Inversion of Robin coefficient by a spectral stochastic finite element approach

Jin, Bangti; Zou, Jun

doi:10.1016/j.jcp.2007.11.042

Cited by 24 publications

(12 citation statements)

References 34 publications

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“…[24]). Since the inverse problem is solved at the discrete level 5 , this yields a finite collection of random numbers mapped to a finite number of points in the physical domain D, and the assumptions may be simplified.…”

Section: Karhunen-loève Expansions Of Random Fieldsmentioning

confidence: 99%

A reduced basis approach for variational problems with stochastic parameters: Application to heat conduction with variable Robin coefficient

Boyaval

Bris

Maday

et al. 2009

Computer Methods in Applied Mechanics and Engineering

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In this work, a Reduced Basis (RB) approach is used to solve a large number of boundary value problems parametrized by a stochastic input -expressed as a Karhunen-Loève expansion -in order to compute outputs that are smooth functionals of the random solution fields. The RB method proposed here for variational problems parametrized by stochastic coefficients bears many similarities to the RB approach developed previously for deterministic systems. However, the stochastic framework requires the development of new a posteriori estimates for "statistical" outputs -such as the first two moments of integrals of the random solution fields; these error bounds, in turn, permit efficient sampling of the input stochastic parameters and fast reliable computation of the outputs in particular in the many-query context.

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Section: Karhunen-loève Expansions Of Random Fieldsmentioning

confidence: 99%

A reduced basis approach for variational problems with stochastic parameters: Application to heat conduction with variable Robin coefficient

Boyaval

Bris

Maday

et al. 2009

Computer Methods in Applied Mechanics and Engineering

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“…where (·, ·) H 1 (D) is the standard H 1 inner product and (·, ·) H 1 ρ is the H 1 inner product weighted by ρ, analogous to (13). The norm · H 1 (D)⊗H 1 ρ (Γ) is that induced by (15).…”

Section: Definitionsmentioning

confidence: 99%

“…Moreover, there is scope to control a system not only for an optimal mean response, but also to include statistics of the response in a cost functional. We note that stochastic PDE-constrained optimisation problems are closely related to stochastic inverse problems, where the control variable corresponds to the parameter to be identified [28,13,21].…”

Section: Introductionmentioning

confidence: 99%

Optimal control with stochastic PDE constraints and uncertain controls

Rosseel

Wells

2012

Computer Methods in Applied Mechanics and Engineering

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The optimal control of problems that are constrained by partial differential equations with uncertainties and with uncertain controls is addressed. The Lagrangian that defines the problem is postulated in terms of stochastic functions, with the control function possibly decomposed into an unknown deterministic component and a known zero-mean stochastic component. The extra freedom provided by the stochastic dimension in defining cost functionals is explored, demonstrating the scope for controlling statistical aspects of the system response. One-shot stochastic finite element methods are used to find approximate solutions to control problems. It is shown that applying the stochastic collocation finite element to the formulated problem leads to a coupling between stochastic collocation points when a deterministic optimal control is considered or when moments are included in the cost functional, thereby obviating the primary advantage of the collocation method over the stochastic Galerkin method for the considered problem. The application of the presented methods is demonstrated through a number of numerical examples. The presented framework is sufficiently general to also consider a class of inverse problems, and numerical examples of this type are also presented

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“…The conjugate gradient method (CGM) in conjunction with the spectral stochastic finite element method (SSFEM) has been applied to study the effect of the uncertainty in thermal conductivity on the accuracy of the inverse solution in Ref. [24]. However, only results for two-dimensional steady-state heat conduction were reported though the approach is quite general in nature.…”

Section: Rmsmentioning

confidence: 99%