and Lipschitz continuous nonlinearities. We analyze the strong error of convergence for spatially…
We analyze optimal complexity of adaptive finite element methods (AFEMs) for general second-order linear elliptic partial differential equations (PDEs) in the Lax-Milgram setting. To this end, we formulate an adaptive algorithm which steers…
Numerically 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…
We propose a new algorithm for Adaptive Finite Element Methods (AFEMs) based on smoothing iterations (S-AFEM), for linear, second-order, elliptic partial differential equations (PDEs). The algorithm is inspired by the ascending phase of the…
We consider an elliptic linear-quadratic parameter estimation problem with a finite number of parameters. A novel a priori bound for the parameter error is proved and, based on this bound, an adaptive finite element method driven by an a…