A fully adaptive multilevel stochastic collocation strategy for solving elliptic PDEs with random data

Abstract

We propose and analyse a fully adaptive strategy for solving elliptic PDEs with random data in this work. A hierarchical sequence of adaptive mesh refinements for the spatial approximation is combined with adaptive anisotropic sparse Smolyak grids in the stochastic space in such a way as to minimize the computational cost. The novel aspect of our strategy is that the hierarchy of spatial approximations is sample dependent so that the computational effort at each collocation point can be optimised individually. We outline a rigorous analysis for the convergence and computational complexity of the adaptive multilevel algorithm and we provide optimal choices for error tolerances at each level. Two numerical examples demonstrate the reliability of the error control and the significant decrease in the complexity that arises when compared to single level algorithms and multilevel algorithms that employ adaptivity solely in the spatial discretisation or in the collocation procedure.

arXiv comment26 pages, 7 figures

References in corpus (5)

• A multilevel Monte Carlo method for computing failure probabilities
• High-Dimensional Stochastic Design Optimization by Adaptive-Sparse Polynomial Dimensional Decomposition
• Stochastic collocation approach with adaptive mesh refinement for parametric uncertainty analysis
• Goal-oriented error estimation and adaptivity for elliptic PDEs with parametric or uncertain inputs
• Goal-oriented error control of stochastic system approximations using metric-based anisotropic adaptations

Cited by in corpus (5)

• Active Learning and Bayesian Optimization: a Unified Perspective to Learn with a Goal
• Convergence of adaptive stochastic collocation with finite elements
• Goal-Oriented Adaptive Finite Element Multilevel Monte Carlo with Convergence Rates
• Two-level a posteriori error estimation for adaptive multilevel stochastic Galerkin FEM
• Error estimation and adaptivity for stochastic collocation finite elements Part I: single-level approximation