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Goal-oriented adaptivity for multilevel stochastic Galerkin FEM with nonlinear goal functionals

Alex Bespalov, Dirk Praetorius, Michele Ruggeri

TL;DR

We develop a provably convergent goal-oriented adaptive SGFEM framework for parametric elliptic PDEs with affine, infinite-dimensional uncertainty, targeting nonlinear QoIs. The approach extends previous linear-functionals work by incorporating continuously Gâteaux differentiable nonlinear functionals and employing a multilevel SGFEM with joint spatial and parametric refinement guided by a posteriori error indicators. A dual-based goal-oriented error estimate is derived and used within a refined adaptive algorithm that converges to zero without relying on saturation for the estimator, with numerical experiments showing rate-optimal decay in 2D problems. The framework advances efficient uncertainty quantification for nonlinear QoIs in high-dimensional parametric PDEs and sets the stage for extensions to 3D and more complex QoIs.

Abstract

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 interest represented by continuously Gâteaux differentiable nonlinear functionals. We consider a class of parametric elliptic PDEs where the underlying differential operator has affine dependence on a countably infinite number of uncertain parameters. We design a goal-oriented adaptive algorithm for approximating nonlinear functionals of solutions to this class of parametric PDEs. In the algorithm, the approximations of parametric solutions to the primal and dual problems are computed using the multilevel stochastic Galerkin finite element method (SGFEM) and the adaptive refinement process is guided by reliable spatial and parametric error reduction indicators. We prove that the proposed algorithm generates multilevel SGFEM approximations for which the estimates of the error in the goal functional converge to zero. Numerical experiments for a selection of test problems and nonlinear quantities of interest illustrate and underpin our theoretical findings.

Goal-oriented adaptivity for multilevel stochastic Galerkin FEM with nonlinear goal functionals

TL;DR

We develop a provably convergent goal-oriented adaptive SGFEM framework for parametric elliptic PDEs with affine, infinite-dimensional uncertainty, targeting nonlinear QoIs. The approach extends previous linear-functionals work by incorporating continuously Gâteaux differentiable nonlinear functionals and employing a multilevel SGFEM with joint spatial and parametric refinement guided by a posteriori error indicators. A dual-based goal-oriented error estimate is derived and used within a refined adaptive algorithm that converges to zero without relying on saturation for the estimator, with numerical experiments showing rate-optimal decay in 2D problems. The framework advances efficient uncertainty quantification for nonlinear QoIs in high-dimensional parametric PDEs and sets the stage for extensions to 3D and more complex QoIs.

Abstract

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 interest represented by continuously Gâteaux differentiable nonlinear functionals. We consider a class of parametric elliptic PDEs where the underlying differential operator has affine dependence on a countably infinite number of uncertain parameters. We design a goal-oriented adaptive algorithm for approximating nonlinear functionals of solutions to this class of parametric PDEs. In the algorithm, the approximations of parametric solutions to the primal and dual problems are computed using the multilevel stochastic Galerkin finite element method (SGFEM) and the adaptive refinement process is guided by reliable spatial and parametric error reduction indicators. We prove that the proposed algorithm generates multilevel SGFEM approximations for which the estimates of the error in the goal functional converge to zero. Numerical experiments for a selection of test problems and nonlinear quantities of interest illustrate and underpin our theoretical findings.
Paper Structure (17 sections, 10 theorems, 102 equations, 2 figures, 1 table, 1 algorithm)

This paper contains 17 sections, 10 theorems, 102 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Multilevel SGFEM discretization
  4. Discretization in the physical domain and mesh refinement
  5. Discretization in the parameter domain and parametric enrichment
  6. Multilevel approximation and multilevel refinement
  7. Multilevel SGFEM approximation
  8. A posteriori error estimation
  9. Goal-oriented adaptive SGFEM with nonlinear goal functional
  10. Dual problem and goal-oriented error estimate
  11. Adaptive algorithm
  12. Convergence analysis
  13. Numerical results
  14. Problem specifications
  15. Results
  16. ...and 2 more sections

Key Result

Theorem 2

Let $d \in \{2, 3\}$ and $\boldsymbol{w} \in \boldsymbol{\mathbb{V}}$. For the multilevel structures $\pmb{\mathbb{P}}_\bullet, \widehat{\pmb{\mathbb{P}}}_\bullet \in {\hbox{\scriptsize$\pmb{\mathbb{REFINE}}$}}(\pmb{\mathbb{P}}_0)$, consider the multilevel approximation spaces $\boldsymbol{\mathbb{V Furthermore, under the saturation assumption eq:saturation, the estimates eq1:thm:estimator are equ

Figures (2)

  • Figure 1: Adaptively refined meshes associated with the zero index at an intermediate step of Algorithm \ref{['algorithm']} for all four setups.
  • Figure 2: Evolution of the error estimates $\mu_\ell (\mu_\ell^2 + \zeta_\ell^2)^{1/2}$, the reference errors $\lvert \boldsymbol{g}(\boldsymbol{u}_\mathrm{ref}) - \boldsymbol{g}(\boldsymbol{u}_\ell) \rvert$, and, for Setups 1--3, the Monte Carlo-based approximations $e_\ell^\mathrm{MC}$ of the error in the goal functional at each iteration of the goal-oriented adaptive algorithm.

Theorems & Definitions (17)

  • Remark 1
  • Theorem 2: bpr2020+
  • Remark 3
  • Lemma 4
  • proof
  • Proposition 5
  • proof
  • Theorem 7
  • Corollary 8
  • Lemma 9: a priori convergence; see, e.g., bprr18++
  • ...and 7 more