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Goal-Oriented Error Estimation and Adaptivity for Stochastic Collocation FEM

Alex Bespalov, Dirk Praetorius, Thomas Round, Andrey Savinov

TL;DR

The paper develops a goal-oriented adaptive framework for quantities of interest $\mathscr{Q}(u)$ in elliptic PDEs with parametric uncertainty, using sparse grid stochastic collocation FEM. It derives reliable, two-level a posteriori error estimates and introduces a correction term to offset the lack of global Galerkin orthogonality in SC-FEM, enabling stable QoI convergence for both affine and nonaffine parameter dependencies and for linear and nonlinear $Q$. A unified adaptive algorithm couples spatial mesh refinement with sparse-grid parametric enrichment, guided by primal and dual error indicators, and is extended from linear to nonlinear QoIs with appropriate modifications to marking and stopping criteria. Numerical experiments on representative problems demonstrate that corrected QoI errors decay at the same rate as the estimators, with larger effectivity in nonlinear cases, and show robust performance up to moderate numbers of random parameters. The work provides a principled, computable approach to controlling QoI errors in surrogate models for uncertainty quantification, with potential extensions to higher-dimensional parameter spaces and dimension-reduction techniques.

Abstract

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 bounded linear or continuously Gâteaux differentiable nonlinear goal functionals, and the approximations are computed using the sparse grid stochastic collocation finite element method (SC-FEM). The proposed adaptive strategy relies on novel reliable a posteriori estimates of the errors in approximating QoIs. One of the key features of our error estimation approach is the introduction of a correction term into the approximation of QoIs in order to compensate for the lack of (global) Galerkin orthogonality in the SC-FEM setting. Computational results generated using the proposed adaptive algorithm are presented in the paper for representative elliptic problems with affine and nonaffine parametric coefficient dependence and for a range of linear and nonlinear goal functionals.

Goal-Oriented Error Estimation and Adaptivity for Stochastic Collocation FEM

TL;DR

The paper develops a goal-oriented adaptive framework for quantities of interest in elliptic PDEs with parametric uncertainty, using sparse grid stochastic collocation FEM. It derives reliable, two-level a posteriori error estimates and introduces a correction term to offset the lack of global Galerkin orthogonality in SC-FEM, enabling stable QoI convergence for both affine and nonaffine parameter dependencies and for linear and nonlinear . A unified adaptive algorithm couples spatial mesh refinement with sparse-grid parametric enrichment, guided by primal and dual error indicators, and is extended from linear to nonlinear QoIs with appropriate modifications to marking and stopping criteria. Numerical experiments on representative problems demonstrate that corrected QoI errors decay at the same rate as the estimators, with larger effectivity in nonlinear cases, and show robust performance up to moderate numbers of random parameters. The work provides a principled, computable approach to controlling QoI errors in surrogate models for uncertainty quantification, with potential extensions to higher-dimensional parameter spaces and dimension-reduction techniques.

Abstract

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 bounded linear or continuously Gâteaux differentiable nonlinear goal functionals, and the approximations are computed using the sparse grid stochastic collocation finite element method (SC-FEM). The proposed adaptive strategy relies on novel reliable a posteriori estimates of the errors in approximating QoIs. One of the key features of our error estimation approach is the introduction of a correction term into the approximation of QoIs in order to compensate for the lack of (global) Galerkin orthogonality in the SC-FEM setting. Computational results generated using the proposed adaptive algorithm are presented in the paper for representative elliptic problems with affine and nonaffine parametric coefficient dependence and for a range of linear and nonlinear goal functionals.
Paper Structure (16 sections, 1 theorem, 75 equations, 4 figures, 3 algorithms)

This paper contains 16 sections, 1 theorem, 75 equations, 4 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Stochastic collocation finite element method
  4. Spatial discretization and mesh refinement
  5. Sparse grid collocation and parametric enrichment
  6. A posteriori error estimation
  7. Goal-oriented adaptive SC-FEM with linear goal functionals
  8. Dual formulations and the goal-oriented error estimate
  9. Adaptive algorithm
  10. Computation of $\widetilde{\mathscr{Q}}(u_{\bullet}^{\rm SC},z_{\bullet}^{\rm SC})$
  11. Goal-oriented adaptive SC-FEM with nonlinear goal functionals
  12. Dual problems and the dual SC-FEM approximation
  13. Goal-oriented error estimation
  14. Modifications to the adaptive algorithm
  15. Numerical experiments
  16. ...and 1 more sections

Key Result

Theorem 4

Suppose the saturation assumptions eq:saturation and eq:saturation:dual:nl hold for the primal solution $u$ and for the dual solution $z$, respectively. In addition, assume that the goal functional $\mathscr{Q}$ satisfies ineq:gateaux. Then, the following error estimate holds: where $\mu_\bullet, \;\tau_\bullet,\; \eta_\bullet,\; \sigma_\bullet$ are defined in eq:spatial:estimate, eq:param:estima

Figures (4)

  • Figure 1: The supports of spatial features and initial meshes for all four setups.
  • Figure 2: Evolution of the goal-oriented error estimates (magenta lines) as well as the reference errors for uncorrected (blue lines) and corrected (red lines) approximations of $\mathscr{Q}(u)$ for setups 1 and 2.
  • Figure 3: Evolution of the goal-oriented error estimates (magenta lines) as well as the reference errors for uncorrected (blue lines) and corrected (red lines) approximations of $\mathscr{Q}(u)$ for setups 3 and 4.
  • Figure 4: Adaptively refined meshes for all four setups.

Theorems & Definitions (4)

  • Remark 1
  • Theorem 4
  • proof
  • Remark 5