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Abstract 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 (BAE) 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 BAE 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. 

Abstract. We consider the problem of inferring the basal sliding coefficientfield for an uncertain Stokes ice sheet forward model from syntheticsurface velocity measurements. The uncertainty in the forward modelstems from unknown (or uncertain) auxiliary parameters (e.g., rheologyparameters). This inverse problem is posed within the Bayesianframework, which provides a systematic means of quantifyinguncertainty in the solution. To account for the associated modeluncertainty (error), we employ the Bayesian approximation error (BAE)approach to approximately premarginalize simultaneously over both thenoise in measurements and uncertainty in the forward model. We alsocarry out approximative posterior uncertainty quantification based ona linearization of the parameter-to-observable map centered at themaximum a posteriori (MAP) basal sliding coefficient estimate, i.e.,by taking the Laplace approximation. The MAP estimate is found byminimizing the negative log posterior using an inexact Newtonconjugate gradient method. The gradient and Hessian actions to vectorsare efficiently computed using adjoints. Sampling from theapproximate covariance is made tractable by invoking a low-rankapproximation of the data misfit component of the Hessian. We studythe performance of the BAE approach in the context of three numericalexamples in two and three dimensions. For each example, the basalsliding coefficient field is the parameter of primary interest whichwe seek to infer, and the rheology parameters (e.g., the flow ratefactor or the Glen's flow law exponent coefficient field) representso-called nuisance (secondary uncertain) parameters. Our resultsindicate that accounting for model uncertainty stemming from thepresence of nuisance parameters is crucial. Namely our findingssuggest that using nominal values for these parameters, as is oftendone in practice, without taking into account the resulting modelingerror, can lead to overconfident and heavily biased results. We alsoshow that the BAE approach can be used to account for the additionalmodel uncertainty at no additional cost at the online stage.