# Convergence of adaptive stochastic collocation with finite elements

@article{Feischl2020ConvergenceOA, title={Convergence of adaptive stochastic collocation with finite elements}, author={M. Feischl and Andrea Scaglioni}, journal={ArXiv}, year={2020}, volume={abs/2008.12591} }

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.

#### One Citation

Error estimation and adaptivity for stochastic collocation finite elements Part I: single-level approximation

- Computer Science, Mathematics
- ArXiv
- 2021

A general adaptive refinement strategy for solving linear elliptic partial differential equation with Random data with random data is proposed and analysed herein and results obtained using a potentially more efficient multilevel approximation strategy will be discussed in part II of this work. Expand

#### References

SHOWING 1-10 OF 27 REFERENCES

A Posteriori Error Estimation for the Stochastic Collocation Finite Element Method

- Mathematics, Computer Science
- SIAM J. Numer. Anal.
- 2018

A residual-based a posteriori error estimate is derived that is constituted of two parts controlling the stochastic collocation (SC) and the finite element (FE) errors, respectively, and is used to drive an adaptive sparse grid algorithm. Expand

A convergent adaptive stochastic Galerkin finite element method with quasi-optimal spatial meshes

- Mathematics
- 2013

We analyze a posteriori error estimation and adaptive refinement algorithms for stochastic Galerkin Finite Element methods for countably-parametric, elliptic boundary value problems. A residual error… Expand

Convergence of Adaptive Stochastic Galerkin FEM

- Mathematics, Computer Science
- SIAM J. Numer. Anal.
- 2019

Novel adaptive algorithms are proposed and analyzed for the numerical solution of elliptic partial differential equations with parametric uncertainty and four different marking strategies are employed. Expand

A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data

- Computer Science, Mathematics
- SIAM J. Numer. Anal.
- 2007

A rigorous convergence analysis is provided and exponential convergence of the “probability error” with respect to the number of Gauss points in each direction in the probability space is demonstrated, under some regularity assumptions on the random input data. Expand

Quasi-Monte Carlo methods for elliptic PDEs with random coefficients and applications

- Computer Science, Mathematics
- J. Comput. Phys.
- 2011

Numerical experiments are reported, showing that the quasi-Monte Carlo method consistently outperforms the Monte Carlo method, with a smaller error and a noticeably better than O(N^-^1^/^2) convergence rate. Expand

A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data

- Mathematics, Computer Science
- SIAM J. Numer. Anal.
- 2008

This work demonstrates algebraic convergence with respect to the total number of collocation points and quantifies the effect of the dimension of the problem (number of input random variables) in the final estimates, indicating for which problems the sparse grid stochastic collocation method is more efficient than Monte Carlo. Expand

Quasi-Optimal Convergence Rate for an Adaptive Finite Element Method

- Mathematics, Computer Science
- SIAM J. Numer. Anal.
- 2008

It is shown that the AFEM yields a decay rate of the energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity. Expand

Stochastic Spectral Galerkin and Collocation Methods for PDEs with Random Coefficients: A Numerical Comparison

- Mathematics
- 2011

Much attention has recently been devoted to the development of Stochastic Galerkin (SG) and Stochastic Collocation (SC) methods for uncertainty quantification. An open and relevant research topic is… Expand

Improved Efficiency of a Multi-Index FEM for Computational Uncertainty Quantification

- Computer Science, Mathematics
- SIAM J. Numer. Anal.
- 2019

A multi-index algorithm for the Monte Carlo (MC) discretization of a linear, elliptic PDE with affine-parametric input and the analysis improves cost estimates compared to multi-level algorithms for similar problems and mathematically underpins the superior practical performance of multi- index algorithms for partial differential equations with random coefficients. Expand

High dimensional polynomial interpolation on sparse grids

- Mathematics, Computer Science
- Adv. Comput. Math.
- 2000

The error bounds show that the polynomial interpolation on a d-dimensional cube, where d is large, is universal, i.e., almost optimal for many different function spaces. Expand