Adjoint Recovery of Superconvergent Functionals from PDE Approximations

By Niles A. Pierce, M. Giles. SIAM Review, 2000

348 citations recorded by the source corpus.

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In Computational Mechanics

Citations in the corpus

Review of Output-Based Error Estimation and Mesh Adaptation in Computational Fluid Dynamics (2011, methodology): “(11) is generalized using a mean-value linearization [14,35,37,38].”
Reduced Basis Approximation and a Posteriori Error Estimation for Affinely Parametrized Elliptic Coercive Partial Differential Equations (2007, methodology): “In actual FE practice —i n which the dual approximation serves to improve output accuracy [ 117 ] or develop output error bounds [23,108] — the FE primal and dual spaces may be different.”
A comprehensive framework for verification, validation, and uncertainty quantification in scientific computing (2011, methodology): “, SRQs) for both finite element [24] and finite volume methods [25].”
Adjoint error estimation and grid adaptation for functional outputs: application to quasi-one-dimensional flow (2000, methodology): “We now briefly mention some differences between the proposed error estimation procedure and the adjoint-based correction method of Pierce and Giles [ 19 ].”
Certified real‐time solution of the parametrized steady incompressible Navier–Stokes equations: rigorous reduced‐basis a posteriori error bounds (2005, methodology): “We shall also need a dual problem [18, 24, 25] associated with our output functional L.”
Reduced Basis Approximation and a Posteriori Error Estimation for Parametrized Parabolic PDEs: Application to Real‐Time Bayesian Parameter Estimation (2010, methodology): “We may pursue primal-dual RB approximations (Grepl and Patera 2005; Pierce and Giles 2000; Rozza et al .”
An implicit, exact dual adjoint solution method for turbulent flows on unstructured grids (2004, methodology): “In addition to its role in design optimization problems, the solution of the adjoint equation can be used to provide error estimates for an output of engineering interest, as well as grid adaptation information that can be used to improve solution accuracy [24–32].”
Local Discontinuous Galerkin Methods for High-Order Time-Dependent Partial Differential Equations (2009, background): “Although there are some works [2,8,28,59,66,76,86,87] on error estimates for hyperbolic conservation law and parabolic problems, there are only a few works in the literature for error estimates of the LDG method for nonlinear wave equations with high order derivatives.”
Certified reduced basis approximation for parametrized partial differential equations and applications (2011, neutral): “The resulting RB approximation uNpr ∈ X N ,pr Npr ,ΨNdu ∈ X Ndu solve a(uNpr (μ), v;μ) = f(v), ∀v ∈ X N ,pr Npr , a(v,ΨNdu(μ);μ) = −`(v), ∀v ∈ X N ,du Ndu ; then, the RB output can be evaluated as [35] sNpr,Ndu(μ) = `(u N Npr )− r (ΨNdu ;μ) where r(v;μ) = f(v)− a(uNpr , v;μ), r(v;μ) = −`(v)− a(v,ΨNd”
Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations (2007, methodology): “In relation to Part I, the key new methodological elements are the development of (i) primaldual (with adjoint) approximation [115] for RB [120], (ii) an a posteriori theory of general (non-compliant, and hence “non-energy”) outputs [5, 22, 110] for RB [120, 139], and (iii) an efficient Offline-Onli”
An Introduction to the Adjoint Approach to Design (2000, methodology): “On the other hand, we prefer the duality viewpoint because it seems more natural for other uses of adjoint variables, such as error analysis [17, 35, 36 , 46], which do not involve constrained optimisation.”
Grid adaptation for functional outputs: application to two-dimensional inviscid flows (2002, methodology): “It should be noted that the correction formula given by (10) is a discrete analogue of the Pierce and Giles correction term [14].”
Analytic adjoint solutions for the quasi-one-dimensional Euler equations (2001, methodology): “…the truncation error in the original nonlinear numerical solution, and using the adjoint solution to estimate the consequential error in the functional of interest, an improved estimate can be obtained with twice the order of accuracy (Giles & Pierce 1998, 1999; Pierce & Giles 1998, 2000).”
Reduced--Basis Output Bound Methods for Parametrized Partial Differential Equations (2002, background): “An analysis of the model-truncation error suggests that the error can be attributed to two sources: first in the inability of the low-order model to reproduce the exact loading; and second, for the approximated loading, in the inability of the low-order model to recover the exact solution [49]; see ”