Modeling errors due to Timoshenko approximation in damage identification

Castello, Daniel A.; Kaipio, Jari P.

doi:10.1002/nme.6175

Cited by 12 publications

(2 citation statements)

References 41 publications

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“…Broadly speaking, the BAE approach propagates all modelling and measurement uncertainties into a single additive total error term which is then approximately (pre)marginalised over the prior model. The robustness and applicability of the approach has been demonstrated in a variety of settings, see for example [13][14][15][16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

On global normal linear approximations for nonlinear Bayesian inverse problems

Nicholson¹,

Petra²,

Villa³

et al. 2023

Inverse Problems

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The replacement of a nonlinear parameter-to-observable mapping with a linear (aﬃne) approximation is often carried out to reduce the computational costs associated with solving large-scale inverse problems governed by partial diﬀerential equations (PDEs). In the case of a linear parameter-to-observable mapping with normally distributed additive noise and a Gaussian prior measure on the parameters, the posterior is Gaussian. However, substituting an accurate model for a (possibly well justiﬁed) linear surrogate model can give misleading results if the induced model approximation error is not accounted for. To account for the errors, the Bayesian approximation error approach can be utilised, in which the ﬁrst and second order statistics of the errors are computed via sampling. The most common linear approximation is carried out via linear Taylor expansion, which requires the computation of (Fréchet) derivatives of the parameter-to-observable mapping with respect to the parameters of interest. In this paper, we prove that the (approximate) posterior measure obtained by replacing the nonlinear parameter-to-observable mapping with a linear approximation is in fact independent of the choice of the linear approximation when the Bayesian approximation error approach is employed. Thus, somewhat non-intuitively, employing the zero-model as the linear approximation gives the same approximate posterior as any other choice of linear approximations of the parameter-to-observable model. The independence of the linear approximation is demonstrated mathematically and illustrated with two numerical PDE-based problems: an inverse scattering type problem and an inverse conductivity type problem.

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Section: Introductionmentioning

confidence: 99%

On global normal linear approximations for nonlinear Bayesian inverse problems

Nicholson¹,

Petra²,

Villa³

et al. 2023

Inverse Problems

View full text Add to dashboard Cite

show abstract

“…The BAE approach has been used in a variety of settings, see for example Kaipio and Kolehmainen (2013); Arridge et al (2006); Castello and Kaipio (2019); Lamien et al (2019), among others, and the references therein. A particularly relevant, and recent, example is the application of the approach to the so-called Robin inverse problem encountered for instance in corrosion detection (Nicholson et al, 2018).…”

Section: Introductionmentioning

confidence: 99%

Inferring the basal sliding coefficient field for the Stokes ice sheet model under rheological uncertainty

Babaniyi¹,

Nicholson²,

Villa³

et al. 2020

Preprint

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Abstract. We consider the problem of inferring the basal sliding coefficient field for an uncertain Stokes ice sheet forward model from surface velocity measurements. The uncertainty in the forward model stems from unknown (or uncertain) auxiliary parameters (e.g., rheology parameters). This inverse problem is posed within the Bayesian framework, which provides a systematic means of quantifying uncertainty in the solution. To account for the associated model uncertainty (error), we employ the Bayesian Approximation Error (BAE) approach to approximately premarginalize simultaneously over both the noise in measurements and uncertainty in the forward model. We also carry out approximative posterior uncertainty quantification based on a linearization of the parameter-to-observable map centered at the maximum a posteriori (MAP) basal sliding coefficient estimate, i.e., by taking the Laplace approximation. The MAP estimate is found by minimizing the negative log posterior using an inexact Newton conjugate gradient method. The gradient and Hessian actions to vectors are efficiently computed using adjoints. Sampling from the approximate covariance is made tractable by invoking a low-rank approximation of the data misfit component of the Hessian. We study the performance of the BAE approach in the context of three numerical examples in two and three dimensions. For each example the basal sliding coefficient field is the parameter of primary interest, which we seek to infer, and the rheology parameters (e.g., the flow rate factor, or the Glen's flow law exponent coefficient field) represent so called nuisance (secondary uncertain) parameters. Our results indicate that accounting for model uncertainty stemming from the presence of nuisance parameters is crucial. Namely our findings suggest that using nominal values for these parameters, as is often done in practice, without taking into account the resulting modeling error, can lead to overconfident and heavily biased results. We also show that the BAE approach can be used to account for the additional model uncertainty at no additional cost at the online stage.

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