A hierarchical duality approach to bounds for the outputs of partial differential equations

Paraschivoiu, Marius; Patera, Anthony T.

doi:10.1016/s0045-7825(99)00270-4

Cited by 78 publications

(42 citation statements)

References 12 publications

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“…To prove this, recall that the null space for the Poisson operator is the one dimensional space of constants, R, and letR = T ∈T h R denote the null space of the broken operator. Consideringĉ ∈R ⊂Û h in the equilibration problem (8) and that anyŵ ∈Û can be represented asŵ +ĉ forŵ ∈Û \R, it is easily shown that L(ŵ +ĉ; λ h ) = L(ŵ ; λ h ). For the Poisson equation, equilibration ensures that the null space of the operator does not cause the minimization to be become unbounded below.…”

Section: B1 Approximate Multipliermentioning

confidence: 99%

“…Paraschivoiu, Peraire and Patera [7], [8] originally proposed this reformulation in the context of two-level output bounding methods which appeal to a second refined but localized finite element approximation for computing the bounds rather than the dual of the infinite dimensional continuum equations. Now that we have our starting point, we can proceed more or less mechanically to apply the ideas from the energy bound to this more general context.…”

Section: Output Boundsmentioning

confidence: 99%

“…Our method differs from that of Kelly in that we obtain bounds for general meshes and more general boundary conditions, and from that Destuynder in that our method is not burdened with the construction of a globally dual admissible field. The work we present here descends directly from earlier work done by Paraschivoiu, Peraire and Patera [7], [8] on two-level residual based techniques for computing output bounds.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Computing Bounds for Linear Functionals of Exact Weak Solutions to Poisson's Equation

Sauer-Budge¹,

Bonet²,

Huerta³

et al. 2004

SIAM J. Numer. Anal.

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Abstract-We present a method for Poisson's equation that computes guaranteed upper and lower bounds for the values of linear functional outputs of the exact weak solution of the infinite dimensional continuum problem using traditional finite element approximations. The guarantee holds uniformly for any level of refinement, not just in the asymptotic limit of refinement. Given a finite element solution and its output adjoint solution, the method can be used to provide a certificate of precision for the output with an asymptotic complexity which is linear in the number of elements in the finite element discretization.

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Section: B1 Approximate Multipliermentioning

confidence: 99%

Section: Output Boundsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Computing Bounds for Linear Functionals of Exact Weak Solutions to Poisson's Equation

Sauer-Budge¹,

Bonet²,

Huerta³

et al. 2004

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

show abstract

“…Finally, in order to obtain an upper bound for × , it is only necessary to derive a lower bound × for × Ë´ µ, following the procedure just outlined, and then set × · × [10], Fig. 3.…”

Section: A Lagrangian Formulationmentioning

confidence: 99%

“…The development of an implicit procedure yielding aposteriori constant-free bounds for linear-functional outputs of partial differential equations was presented in [10], [11], [12]. The method is applicable to elliptic coercive problems including non-symmetric terms.…”

Section: Introductionmentioning

confidence: 99%

The efficient computation of bounds for functionals of finite element solutions in large strain elasticity

Bonet

Huerta

Peraire

2002

Computer Methods in Applied Mechanics and Engineering

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Abstract-We present an implicit a-posteriori finite element procedure to compute bounds for functional outputs of finite element solutions in large strain elasticity. The method proposed relies on the existence of a potential energy functional whose local minima, over a space of suitably chosen continuous functions, corresponds to the problem solution. The output of interest is cast as a constrained minimization problem over an enlarged discontinuous finite element space. A Lagrangian is formed were the multipliers are an adjoint solution, which enforces equilibrium, and hybrid fluxes, which constrain the solution to be continuous. By computing approximate values for the multipliers on a coarse mesh, strict upper and lower bounds for the output of interest on a suitably refined mesh, are obtained. This requires a minimization over a discontinuous space, which can be carried out locally at low cost. The computed bounds are uniformly valid regardless of the size of the underlying coarse discretization. The method is demonstrated with two applications involving large strain plane stress incompressible neo-hookean hyperelasticity.

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Non‐parametric‐validated computer‐simulation surrogates: a Pareto formulation

Kambourides¹,

Yeşilyurt²,

Patera³

1998

Int. J. Numer. Meth. Engng.

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In the surrogate approach to simulation-based optimization, the large-scale simulation is evoked only to construct and validate a simpliÿed input-output model; this simpliÿed input-output model then serves as a simulation surrogate in subsequent engineering optimization studies. We present here 'basic' and Pareto surrogate formulations through an illustrative application from uid dynamics.The critical ingredient of both formulations is a non-parametric statistical validation and error estimation procedure which, based on veriÿable hypotheses, precisely quantiÿes the e ect of surrogate-for-simulation substitution on system predictability, stability, and optimality. The Pareto formulation improves upon the basic approach by operating only in the vicinity of the e cient frontier of the output achievable set A; for problems with many inputs and few outputs, this considerably reduces the dimensionality of the problem, and correspondingly improves the surrogate error estimates. ? 1998 John Wiley & Sons, Ltd.

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A hierarchical duality approach to bounds for the outputs of partial differential equations

Cited by 78 publications

References 12 publications

Computing Bounds for Linear Functionals of Exact Weak Solutions to Poisson's Equation

Computing Bounds for Linear Functionals of Exact Weak Solutions to Poisson's Equation

The efficient computation of bounds for functionals of finite element solutions in large strain elasticity

Non‐parametric‐validated computer‐simulation surrogates: a Pareto formulation

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