Approximation errors and truncation of computational domains with application to geophysical tomography

Anomaly Detection in Random Heterogeneous Media

“…Indeed, the estimate corresponding to α = 10 −4 gives quite precise information about the center of the anomaly but not about its size. Moreover, we could confirm the observation made in [6,100] concerning the robustness of the approximation error approach with respect to misspecified priors. The computation time for the MAP estimates was around 120 seconds.…”

“…On the other hand, Gaussian smoothness priors are appealing in practice due to their differentiability which is necessary if one wants to apply Newton-type methods. Moreover, it has been shown in [6,100] that the approximation error approach is to some extent tolerant when it comes to misspecification of the prior. We would nevertheless like to point out, that in the Bayesian framework it would be possible to use priors that allow moderate discontinuities of the conductivity, however, leading to non-differentiable prior functionals, cf.…”

“…That is, we construct a statistical model describing both the modeling error due to homogenization as well as the approximation errors caused by truncation of the computational domain and coarse finite element discretization. While in the context of EIT and optical tomography, the approximation error approach has been successfully applied to cope with the latter, cf., e.g., [6,100,128,139], the model reduction by using homogenized equations in conjunction with the statistics of the resulting homogenization error is a novel aspect. The Monte Carlo sampling which has to be carried out in order to precompute these statistics is, however, computationally expensive since resolving the highly heterogeneous microstructure requires an extremely fine finite element discretization.…”

Correction of approximation errors with Random Forests applied to modelling of cloud droplet formation

“…This far, the approach has mainly been applied to so-called soft field tomography imaging problems that are related to estimation of spatially distributed parameters of partial differential equations from boundary measurements. In such problems, the approach has been successful, for example, in compensation of approximation errors due to coarse finite element discretization (Arridge et al, 2006;Nissinen et al, 2009), unknown nuisance parameters (Nissinen et al, 2009(Nissinen et al, , 2011Kolehmainen et al, 2011), and the truncation of the computational domain (Lehikoinen et al, 2007;Kolehmainen et al, 2009).…”

State estimation in process tomography—Three‐dimensional impedance imaging of moving fluids

“…In real measurement situations, the pipe should be longer in order to avoid modeling errors in the observation model resulting from the incorrect boundary condition. Alternatively, it is possible to construct a statistical model for the modeling errors due to domain truncation as described in [24,26]. By including this model to the state-space model, it is possible to decrease the effect of the erroneous boundary condition.…”