Finite element solution of boundary value problems: theory and computation
Finite element solution of boundary value problems: theory and computation
Computational Methods for Inverse Problems
Computational Methods for Inverse Problems
Inverse Problem Theory and Methods for Model Parameter Estimation
Inverse Problem Theory and Methods for Model Parameter Estimation
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Variational mass consistent models (MCM's) provide a solenoidal field V in a region Ω by minimizing a metric $||V-V^0||_{ΩS}^2$ subject to ∇ · V=0, where $S_{ij}=α _{i}^{2}δ _{ij}$ is a symmetric and positive definite matrix. A least-squares approach suggests that $S^{-1}$ can be estimated with a Gaussian statistics. This approach is not valid in general, instead, it is shown that MCM's constitute a general scheme to get solenoidal fields V where the $α _i $'s are distributed parameters that can be estimated by means regularization methods. The range of values of $α ^{2}=α _1^2/α_3^2$ considered in the literature is $[10^{-12},∞)$. It is shown that V becomes singular as $α ^{2}→ 0$ or ∞ and this behavior together with the loss of regularity of λ on the boundary of Ω produce a spurious sensitivity of the residual divergence, which was proposed to estimate the optimal ratio $α^2$. Several simulations with MCM's use the boundary condition ∂ λ/∂ n=0 but it is not valid general. A deduction of MCM's in terrain-following coordinates is given and it is shown that such coordinates may hide the use of inconsistent boundary conditions. Numerical examples illustrate significative changes in the field V when the condition ∂ λ/∂ n=0 is used.