Filters

2020 - 20212

More filters

20222
Reset filters

Settings

Author: Eigel, Martin × Start date: 2020 × End date: 2022 × Start date: 2020 × Subject: residual error estimator × WGL Institute: WIAS ×

Search Results

Now showing 1 - 2 of 2
  • Thumbnail Image
    Item
    Adaptive non-intrusive reconstruction of solutions to high-dimensional parametric PDEs
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Eigel, Martin; Farchmin, Nando; Heidenreich, Sebastian; Trunschke, Philipp
    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 concerned with a non-intrusive generalization of the adaptive Galerkin FEM with residual based error estimation. It combines the non-intrusive character of a randomized least-squares method with the a posteriori error analysis of stochastic Galerkin methods. The proposed approach uses the Variational Monte Carlo method to obtain a quasi-optimal low-rank approximation of the Galerkin projection in a highly efficient hierarchical tensor format. We derive an adaptive refinement algorithm which is steered by a reliable error estimator. Opposite to stochastic Galerkin methods, the approach is easily applicable to a wide range of problems, enabling a fully automated adjustment of all discretization parameters. Benchmark examples with affine and (unbounded) lognormal coefficient fields illustrate the performance of the non-intrusive adaptive algorithm, showing best-in-class performance
  • Thumbnail Image
    Item
    On the convergence of adaptive stochastic collocation for elliptic partial differential equations with affine diffusion
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Eigel, Martin; Ernst, Oliver; Sprungk, Björn; Tamellini, Lorenzo
    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.