Rate-optimal goal-oriented adaptive FEM for semilinear elliptic PDEs

Becker, Roland; Brunner, Maximilian; Innerberger, Michael; Melenk, Jens Markus; Praetorius, Dirk

doi:10.1016/j.camwa.2022.05.008

Cited by 11 publications

(5 citation statements)

References 40 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Further possible extensions are the generalization to adaptive step-sizes and the incomplete solution of the discrete state equation. Adaptive step-sizes are important to treat nonlinearities that are only locally Lipschitz-continuous [10], and avoid the explicit knowledge of a Lipschitz-constant. Although the presented algorithm avoids solution of the optimization problem on each finite element space, it supposes solution of the nonlinear state equation.…”

Section: Discussionmentioning

confidence: 99%

Gradient method with AFEM for parameter-estimation

Becker¹

2023

Sel.mat

View full text Add to dashboard Cite

We consider the adaptive finite element discretization of parameter estimation problems for nonlinear elliptic partial differential equations. The idea is to use a gradient method on the finite-dimensional parameter space for the minimization of the least-squares residual. Since the gradient involves solution of partial differential equations, it is not accesable, and is replaced by an approximation obtained by finite elements. This results into a perturbed gradient method. We use an (a posteriori) error estimator to control the accuracy of the gradient approximation and propose an algorithm, which links the estimator to the progress of the iteration. We show convergence of the algorithm under typical structural assumptions.

show abstract

Section: Discussionmentioning

confidence: 99%

Gradient method with AFEM for parameter-estimation

Becker¹

2023

Sel.mat

View full text Add to dashboard Cite

show abstract

“…Moreover, the theory has been generalised to control the overall work of the adaptive algorithm, and not just the dimension of the discrete spaces, see, e.g., Haberl et al (2021); Becker et al (2023); Févotte et al (2024). This is particularly important from a practical viewpoint, since the various stopping criteria in the different nested loops can then be harmonised.…”

Section: Other Recent Achievementsmentioning

confidence: 99%

A short perspective on a posteriori error control and adaptive discretizations

Becker,

Bordas,

Chouly

et al. 2024

Advances in Applied Mechanics

View full text Add to dashboard Cite

“…) denotes the likelihood with the observation noise covariance-weighted, data-to-observation misfit, where |x| 2 Γ := x Γ −1 x and Z is defined in (1). As Θ(y) > 0 for all y ∈ U , Z > 0.…”

Section: Bayesian Inverse Problem (Bip)mentioning

confidence: 99%

“…For K = 1, i.e. for a single observation functional, goal-oriented AFEM results from [1] and the references there, can be used to obtain sharper a-posteriori error bounds.…”

Section: Fem Error Estimators For Bipmentioning

confidence: 99%

A-posteriori QMC-FEM error estimation for Bayesian inversion and optimal control with entropic risk measure

Longo¹,

Schwab²,

Stein³

2023

Preprint

View full text Add to dashboard Cite

We propose a novel a-posteriori error estimation technique where the target quantities of interest are ratios of high-dimensional integrals, as occur e.g. in PDE constrained Bayesian inversion and PDE constrained optimal control subject to an entropic risk measure. We consider in particular parametric, elliptic PDEs with affine-parametric diffusion coefficient, on high-dimensional parameter spaces. We combine our recent a-posteriori Quasi-Monte Carlo (QMC) error analysis, with Finite Element a-posteriori error estimation. The proposed approach yields a computable a-posteriori estimator which is reliable, up to higher order terms. The estimator's reliability is uniform with respect to the PDE discretization, and robust with respect to the parametric dimension of the uncertain PDE input.

show abstract

Rate-optimal goal-oriented adaptive FEM for semilinear elliptic PDEs

Cited by 11 publications

References 40 publications

Gradient method with AFEM for parameter-estimation

Gradient method with AFEM for parameter-estimation

A short perspective on a posteriori error control and adaptive discretizations

A-posteriori QMC-FEM error estimation for Bayesian inversion and optimal control with entropic risk measure

Contact Info

Product

Resources

About