A posteriori error estimation for elliptic partial differential equations with small uncertainties

Guignard, Diane; Nobile, Fabio; Picasso, Marco

doi:10.1002/num.21991

Cited by 15 publications

(5 citation statements)

References 46 publications

(112 reference statements)

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“…However, in the late 90s, an adjointbased (goal-oriented) error estimation theory devoted to the error in functional outputs of computer simulations-so-called quantities of interest (QoI)-was developed by Oden and Prudhomme [19,24], Becker and Rannacher [4], Patera and Peraire [21], and Giles and Süli [12]. More recently, sophisticated a posteriori error estimation techniques have also been developed for stochastic problems with parametric and domain uncertainties [13,14,20]. However, this theory is still limited in nonlinear or transient settings.…”

Section: Adjoint-based Strategies and Other Extensionsmentioning

confidence: 99%

D2.3. Adjoint-based error estimation routines

Keith¹,

Apostolatos²,

Kodakkal³

et al. 2021

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This document presents a simple and ecient strategy for adaptive mesh renement (AMR) and a posteriori error estimation for the transient incompressible Navier{Stokes equations. This strategy is informed by the work of Prudhomme and Oden [22, 23] as well as modern goal-oriented methods such as [5]. The methods described in this document have been implemented in the Kratos Multiphysics software and uploaded to https://zenodo.org [27].1 This document includes: A review of the state-of-the-art in solution-oriented and goal-oriented AMR. The description of a 2D benchmark model problem of immediate relevance to the objectives of the ExaQUte project. The denition and a brief mathematical summary of the error estimator(s). The results obtained. A description of the API.

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Section: Adjoint-based Strategies and Other Extensionsmentioning

confidence: 99%

D2.3. Adjoint-based error estimation routines

Keith¹,

Apostolatos²,

Kodakkal³

et al. 2021

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“…In such a setting, it allows to combine sensitivity analysis with goal-oriented a posteriori error estimation, see [15]. In the same spirit, the interplay between a posteriori error estimation and uncertainty quantification has been object of recent research interests [40,56]. We also note that if users can obtain some estimate, even rough, of modeling errors, they will also be able to compare discretization and model errors.…”

Section: Applicability For Patient-specific Biomechanics?mentioning

confidence: 99%

Quantifying discretization errors for soft tissue simulation in computer assisted surgery: A preliminary study

Duprez

Bordas

Bucki³

et al. 2020

Applied Mathematical Modelling

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Errors in biomechanics simulations arise from modeling and discretization. Modeling errors are due to the choice of the mathematical model whilst discretization errors measure the impact of the choice of the numerical method on the accuracy of the approximated solution to this specific mathematical model. A major source of discretization errors is mesh generation from medical images, that remains one of the major bottlenecks in the development of reliable, accurate, automatic and efficient personalized, clinically-relevant Finite Element (FE) models in biomechanics. The impact of mesh quality and density on the accuracy of the FE solution can be quantified with a posteriori error estimates. Yet, to our knowledge, the relevance of such error estimates for practical biomechanics problems has seldom been addressed, see [25]. In this contribution, we propose an implementation of some a posteriori error estimates to quantify the discretization errors and to optimize the mesh. More precisely, we focus on error estimation for a user-defined quantity of interest with the Dual Weighted Residual (DWR) technique. We test its applicability and relevance in two situations, corresponding to computations for a tongue and an artery, using a simplified setting, i.e., plane linearized elasticity with contractility of the soft-tissue modeled as a pre-stress. Our results demonstrate the feasibility of such methodology to estimate the actual solution errors and to reduce them economically through mesh refinement.

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“…In what follows, the only restriction on I will be that it is a downward closed set (a.k.a. lower set), i.e., it satisfies (10) \forall…”

Section: \Biggl\{mentioning

confidence: 99%

“…Moreover, exploiting the possible regularity of the solution with respect to the random parameters, the SC method has the advantage of a potentially much faster convergence rate than the Monte-Carlo method. It is also suitable for large uncertainties, contrary to perturbation-type methods as considered in our previous works [10,11].…”

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confidence: 99%

A Posteriori Error Estimation for the Stochastic Collocation Finite Element Method

Guignard¹,

Nobile²

2018

SIAM J. Numer. Anal.

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In this work, we consider an elliptic partial differential equation (PDE) with a random coefficient solved with the stochastic collocation finite element method (SC-FEM). The random diffusion coefficient is assumed to depend in an affine way on independent random variables. We derive a residual-based a posteriori error estimate that is constituted of two parts controlling the SC error and the FE error, respectively. The SC error estimator is then used to drive an adaptive sparse grid algorithm. Several numerical examples are given to illustrate the efficiency of the error estimator and the performance of the adaptive algorithm.

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A posteriori error estimation for elliptic partial differential equations with small uncertainties

Cited by 15 publications

References 46 publications

D2.3. Adjoint-based error estimation routines

D2.3. Adjoint-based error estimation routines

Quantifying discretization errors for soft tissue simulation in computer assisted surgery: A preliminary study

A Posteriori Error Estimation for the Stochastic Collocation Finite Element Method

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